Lecture Notes in Mathematics
Edited by A. Dold and B. Eckmann
883
Cylindric Set
Cylindric Set Algebras and Related Structures By L. Henkin, J. D. Monk, and A Tarski
On Cylindric-Relativized Set Algebras By H. Andreka and I. Nemeti
Springer-Verlag Berlin Heidelberg New York 1981 Authors Leon Henkin Department of Mathematics, University of California Berkeley, CA 94720, USA
J. Donald Monk Department of Mathematics, University of Colorado Boulder, CO 80309, USA
Alfred Tarski 462 Michigan Ave. Berkeley, CA 94707, USA
Hajnalka Andreka Istvan Nemeti Mathematical Institute, Hungarian Academy of Sciences Realtanoda u. 13-15, 1053 Budapest, Hungary
AMS Subject Classifications (1980): 03C55, 03G15
ISBN 3-540-10881-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10881-5 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 Introduction
This volume is devoted to a comprehensive treatment of certain set- theoretical structures which consist of fields of sets enhanced by addi- tional fundamental operations and distinguished elements. The treatment is largely self-contained.
Each of these structures has an associated dimension a , a finite or infinite ordinal: their basic form is well illustrated in the case a 3
R be an arbitrary set, and let be afield of subsets of the set
of all triples of elements of R. Thus is a non-empty collection of subsets of 3R closed under union, intersection, and complementation 3 relative to R• We shall assume that is closed under the three opera- tions CO' C C of cylindrification, where CO' for example, is the l' 2 operation given by:
COX = £(x,y,z): (u,y,z) E X for some u, with xER};
COX is the cylinder formed by moving X parallel to the first axis. ClX and C are similarly related to the second and third axes. We also assume 2X that the diagonal planes DOl' D D are in here, for example, 02' 12
£(x,x,y) x,y E R} •
Similarly D (resp., D consists of all triples of 3 whose first 02 12) R and third (resp., second and third) coordinates coincide. A collection satisfying all of these conditions is called a cylindric field of sets
(of dimensicn 3). Cylindric fields of sets and certain closely related structures are the objects of study in this volume. Considered not merely as collections of sets, but as algebraic objects endOWed with fundamental IV operations and distinguished elements, cylindric fields of sets are called cylindric set algebras. Cs is the class of all cylindric set algebras of dimension a , and ICs is the class of algebras isomorphic to them. a In much of the work, general algebraic notions are studied in their application to cylindric set algebras. We consider subalgebras, homomorphisms, products, and ultraproducts of them, paying special attention, for example, to the closure of ICs and related classes under these operations. In a addition, there are natural operations upon these structures which are specific to their form as certain Boolean algebras with operators, such as relativization to subsets of 3R and isomorphism to algebras of subsets of
38 with SiR, and there are relationships between set algebras of different dimensions.
Although, as mentioned, the volume is largely self-contained, we shall often refer to the book Cylindric Algebras, Part I, by Henkin, Monk, and
Tarski. Many notions touched on briefly in the present volume are treated in detail in that one, and motivation for considering certain questions can be found there. Indeed, the present work had its genesis in the decision by Henkin, Monk, and Tarski to publish a series of papers which would form the bulk of Part II of their earlier work. Their contribution to the present volume is, in fact, the first of this proposed series. As their writing
I proceeded, they learned of the closely related results obtained by Andreka and and invited the latter to publish jointly with themselves.
Thus, the present volume consists of two parts. The first, by Henkin,
Monk, and Tarski, contains the basic defintions and results on various kinds of cylindric set algebras. The second, by and is organized parallel to the first. In it, certain aspects of the theory are investigated more thoroughly; in particular, many results which are merely formulated v in Part I, are provided with proofs in Part II. In both parts, many open problems concerning the structures considered are presented.
The authors
Berkeley Boulder Budapest Table of contents
First Part: Cylindric set algebras and related structures, by L. Henkin, J.D. Monk, and A. Tarski 1
1. Various set algebras 4
2. Relativization 12
3. Change of base 33
4. Subalgebras 56
5. Homomorphisms 66
6. Products 73
7. Ultraproducts ...... 86
8. Reducts 122
9. Problems 126
References 12tl
Second Part: On cylindric-relativized set algebras, by H. Andr§ka and I. N§meti 131
O. Basic concepts and notations ...... 132
1. Regular cylindric set algebras 145
2. Relativization 153
3. Change of base 155
4. Subalgebras ...... ltl5
5. Homomorphisms ...... 209
6. Products 222
7. Ultraproducts 229
8. Reducts 261
9. Problems 310
References 314
Index of symbols ...... •...... 317
Index of defined terms 322 Cylindric set algebras and related structures 1)
by L. Henkin, J.D. Monk, and A. Tarski
The abstract theory of cylindric algebras is extensively developed in
the book [HMT] by the authors. Several kinds of special set algebras were mentioned, primarily for motivational purposes, in that book. It is the purpose of this article to begin the examination of these set algebras in more detail. The simplest and most important kind of set algebras are
the cylindric set algebras introduced in 1.1.5. (Throughout this article we shall refer to items from [HMT] by number without explicitly mentioning
that book). Recall that the unit element of any a - dimensional cylindric set algebra (Cs ) is the Cartesian power aU of a set U (the base), a and the other elements of A are subsets of U• The diagonal element a of is the set [x E U·.Kx = x } for each < a fundamental Boolean operations of are union, intersection, and comple- mentation; and for each K < a the fundamental operation C consists K of cylindrification by translation parallel to the Kth axis of the space.
Several other kinds of set algebras were briefly considered in [HMTJ, and
their definition is similar to that of a cylindric set algebra: weak
cylindric set algebras (cf. 2.2.11), generalized cylindric set algebras
(cf. 1.1.13), and shall now call generalized weak cylindric set
algebras (cf. 2.2.11). The algebras of each of these kinds have for their unit element! subsets of a special kind of some Cartesian space au, while
1) This article is the first in a series intended to form a large portion of the second volume of the work Cylindric Algebras, of which Part I has appeared ([HMTJ in the bibliography). The research and writing were sup- ported in part by NSF grants MPS 75-03583, MCS 77-22913. the fundamental operations of any such algebra are obtained from those of a Cs , with unit element aU , by relativization (in the sense of 2.2) a to the unit element of the algebra discussed. To unify our treatment of these several classes of cylindric set algebras we use here as the most general class of set algebras that of the cylindric-relativized set algebras, in which the unit elements may be arbitrary subsets of a Cartesian space. These algebras are simply subalgebras of those algebras that are obtained by arbitrary relativizations from full cylindric set algebras.
We shall not discuss here, however, the class of all cylindric-relativized set algebras in any detail, restricting ourselves to the aspects of relativization directly relevant to our discussion of those set algebras which are GA's
Much of the importance of cylindric set algebras stems from the following construction. Given any relational structure and any first-order discourse language A for , the collection c;f: CjJ a formula of A} forms an a - dimensional cylindric field of sets, where a is the length of the sequence of variables of A (for the notation used here, see the Preliminaries of [HMT]). Thus the above collection is the universe of a cylindric set algebra . This algebra is locally finite dimensional in the sense of 1.11.1 since K except possibly for the finitely many K < a such that the Kth variable of A occurs free in CjJ Furthermore, has an additional property of a regularity: if x E B f E x g EC and flx1f !:,x1g , then g E x (Here fix the dimension set of x ,is LK E a : cKx I x} see 1.6.1.) Regular set algebras will be discussed extensively later.
The article has nine sections. In section I we give formal defini- tions of the classes of set algebras which are studied in this article and we state the simplest relationships between them; the proofs are found in later sections of the paper. In section 2 some deeper relation- ships are established, using the notion of relativization. Section 3 is concerned with change of base, treating the question of conditions 3 under which a set algebra with base U is isomorphic to one with a different base W; the main results are algebraic versions of the
Lowenheim, Skolem, Tarski theorems (some results on change of base are also found in section 7). In section 4 the algebraic notion of subalgebra is investigated for our various set algebras, paying parti• cular attention to the problem about the minimum number of generators for a set algebra. Homomorphisms of set algebras are discussed in sec• tion 5, and products, along with the related indecomposability notions, are studied in their application to set algebras in section 6. Section
7, devoted to ultraproducts of set algebras, gives perhaps the deepest results in the paper. In particular, it is in this section that the less trivial of the relationships between the classes of set algebras described in section 1 are established. Reducts and neat embeddings of set algebras are discussed in section 8. Finally, in section 9 we list the most important problems concerning set algebras which are open at this time, and we also take this opportunity to describe the status
of the problems stated in [HMT] •
For reference in later articles, we refer to theorems, definitions, etc., by three figures, e.g. 1.2.2 for the second item in section 2 of paper number I, which is just the present paper (see the initial footnote).
The very most basic results on set algebras were first described in the paper Henkin, Tarski [HT]. Other major results were obtained in Henkin, Monk [HM]. In preparing the present comprehensive discus• sion of set algebras many natural questions arose. Some of these ques• tions were solved by the authors, and their solutions are found here.
A large number of the questions were solved by H. and I.
Where their solutions were short we have usually included the results here, with their permission, and we have indicated that the results are theirs. Many of their longer solutions will be found in the paper [AN3] 4 Various set algebras 1.1. 1 following this one, which is organized parallel to our paper; a few of their related results are found in [AN2], [AN4] , or [Nl • In the course of our article we shall have occasion to mention explicitly most of their related results. We are indebted to AndrJka and Nemeti for
their considerable help in preparing this paper for publication.
The following set-theoretical notation not in [HMT] will be use ful.
If f E cru , K < a and u U then fK is the rrember of aU such E , u that (fK)I, = fA if II 1- K, while (fK )K = u For typographica 1 u u reasons we sometimes write f(K!u) in place of l u
Definition 1.1.1. (i) Let U be a set, a an ordinal, and
For all K,"A < a we set
and we let be the mapping from SbV into SbV such tha t, for
every X>;; V ,
K {y E V Yu E X for some u E U} •
(When V is implicitly understood we shall write simply D or C .) K"A K (ii) A is an a - dimensional cylindric-relativized field of sets
iff there is a set U and a set V aU such that A is a non-empty
family of subsets of V closed under all the operations U, n,
and c[V) (for each K < et ), and containing as elements the subsets K (for a 11 K, II < et ). The base of A is the set UxEvRgx .
(iii) is a cylindric-relativized set algebra of dimension a
with base U iff there is a set V aU such that 1.1.1 Various set algebras 5 u = (A U n 0 V C[V] D[V]) where ,, 'V ', 'K 'K;\ K, A < Q A is an C( - dimensicnal cylindric-relativized field of sets with unit element V and base U.
Crs is the class of all cylindric-relativized set algebras of dimension ct
C(. In case A = SbV , the set A and the algebra are called res- pectively a full cylindric-relativized field of sets and a full cylindric- relativized set algebra.
(iv) Let U be a set and a an ordinal. aU is then called the
Cartesian space with base U and dimension C( Moreover, for every p E aU we set
au(p) = [x E aU : a : x f p} is finite} i; ':; and we call aU(p) the weak Cartesian space with base U and dimension a determined by p
For the following parts (v) - (ix) of this definition we assume that
A is a cylindric-relativized field of sets and is a cylindric-relativized set algebra, both with dimension a ,base U ,and unit element V
(v) A, respectively , is called an a - dimensional cylindric field of sets, respectively set algebra, if V = aU. The class of all cylindric set algebras of dimension a is denoted by Cs 0:' (vi) A, respectively , is called an a - dimensional weak cylindric field of sets, respectively set algebra, if there is a p E Cl'U such that V = au(p). The class of all 0:' - dimensional weak cylindric set algebras is denoted by Ws a (vii) A, respectively , is called an a - dimensional genera-
lized cylindric field of sets, respectively set algebra, if V has
the form iEI Cl'y i ,where Y 0 for each i E I ,and Y. nY. = 0 for any twc U i f J distinct i,j E I The sets Y are called the subbases of The i symbol Gs denotes the class of all generalized cylindric set algebras Ci 6 Various set algebras 1.1.2 of dimension 01
(viii) A, respectively , is called an 01 - dimensional generalized weak cylindric field of sets, respectively set algebra, if V has the form
01 (pi) Y ' where for each i EI and Ui EI i for any two distinct i,j EI The sets Y are called the subbases of i We use Gws for the class of all generalized weak cylindric set 01 algebras of dimension 01.
(ix) An element x E A is regular provided that g E x whenever f Ex, g E V ,and CftxU 1)1 f = ({;xU 1)1 g A and
are called regular if each x E A is regular. (We assume here a > 0
if a = 0 ,all elements, as well as A and ,are called regular.) r e g K being any class included in ers we denote by K the class of 01 all regular algebras which belong to K
(x) Let be a cylindric algebra and b E A We denote by
the algebra obtained by relativizing to b (see 2.2.1 for a
formal definition). K being a class of cylindric algebras, we let
Remark 1.1.2. The definition of cylindric set algebras in Ll.l(v) coincides with definition 1.1.5. Other parts of 1.1.1 are consistent with the informal definitions in 1.1.13 and 2.2.11. Originally,
regularity was defined only for cylindric set algebras, in the form given
in the introduction. The general definition in 1.1.1 (ix) is due to
H. and I. NJmeti and turns out to have the desired meaning for
cylindric set algebras as well as Ws IS and Gs IS ; see I . 1 . 13 - I . 1 . 16 . 01 01 The inclusions holding among the various classes of set algebras
introduced in 1.1.1 are indicated in Figure 1.1.3 for a > 0 ; if we
conSider the classes of isomorphic images of the various set algebras,
then the diagram collapses as indicated in Figure 1.1.4. The inclusions
in each case are proper inclusions. 1.1.3 Various set algebras 7
crlsa---- r eg , --- Crs 0/ RtCs l 01 Gws eg I Ct Gws:
GsQ' Ireg ------Ws reg Ws a I GsO/ a CSO/ [reg Cs Ct
Figure 1.1. 3
ICrs I 01 IRtCs [ 0/ /'" 01 r eg IGws IGws 01 [0/ 01 0/ ICs r :eg ICs I III reg IWs = IWs 01 Ct
Figure 1.1.4
In case III < W, the classes Ws and Cs coincide, and so do Gws Ct 01 01 and Gs ; furthermore, under this assumption each member of any of 0/ these classes is regular. In the general case, GS's are isomorphic 0/ to subdirect products of Cs 's and conversely; similarly for 01 Gws 's and Ws 's Every Ws is regular. Every Gs is isomor- 0/ Ct 0/ 0/ phic to a regular Gs ' and to a subdirect product of regular Cs IS. Ol 01 Proofs of these facts and the relationships in the diagrams will be found at the appropriate places in this paper.
We begin our discussion by describing some degenerate cases of the 8 Various set algebras 1.1.5 notions in 1.1.1, and giving those inclusions between the classes which follow easily from the definitions. Throughout the paper we
omit proofs which seem trivial.
Because of this theorem we will frequently make such assumptions as
> 0, U 1 0 , or V 1 0 •
Corollary 1.1.6. If a < ill, U is any set, and p E aU , then
= aU. Hence for < ill we have Gs = Gws , if 0 < a < ill (1 we have Cs = Ws U}, where is the unique Cs with universe a Ci a a a 1 , and finally Cs WS • O = O
Corollary 1.1. 7. If m is a Gws with every subbase having only one element, then m is a discrete Gs a
Corollary 1.1.8. Crs Gws GS l = l = I is the Crs with universe 1; Crs Gws = Gs = {m } and l O O O l,m2 es = Ws = , where and are the unique ersO's with O O ml m2 universes 1 and 2 respectively. Furthermore, every ers is O full, and the base of any Crs is O. O
Corollary 1.1.9. (i) For any a Cs U Gs U Ws U Gws <;;; CA a (ii) For ex s 1 Crs <;;; CA a (iii) For a :0: 2 Crsa 1- CAa 1.1.10 Various set algebras 9
Proof. Both (i) and (ii) are trivial. To establish (iii), we construct 2I E Crs "" CA for 01 2 by choosing any set U with 01 01 lUI> 1 , taking V = OIU , and letting 2I be the full Crs 01 with unit V (1.l.l(iii». We have = 0 , so that v =
C[VJ (D[VJ n D[VJ) = O. Thus 2I fails to satisfy axiom 1 01 10 whence 2I rt CAa and the proof is complete.
Corollary I.1.9(iii) explains why we shall not give many results con- cerning the class Crs in these papers; as indicated in the introduc- a tion, this class is introduced just to unify some definitions and results, and plays an auxiliary role in our discussion.
Corollary 1.1.10. If a 2 2 , then Cs C Gs and R.{,Cs S; Crs Q( C( Q( a
We shall be able to strengthen these results below by showing that
Gws a C RtCsa for a 2 2 (see 1.2.12 - 1.2.13). It is known that we have
RtCs aC Crs a for a > 1 , but we shall not give an example here; see [RR] and Prop.2.3 of [AN3] (p.155).
1.1.11. If a W, then Ws C Gws C Crs and Cs a a a a C Gs c: Gws a a
Q'u(pi) Theorem 1.1.12. Let be a Gws with unit element CI UiEI i ' where aU(pi) n au(pj) 0 for all distinct i,j E I . Assume that i j ClU(pi) EA for all i E I Also assume a r 1 i Then for any .X E A the following conditions are equivalent: 10 Various set algebras 1.1.13
(i) x = O!U(pi) for some i EI i (ii) X is a minimal element of 2.l (under ,;;; ) such that X r 0 and CoX = 0
Proof. (i) (if). Clearly for any i E I we have o r ClU(pi) i and CoO!u(pi) = 0 Now suppose that 0 r y aU(pi) and O. Fix i i = Pi) y E Y , and let x E O!ui be arbitrary. There is then a finite r O! such that = Thus x E ccnY = Y. and hence Y = O!uiPit
(ii) (i) . Assume (if). and choose i E I such that X n au(pi) i Pi 1- 0 we have nO!ui » = 0 by 1.6.6 Hence
, while by the implication (i) (ii) already
Thus (i) holds.
Now we discuss regular set algebras. In the case of the classes Cs O! Ws Gs the definition assumes a simpler form mentioned in the O! O! introduction.
Corollary 1.1.13. Let 2.l be a Cs with base U. and let O! x EA Then the following conditions are equivalent:
(i) X is regular;
(ii) for all f E X and all g E aU • if £lX1 f = £!X1 g then g EX
Proof. (i) Trivial. (i) (ii) Assume (i) and the hypotheses of ( ii) If o E-- 6X , the desired conclusion g EX is obvious. Suppose therefore o CoX. Then fO EX since and ",. gO 0 (CoX U1)1 f 0 = (9,X U1)1 g Hence g E X by (i) v>' g
The next two Corollaries are proved in the same way as 1.1.13. 1.1.14 Various set algebras 11
Corolla ry 1. 1. 14. Let \U be a Ws with unit element au(p) a and let X EA. Then the following conditions are equivalent:
(i) X is regular;
(ii) for all f E X and all g E Q'U(p) , if kX1f ,.6X1g then g EX.
C1 Corollary 1.1.15. Let be a Gs with unit element 0' UiEI Yi ' where Y n Y = 0 for i j , and let X EA. Then the following i j * conditions are equivalent:
(i) X is regular;
(ii) for all i E I ,all f E X n Q'Y ' and all g E Q'Y ' if i i 6X1f = 6X1g then g EX
No analogous simplification of the notion of regularity for arbitrary Gws 's is Q' known. Weak cylindric set algebras are always regular:
reg Corollary 1.1.16. Ws Ws a Q'
Proof. Suppose that \U is a Ws with unit element aU(p) Q' Also assume XEA , f E X , g E ClU(p) , and 6Xlf = 6X1g Since h' f,g E au(p) , there is a Hni te r <;;; a with =
Hence so that g E c X = X. Thus by - (f-AX) 1. 1. 14 X is regular.
r e g r e g Corollary 1.1.17. If a < w , then Cs Cs Ws Ws Q' a Q' a r e g r eg Gs Gs ,and Gws Gws a y a Ct
Corollary 1.1.18. If C1
r e g r e g Proof. To produce a member of Gws _ Ws , let U and V be 01 rY. rY. rY. two disjoint sets with at least two elements, and let p E U , q E V be arbitrary. Let be the full Gws with unit element rY.U(p) U rY.V(q) 01 Then it is easily checked, along the lines of the proof of 1.1.16, that r e g E Gws Ws It is also clear that Gs If we let i.!l be 01 01 ct the two-element Crs with unit element [p} , we see that i.!l E Crs reg ct rY. _ Gws (provided that p has more than one element in its range). 0: Finally, if U and V are as above and is the minimal subalgebra of the full Gs with unit element 0:
r e g re g Corollary 1.1.19. For 01 W we have Cs C Cs ,Gs C Gs 01 1'1 f't 01 r eg and Gws C Gws ct a
Proof. By 1.1.11 it suffices to exhibit a Cs which is not 01 regular. Let 91 be the full Cs with base 2 Let p = <0 : K < Cf) 01 01 (p) l\CI'2(P) 0I (p ) Then 2 E A and = 0 Hence 2 is not r e guLa r-, since
in a regular es the only elements X such that l\X o are the Oi zero element and the unit element.
2. Relativization ... co
Our basic kinds of set algebras have been defined in terms of relativiza-
tion of full cylindric set algebras. We want to present here some results,
1.2.5, 1.2.9, and 1.2.11, which involve relativization and exhibit impor-
tant connections between Cs 's and Gws 'so First we establish an ele- 01 Ci mentary characterization of unit sets of Gws 's, which was mentioned Ci in 2.2.11. 1.2.1 Re1ativization 13
Theorem 1.2.1. Let be a Csa with base U, and let
F ( XEA ·lXnsAX=X•AK for all K,A Under these premises, dependent on an additional assumption imposed on a , each of the following conditions (i)-(iii) is necessary and suffi- cient for any given set V to belong to F (i) V belongs to A ,assuming a 1 (so that in this case FA) ; (ii) V belongs to A and is a Cartesian space, assuming a = 2 ; (iii ) V belongs to A and is the unit element of a Gws ,assum- Ct ing Ct;;' 3 . Under the same premises we obtain an additional conclusion: (iv) assuming 2 a < lJ) ,the conditions C(cr-l)V = V and VEF are jointly equivalent to the condition that V is the unit eEment of a Cs in case a = 2 , or a Gs in case Ct;;' 3 , with base U. Ct a Proof. The necessity and sufficiency of (i) is obvious, and so is the sufficiency of (ii) To establish the sufficiency of (iii), assume a 3 and consider a set VEA which is the unit element of a Gws ,say V = U CtW(pi) where n OW(pj) = 0 for any o iE1 i i j i,j E I with i l' j Given any K,A < Ct , we clearly have V c n sAy Now suppose f E SKy n sAy ; thus, say, fK E and fA E K A K fA i fK Pick a and notice tha t f E Wi n W. There J f.I u J is a finite r G Ct such tha t f = (a_r)1 pi = (a_r)1 pj If therefore we define g by stipulating that gv = f.,. for all v E r ,and gv = fv for every \i E Ct .... r , we get g E awipi) n Ctwjpj) Conse- quently i = j and f E awipi) ;;; V Hence V = n and we conclude that VEF, as desired. 14 Relativization 1. 2.1 We wish now to demonstrate the necessity of both (ii) and (iii). To this end, we show first that every set VEF is the unit element of a Gws (This portion of the proof is due to and a For each f E V let We need the following facts about 0) If f E V, then For,if 1«Cl',then f EV c S V, so fOE V and hence I(° fl( (2 ) If f E V and u E Yf ' then Yf = Yf(O/u) . (3) If f E V , u E Yf ,and I( < Q' , then EV I( For we may assume tha t I( #- Then f ° V c , so f°l( E V ° u E sOV uu I( I( 0 I( Also, f E V ;: Hence E V as sOV , whence ffO E V. lu sl(V n sOV = desired. (4) If f E V, I( < Q' ,and u E Y ' then f Indeed, (4) clearly follows from (2) if I( = 0 Assume I( to, and assume the hypothesis of (4). Then E V by (3). Consider now any v EY . We f 01( 0\ have f , f E V. Hence fO{ E sKV nsOv =V Thus uu vv vu 0 I( , as desired. From (4) we get (5) and (6) : o' (f) c (5) y f - V (6) If f,g E V and f E O'y(g) , then Y c Y g g f :l ( f) c Cl (f) Ct'y(g) (7) If f,g EV and y , then Y f - g f g 1.2.2 Relativization 15 E Q'y(f) Q'y (£) Q'y(g) In fact, f so that Yg yf by (6) , whence f f g Q' (f) IX (g) (8) If f,g EV and Q'y(£) n ay(g) t- O, then y f g yf g Let indeed h Q'y(f) c ay(h) O:y (g) . f h g By (8) , V is the unit element of a Gws , as desired in (iii). 0: In case IX = 2 , we have O:y(f) = C:V y for all f EV f f . Furthermore, if 1 f ,g EV then n sOV V , and consequently, gl EYf' Thus by (1) , yf n yg t 0 and hence Therefore cty = IXy by (8). f g It follows that V is a Cartesian space, as desired in (ii) Finally, (iv) is clear since the condition is equivalent to Theorem 1.2.2. Assume that Q' 3. Let be the full Cs Oi with base a U and let l!l be a ers with unit set V Q'u . Oi Then the following conditions are equivalent: (i) is a Gws 0: (ii) n = V for all < 0: (operating in ). Proof: By 1.2.1. Remark 1.2.3. For Oi 3 Theorem 1.2.2 gives a simple elementary characterization of those unit elements of Gws 's which are members of a Oi Cs For Q' $ 1 the corresponding result is trivial, in view of 1.1.8. Ci. On the other hand, for Ci. = 2 , there is no elementary characterization of those elements of a CS which are unit elements of a Gs In fact, 2 2• let U and U be disjoint sets each having at least two elements. Let o l be the full CS with base U UU ' let X = 2 U 2 ' and let = 2 o l uO Ul 66J(2J)X. One easily checks that III is atomic am actually possesses just 16 Relativization 1.2.4 2 three atoms , and X and that its structure is DOl'X DOl (VOUVl) completely determined by the fact that C y = 1 for each atom y CoY 1 It follows directly that there is an automorphism of 58 which inter- 2 changes the atoms DOl and (V X , and which therefore OUV2) 2 interchanges the elements X and DOl U (Va UV .... X. As X is the l) 2 unit element of a Gs2 ' this implies tha t DOl U (Va UVI) .... X sa tisfies all first-order (and indeed higher-order) sentences that hold for all unit elements of Gs (in arbitrary Cs As 2's 2's). is not itself such a unit element, no elementary (or even higher-order) characterization of such elements of Cs is possible. It is also 2's of interest that if is a full CS with base V, then an element 2 VEA is a unit element of a GS iff V is an equivalence relation 2 with field included in U. In [AN3] it is shown that for a w there is no elementary charac- terization of the unit elements of Gs 's which are members of a given Cs a a The other main results of this section are related to the following obvious theorem. Theorem 1.2.4. Fa r any algebra Ql similar to CA 's the following 0' three conditions are equivalent: (i) Ql E Cr s a (ii) for some full Cs \lJ and some VE B u c "RLV\lJ a (iii) Ql c "RLV\lJ for some Cs 18 and some V E B. 0' The question naturally arises whether the condition Ql "RLVB in (iii) can be replaced by Ql = "RL 'i8 • For arbitrary ere's the answer in v 01 general is negative. Here we want to carefully consider this question for our classes Ws Gs • and Gws We begin with the following 0: 0' 0' 1.2.5 Relativization 17 simple result. Theorem 1.2.5. Ws Rtcs a a More specifically, assuming is a Ws with base U and unit set V , we have = ry(L for some Cs 18 with a V18 a VEB ; in fact, if we let (£: be the full Cs with a base D, we can take 65j([)A for 18 Proof. Say V = CiUCp). Obviously For the converse first note that = 0 , and hence A c (XEC: XnVEA} E SIllS: Hence B {X EC:X nv EA} so Remarks 1.2.6. TIle last part of 1.2.5 does not extend to Gws 's with a :2: W• In fact, let "o w D =w-l, P = (O:K is a Gws It has characteristic 0 (cf. 2.4.61), and hence Ci is simply the Boolean subalgebra of generated by its diagonal ele- ments. On tre other hand, if is any Cs with V E D and with Q' base w , then in so that is in but not in A• Hence The Gws just constructed is unusual, in that the subbase U Ci 1 is included in the subbase U o Usually a GwsQ' is normal, in the following sense. Let 18 be a Gwe , Q' :2: W , and suppose the unit Q' Ci (Pi) Q' (Pi) (p .) element of 18 is with n J =0 whenever i, j E I and Ui E1 Wi Wi i i "f j We ca 11 18 norma 1 if Wi = W. or Wi n W. = 0 for all i,j J J widely-distributed if Wi n W = 0 whenever i"f j ; compressed if j 18 Relativization 1.2.7 for all i,j Thus every Gs is a normal Gws ' and every Q' a Cs is a compressed Gws It follows from results which will be esta- a a blished in section 7 of this paper that eveDT Gws is isomorphic a to a widely-distributed Gws It is also clear that every compressed a Gws \B is a direct factor of a Cs which has the same baae as Q3 ; so a a Q3 is a RtCs with that same base. Q' We do not know whether a normal Gws with a w can always be a obtained by relativization from a Cs haVing the same base as • We a show below in 1.2.9 that this is true in case 3 a < w and is a Gs On the other ham, we show in 1.2.11 that in case a w we have Q' at least Gws I;; RtCs (Thus, of course, the last part of 1.2.5 holds 0' Q' for Gws ' s with a < w since we have then Gws Gs We also have a a a Gs c RtCs in case Q' W , since Gs Gws .) 1.2.9 is dee to Ci a Q' a Henkin, while 1.2.11 has been obtained jointly by Henkin and Monk. We need two lemmas about arbitrary CA 's. In connection with 0' these lennnas recall that, by 2.2.10, ')lL is a CA for every b ,Ci \ ' E CA and every b E A having the property that /b • = b for a A all K,A < a Lemma 1.2.7. Let be any CA , let b EA , and assume tha t Ci sKb • sAb = b for all K,A ( iii) If rea, then b c (r)x = c Cr)x c' - (iv) c(f) (f)x - c(f)x (v) If f U (:. C CY. , then c (r)x . c(l,)y 1.2.7 Relativization 19 (vi) If f C a , then -c(f)x = c(n(b. -c Furthermore, assume that a < w and that = 1. For all K,A. < Q' let e = C b. Then t'le following conditions (vii)-(xii) hold. \Oc (ry,..{ K, A. }) (vii) c(Cl'-{K})b = 1 (viii) C b = n e (f) (ix) C e = e whenever < Q' and 1 K,A. • KA. KA. (x) d s: e . KA. KA. (xi) dKA. = C(Cl""{ II: ,A.}) (b . dn ) (xii) T U rA.o} C Q' ,and w = c(r)x • . Then X \0(C(n • IT - = c(fUrA.O})X • • Proof. (0. We prove (i) by induction on If U 61 It is ob- vious for f = 0 or 6 = 0 , so suppose r r c » s If If U 6 \ = 2 , then \r \ = 1 = \6 \ ; say f = Cd and 6 = CA.} Then b s: c b C b s; c sKb C sA.b Kb sA.b = b K A. K I. A. K sA. K so (i) holds in this case. Now suppose that \fU6\>2 and that (i) holds for sets r ' ,6' with \f'U6'1 < \rU61 . Say 161> 1. Choose K E 6 and A. E Q' (f U6) Then K b s: c(nb c(6)b s; c(f)b . c(f)sA.b K K c(nb sAc(f)b . SAC(li-{K})b c(nb • c(t.--{K} )b) c(f)b s: c(f)b • cKb = b 20 Relativization 1.2.7 Thus Ci) has been established. Cii). Assume f U (K} ca. We may assume that Kef Then b . cCfU(K})x cKb . CKCCf{ = cKCcKb . cCf)x) cKCcKb' ccnb. cCf)x) cK(b. ccnx) (by (i» , and Cii) holds. Ciii). This follows from (ii) by an easy induction on If\ . (Lv), Obvious, by direct calculation. (v). Assume that r U /c, Ca. Then CCf)X . cC/c,)y c(f)x, c(/c,)cC/c,)y (by (iv» cCf06) CC(nx . cCD,-f/(6)y) cCfrt,)(C(nx' cCtr-r)cC6)y ·b) (by (i» C(frt,)(CCr)X' cCt,-l')CC6/) (by (iii» c(f06)(c(r)x. cC6/) Cvi) Assume f Ca. Then c(n(b. -c(nx) = cCf)(b. -(b· c(f)x» (by Ciii» = cCf)Cb . -c(r)x) = c(r)b . -c(f)x , and (vi) follows, since cCf)x C(f)b Now we assume the additional premises for (vii)-(ix). (vii) We may assume that K + O. Then 1.2.7 Relativization 21 o 0 1 sKCCcr-1)b '" SKC(cr{O,K} )cKb o 0 K c(Cl"'{O,K})s, K cCO'-(K}) (dOK • sOb. cKCO(dOK • b» = cCcr-(,<})(d • b· cKcO(oOK • b ) OK C(O"'{K})b . (vI Ld) . Since c' b a b for all finite 6 a , this is an 1mmedi- (tl) ate consequence of Cv) and (vii). (ix). Obvious. (x). We have, assuming K fA, d (by (vii» KA dKA c(cr..{K})b diO. C (O'....(!c ,A} ) \ b cCa--{K ,A}) (d • s: KA cCa-(K,A}) (d • b) e IC A s: KA (xi). It follows immediately from (x) (xii). Assume the hypothesis of (xii) Also let z Hence it suffices to show that C z CW = 0 , and for this it is 1.. 0 AO enough to show that z . C w o or, by the definition of w , to take 1.. 0 any E v-I and show that And in fact 22 Relativization 1.2.8 z . (by (v)) o This completes the proof of 1.2.7 . Lemma 1. 2 . 8 . Let 3 :s: a < W, a nd Le t be a s imp Le CA Let a b E A satisfy the following conditions: (i) sA. b c b for all K,A. < a Cii) cCa-l)b = 1 ( iii) Setting for all < ex , we have, for every x:S: b , every v E W 1 , and every sequence A. E va without repeating terms, c (c x'l1 -e ) z +!: (z· - z . w ) V K K+1 K A.O (e»-[).,O}) A.O\ K stipulations: s (6 : 6 <;;; v-I, I 61 = K} for each K < \) K w !: en c } c '. x·n -e) (with K 6fS ) (a) -K denoting the genera1izec cylindrification in ) for each K < v . 1.2.8 Relativization 23 Proof. By the assumptions (i) and (ii) we have all the parts of 1.2.7 Again, we denote operations of by etc. Now let A* = {c(r)x:xEB and fl;;:a} U (-e :1C,,, of A'" Obviously A** va . I n vi ew 0 f I•• 2 7 (iii) iiIt snow c' ear1y enough to see how to prove the converse, and in fact to show that B A** E StiU. Obviously BI;;A**. For any K,A -, it suffice to show that -y E A** for every y E A* If y= -e ' then "Y KA elC A = E A*. Suppose that y = c(f)x If T = a , then y 1 or y = 0 by simplicity, so -y E A**. If f f a then by 1.2.7 (vi) we have and obviously c(n(b. -c(r)x) E A* c A, while -c(r)b E A** by 1.2.7 (viii) . Finally, we must show that A** is closed under each c (Here A we give an algebraic version of an elimination of quantifiers.) From the definition of A** we see that it suffices to show that c y E A A** for every y which is a product of elements of A* . Say IT - e z- with each x E B, where S,y < w . Clearly we may assume that t. for each z- < S , and for each t. < y Thus by 1.2.7 (v) "E {IC z- ,>J. z- } we may assume that S S 1 Since c } b = 1 , by 1.2.7 (vii) , we (a...{ A) 24 Relativization 1.2.9 may assume tha t 13 1. Thus we are concerned with y of the form y = c x· n - e (6) where x E B, A e 6 , and for each < Y, and in case < e < Y If 6 UtA} c a , then by 1.2.7 (xii) we have -c w A where w C(A)X • r; e and is < w c ( c ( A) x· r; e ) A A u -. cAr; so that c y E A** On the other hand, if 6 U {A} = a , then c y E A** A A by premise (iii) of our lemma, since under the notation used there each z is closed and hence is either 0 or 1 by the si mp lie!ty of 21 while K obviously each w is in A** • K This completes the proof. Theorem 1.2.9. In case 3 S a < ill we have Gs R1,Cs More a a specifically, assuming 2l to be a Gs with base U and unit element a v , we have 21 = for some Cs 'i8 with VEB in fact, if we a let be the full CSa with base U, we can take for 'i8 Proof. It suffices to verify the hypotheses of 1.2.8 (with and V in place of 2l and b). It is easy to see that is simple; see also 2.3.14 or 2.3.15. Condition 1.2.8 (i) follows from 1.2.2, while 1.2.8 1.2.9 Relativization 25 (ii) is obvious. Assume the notation of 1.2.8 (iii) (with V in place of b ). Say V '" U a where Wi W 0 for i #- j Thus i El Wi n j = For each u E U there is a unique i E I denoted by U '" UiEIWi in(u) , such that u E Wi Thus clearly (1) for every K,A < a and f E aU , we have iff in(f ) K in(f )• A Choosing any x V, we then get (2) for every K < QI and iff f K (Recall from the Preliminaries that = egA : g Ex} .) Hence for every f E aU , (3) f E c 1.. (c(.J 1.. })x iff there is a u E pj* x 0 0 AO such that in(u) f in(f for a 11 z, E \) _ 1 . A z, On the other hand, obviously (4) for each z '" aU or Z = 0 ; Z iff there are K K K elements W pjAox,* ... ,wK_l E pjAK_I* X such tha t in(w ) f- in(w ) o E q whenever # 8 . • "k • -ic -k .* Since 'P) x '" in PJ x for all K,p < Q' , we may use the fact K p that x s V to rewrite (4) as (5) for each K s; \), zK or o iff \( in(u) : u E pj'" x} 1 K 1.. 0 (6) cCa)X = U {Q'W iEl, nx;/'o} . i: OW i 26 Relativization 1.2.10 One can easily prove (6) by observing from 1.2.7 (iii) that c(a)X = cO(c(a-l)x . V) . V and using (2). From (6) we easily get (7) if K < v, 6. c v 1, \ 6. \ = K and f E aU , then f EI1l,ElIc [ })c( /'11 - e iff in(f )'f-in(f for all Al, a Al,A e Al, Ae distinct l"eE6., and f EU {Wi: iEI,aWi nx'f-O} for all (, E6. . Al, Hence for any f E aU we have (8) fEw iff K Now let us compare (3), (5), and (8) If z au, we see that v CAO (c (0)-{ AO} )x . 11l, Ev--- 1 - 1.2.8 (iii) holds. Suppose z = 0 and Le t K • By (5) we have K z + z: z \) TI1is completes the proof. Remark 1.2.10. The preceding theorem can be extended to the case a = 2. To cover that case we would first establish an analog of Lemma 1.2.8 having the same conclusion and same premise (iii) , but with premises (i) and (ii) replaced by the following: (1) (3) CK(C,X 'b) . b s; C (c x v b ) for any A,K < 2 and x S; b . " AK 1.2.11 Relativization 27 The proof of such a lemma for the case a = 2 is merely a simplified version of the proof given for 1.2.8 With such a lemma at hand, there is no difficulty in extending 1.2.9 to cover the case a 2, by verifying that the above hypotheses (1), (2), and (3) hold under the conditions assumed in 1.2.9. For a = 2 , I.Z.9 specializes to GS In a later Z paper of this series we shall show that CA IFl.Cs In view of Z Z results from [HR] this that perhaps Crs RtCs ' but 2 = Z this has been disproved by And reka and see 2.3 of [AN3] (p.155). Theorem I.Z.l1. In case a w we have Gws RtCs a a Proof. Let be a Gws with unit set V a (pi) (pj) where a 0 for distinct i,j E I. Let U UiE1W Wi n OW j = = i be the base of and choose U' such that U' -U is infinite . Let be the full CSa with base U' , and let = 6 Let c'" = {C(r)x: x E A, r a finite subset of a} U {-C(r)V: T a finite subset of a} U {DK:1e : K: ,Ie < a} U {-D : K: ,Ie < a} • KIe Let C** be the set of all finite unions of finite intersections of members of C*. Obviously C** and it suffices to prove 28 Relativization 1.2.11 the converse, i.e., that A c** E Obviously A C** , D E C** KA for all K,A < a ,and C** is closed under unions. To show that C** is closed under it suffices to show that -y E C** for each y E This is obvious for y of the form -C(r)V,D or -D ' KA KA while for y = C x by 1.2.7 (vi) , which applies by 1.2.1 (r) So it remains to show that eyE c** whenever K < a and K y E C**. We may assume that y is merely a product of members of C* , say y C(r )x n ... n C(r o O n Vn '" n -C V n » n···nD Ao»O A5-liJ.()-1 n -D n ... n -D oP 0 SE -1PE- 1 where < w , r and are finite subsets of a for z, < f3 , 8 < , and for 8 < E Further, we may assume that K l r for all < , K for all < , o 0 KE [\,\.10) for all < 0 , and K E for all 8 < E Say K = AO = '" = AO-l = Po = PE- l Further, we may assume that the IJo 's are distinct from each z, other and from K , and similarly for the 6 's . The following easily z, verified facts show that we can assume that () = 0 Chere K f ): C (X n y n D ) = C CX n D ) n c (y n D ) K K K D Ccr D if r n K) n K) 0 0U{K})C\, C/Ccr n DK) Ccr n n V) 0 if E r z, 1.2.11 Relativization 29 n = n V) E c** by closure under -, proved above; C (D n D) = D; K C (-D n D "'-D K Also, by 1.2.7 (v) we can assume that 8 s 1. Thus we have reduced our considerations to y of the form C(f)X n -C(6 n '" n -C(6 0)V y_l)V n -D n··· n -D Kpo KPE-l where K f , K !::. for each < y , K t- P for each < E , all 'the P 's are distinct, and where perhaps the factor does not t. C(r)x appear, and possibly y '" 0 or E = 0 Now we consider two cases. C(f)X is a factor of y. We proceed by induction on E• First take E O. Clearly we may assume that y > 0 Then, as in the proof of 1.2.7 (xii) we see that so by 1.2.7 (v) and (vi) CKY E C**. Now assume, inductively, that E > 0 For each < E we have Thus (9) CY K nII, z.. Now we have case E = 0 just disposed of. And for each < E we have 30 Relativization 1.2.12 by 1.2.7 (v) to which the induction hypothesis applies. 2. C(f)x is not a factor of y. Then (*) Qlu I• l In fact, let Choose s E U (U U (up : 1., Then 1., clearly -D ), and (*) follows. KPl., By (*) we clearly have CKy E C** in this case also, so the proof of 1.2.11 is crnnplete. Corollary 1.2.12. For every QI 3 we have Gs Gws s;; UCs CI C'i CI Remark 1.2.13. For each a 2 there is a Cs and a V E A Ci such that is not a Gws , and in fact is not even isomorphic to Ci a Gws Thus Gs C R{,Cs 1Gs C IR.{"Cs , and IGws CIlV"Cs It QI Ci QI 0' CI Ci Ci is a trivial matter to construct such and V. For example, let Cl2: 'U be the full Cs with base 2, and let V = (fE fO#-fl} • Then Ci is not a CA , since DOl n V = 0 (see 2.2.3 (ii)). So the Ci above facts are trivial. It is of more interest to construct an example is a CA (Since 1Gs is the c lass of a 11 repre sentab Le ex' a CA IS, this will provide another example of a non-representable CA CI CI we gave one in 2.6.42. See 1.1.13 for the notion of a representable 1.2.13 Relativization 31 ; it will be proved in 1.6.3 (rather easily) that 1Gs is the c class of all such.) The example which we shall now give is due to Henkin. Theorem 1.2.14. For each a 2 there is a CS m and a V E A et such that 9\£ 91 E CA et- IGws et Proof. Let W be a set of cardinality let\ with et n W m 0 and let be a one-to-one function mapping a onto W The base U of our Cs m is to be a U W Let G be the set of all permutations a f of U such that for some permutation T of a we have a f S T while fK' - for each K < a Clearly G is the universe of a group of permutations of U Now for g,h E aU define g =h iff fog s h for some f E G It is easily verified that = is an equivalence relation on U Let A be the collection of all sets x aU which are unions of equivalence classes under = It is easily verified that A is a cylindric field of subsets of etu We let m be the cylindric set algebra with universe A For each u E U let gu u if u E et and gu s K if u - K' K < a We now set v (fEaU:gof is not one-to-one} U «0',1,2, ...)(=) . Clearly V EA. Now we want to show tha t 'Jl.l is a CA and to do a this we shall apply 2.2.3. Since for all distinct K,A < ct , condition 2.2.3 (ii) is clear. Now suppose that K,A < a , XEA, X c V , and h E c (C X nV) n V We may assume that K l' A Now KA our assumption on h implies that there is an a E U with hi<: ECX a A KA A n V , and so there is a b E U with h EX If h E V , then ab b 32 Re1ativization 1.2.14 hA. E C X n V and hence h E C (C X nV) , as desired. Hence suppose b K A. K E V. Thus (1) is one-to-one. (2) hA."# b For, otherwise h E V • (3) hK -# a . For, otherwise Now we consider several cases. Case 1. ga = gb. Let c ghK: or (gbx ), according as b E a or b E w , and let d = ghK: or (ghx ), according as a E a or a E W. hA.K Then E X since hKA. E X and hA.K: - hKA. , and hA. V since cd ab cd ab c E gc = ghK , so h E C (C XnV) A. K Case 2. gb "* ga f- ghK Then by (1) , gb f- ghtJ. for all f- K,A. and gb "* ga And also ghK -# g4J. for all f- K,A. and ghK f- ga Let c = ghK if b w , c = (ghK) I if b E W Then hA.K E X since E ca hA.K E X , and hA. E V since gc = ghK , so h E C (C X nv: ba c A. K Case 3. ga ghK 0 Thus since hK:A. is one-to-one gb "* = , {K, A.} g 0 ab by (1) , we have a,b Hence hA.K X , and hA. V since E a ab E a E ga = ghK , so h E C (C X nV) A. K Case 4. gb f- ga ghK , K = o . Again by (1) , a E W and b Ea. A. K hA. Hence h (ga)b' E X and E V , so hEC(CXnV) ga A. Ie Case 5. gb "* ga ghK , A. = 0 Then a E a and b E W , so h A. K A. a'(gb) EX, ha' E V, hEC(CXnV) A. K Thus is a CA It is obvious that V is not the unit set ix of a Gws To show that is not even isomorphic to a Gws we a a 1.3.1 Change of base 33 shall use the equation of 2.6.42 (4 ) o It is easy to check that this equation holds in every Gws To say CI that (4) holds in is to say that the following equation holds in itself, for any XV such that X E A: Now let X = (0',0,2,3,4, ...)/'=. Let f = (0',0',2,3,4, .•.). Then 1 10 101 (1',0,2,3,4, ...) V and f O EX, and f 01' n , f 01'1 = (1',1,2,3,4, ...)E X Thus f E Hence if 0 (5) holds in III there is an a E U with fa n V , and a b EU 01 with f ab r:- X From the form of X it follows that a = 0' and b = 0 , or a I' and b = 1. But the first possibility yields and the second possibility yields (1',0',2,3,4, ... ) V Thus (5) does not hold in IU • So L. vlU is not isomorphic to a Gws CI Given a set algebra IU with base U, and given some set W, is III isomorphic to a set algebra with base W? We first consider the case lu\ = \W\ ' where the answer is obviously yes. Theorem 1.3.l. (i) Let IU be a Crs with base U and unit Q' element V Suppose f is a one-one function from U onto a se t W a -1 For any X EA let FX= {y E W:f oy E X} Then F is an isomor- phism from onto a Crs wi.th base and unit element IU a W FV 34 Change of base 1.3.2 (ii) If in (i) is a Cs , then 58 is a Cs m 01 01 (iii) If in ( i) m is a Ws with unit elenent CYU(p) , then 01 OIW(fop) i8 is a Ws with unit element 01 (iv) If in (i) is a Gs with unit e lemen t 01 , where 01 Ui EI Si Si n Sj 0 for i '! j then i8 is a Gs with unit element = 01 U 01 f *Si ' where fS.nfS.** o for i '! j . i EI 1. J (v) If in (i) m is a Gws with unit element OIS(pi) 01 UiEI i ' where n = 0 for i '! j , then 58 is a Gws with unit 1. 1. Oi 01 * (fopi) opj) element (f Si) , where OI(f*Si)(fopi) n OI(f*Sj)(f = 0 Ui EI for i '! j • If we apply 1.3.1 to the special case U = W, in some cases the function F is an automorphism of the algebra m This is always true, e.g., if m is a full Cs In this way one can develop a 01 general kind of Galois theory. We shall not go into this theory, which is rather extensive. See, e.g., Daigneault [D] • Remark 1.3.2. The less trivial question concerning change of base arises when the two bases have different cardinalities. To begin our discussion of this case we shall show that for each cy 3 there is a Cs with an infinite base not isomorphic to a Cs with a finite 01 01 base. First suppose that 3 01 < w Then, we claim, the following equation E holds identically in every CS with a finite base, but OI fails in some finite Cs with an infinite base (this equation is due 01 to and and replaces a longer one originally found for this purpose): 1.3.2 Change of base 35 1 o c(a c(a)[-c(a_l)x + c(a_ 2)x' dOl + c(a .... 2f· s ls2 .... 2/ 1 . -s2c(a .... 2)x] = 1 . To show this, let be a Cs with base U in which does not m a hold identically. Then it is easy to see that there is an XEA such that the following conditions hold: aU ; ( 1) C(a .... ll = (2 ) ""D C(a ....2)X ,-;; 01 Ole 1 (3) C(a_2)X C C(a .... 2l n s ls2 - s2 (a _ 2l Now let R = {21u: u E X} Then R is a binary relation on U satis- fying the following conditions: (4) for all u E U there is a v E U with uRv (5) for all u E U ,not (uRu) (6) R is transitive. holds in every Cs with a It follows too t U is infinite. Thus a finite base. Now we construct a finite Cs 'll with an infinite base such that a fails to hold in m Le t 18 = (B ,<) ,where B is tbe set of ra- tional numbers and < is the usual ordering on B. Let A be a discourse language for 18 , with a sequence (v :S < a) of variables . S .... (18 ) Then {cp : cp a formula of i\} is the universe of a Cs 'll with in- a finite base B. From the usual decision procedure for sentences holding in 18 we see that 'll is finite. Letting X = -;p('8) ,cp the formula V < v ' we see that fails in'll. o l 36 Change of base 1.3.3 For a w , let m be any Cs with an infinite base. By a Theorem 1.3.3, is not isomorphic to a Cs with a finite base. m a Finally, some remarks on the case Ci 2 Any crs has base O If is a finite Cs , then is isomorphic to a CS with o m 1 m 1 a finite base. In fact, say U is the base of m. For each non-zero a E A choose u U so that E a Let U (u : a E A} , and a E (ua > ' a for any a E A let fa = a n 1U' Then f is an isomorphism of m onto a CS with finite base U' In a later article in this series 1 we shall show that any finite GS (resp. Cs ) is isomorphic to a 2 2 GS (resp. Cs ) with a finite base. 2 2 Theorem 1.3.3. Let m and m be Gws 's and assume that a ' If IW\ < a n ill for some subbase W of 58 ,then !W\ = \W'\ for some subbase Wi of m. Proof. Let K: Then (1) in 58 In fact, Ciw(p) is included in the unit element of 58 for some p E Ciw , and any q E aw(p) such that K1q maps one-one onto W will be a member of the left side of (1) It follows that (1) also holds in m, and this gives the desired set Wi • Remark 1.3.4. The implication in Theorem 1.3.3 does not hold in the other direction. Namely, for each a 2 there are a Gs a m, a Cs m with m;;. 58 and a subbase W of m with \WI < a n ill , Ci such that the base of m has cardinality f IWI To construct these objects, let U and W be disjoint sets with \ul > \WI = 1. Let 1.3.5 Change of base 37 be the full Gs with unit element aU U OW , and j!J the full Cs a a with unit element aU. The mapping f from A onto B such that fX = X n aU is a homomorphism by 2.2.12. Clearly the above properties hold. We can strengthen a part of 1.3.3 by use of the following notion of base-isomorphism. Definition 1.3.5. (i) With f as in Theorem 1.3.1, we denote by f the function F defined in 1.3.1. (it) Let and j!J be Crs 's with bases U and Wand unit a elements V and Y respectively. We say that and j!J are base- isomorphic if there is a one-one function f mapping U onto W such f is an isomorphism from al onto j!j • The following result is an algebraic version of the logical result according to which any two elementarily equivalent finite structures are isomorphic; it is due to Monk. Theorem 1.3.6. Let and j!J be locally finite-dimensional regular Cs 's , and assume that If the base of either or a j!J has power < 01 nUl, then and j!J are base-isomorphic. Proof. Assume the hypotheses, and suppose that and j!J have bases U and W respectively, and that lUI = < 01 nUl Let G be the given isomorphism from onto j!J Note that by 1.3.3, \w\ = lUI . Now we need the following For each regula r Cs having a base T of power (S: Dl there is a function s which assigns to each T E 0/ a mapping sIS: of C into C such that: T IS: (a) for any x EC and any f E O/T we have f E s x iff T f 0 T E x ; (b) if F is an isomorphism from (S: onto any regular CSO/ with base of power < 0/ n w , then for all x E C and TT This claim is of course related to our considerations in section 1.11. For 0/ w it follows easily by the methods of section 1.11. In fact, for any x E C let t::.x: (1<:0"" ,I<:z.,_l} , with 1<:0 < .•. < I<:z.,_l ' let T E 0/0/ , and let 11. < ... < At. -1 be the first z., ordinals < 0/ 0 not in t::.x U tTI<:O, ••• ,TK:z.,_l} • Then we set Now note that f E iff E y for all < 01 and all y E C \I fv The claim now follows easily, using the regularity of (S: in checking (a) To establish the claim in the case 01 < w it suffices to con- sider the case in which T is a transposition [I<: IA,t../I<:] with K: t- A (see the preliminaries), since every transformation of 0/ is the com- position of a finite sequence of replacements and transpositions. In this case we let 1.3.6 Change of base 39 Then condition (b) of the claim is clear. For condition (a), let x E C and fEaT. First note: (1) if f E DK'A ' then f = f 0 T (2 ) if f E D'AIJ. with ". K,'A then f 0 T f 0 0 ['A/K] 0 (3) if fED with IJ. r K,'A then f 0 T f 0 0 [KIA] 0 (4) if fED with K,'A and iJ. r \I , then f 0 T = f 0 [iJ.IA] 0 IJ.\I r Now to verify (a), first suppose f E s x • If f E x DK'A ' then 'i n f 0 T E x by (1) ; the other possibilities are taken care of by (2)- (4). Second, suppose f 0 T Ex. Since \T\ < a , we have U D = aT , and so fED for some distinct IJ.,\I < a. Then iJ.\i (1)-(4) yield f E , as desired. We have now fully established T the claim. Now let u be a one-to-one function mapping U onto • To shoe that III and l8 are base-isomorphic we shall first take the case in which III is finitely generated. Say III = SgX , where 0 r XA and X is finite. For each w mapping a subset r of a into U we set Wi = W U the following eLerre n t of A: 40 Change of base 1.3.6 (5) E w' E x} n(suow IX : x E X, w n n{",suowIX:X E X, w E w' !/:. x} n n D n CO D K (This element is an algebraic expression of a complete diagram of a -1 -1 finite structure.) Le t h = u U Then = w' for each w in (5), so it follows from (a) of the claim that h is a member of the element (5), which is thus shown to be non-zero. Applying G to (5) and using (b) of the claim we conclude that the following element of B is non-zero: (6) n(s58uowIGX:X E X, w E w' EX} n E X, w E w' !/:. x} Let g E be any member of (6). It is easily checked that f go u is a one-to-one function mapping U onto W By 1.3.1, 1 is an isomorphism of into the full Cs with base W. Thus by CI 0.2.14 (iii) it suffices now to show that X1 f X1G (hence f = G , ,., as desired). So, let x E X , and suppose that k E fx Thus -1 1 f ok E x . Let w = 1 (i o k) Since x is regular, w' E x Thus because g is a member of (6) we infer using the claim that go U 0 w' E Gx Now 1 go u 0 w' = !i:.x 1 k , so by the regularity of Gx we see that k E Gx. Similarly, k!/:. Fx implies k!/:. Gx , so Fx Gx. Having taken care of the finitely generated case, we turn to the general case. Let = {C : 0 f C A, le\ < w} . For each 1.3.7 Change of base 41 e E' let Ie = {f : f is a one-to-one function mapping U onto W , and 7 is a base-isomorphism of onto a subalgebra of 18} . Thus the finitely generated case treated above shows that Ie t- 0 for all e E Since each set Ie is finite, choose eO E with \I minimum. For each DE we have e I Ie UD I ,and hence o o C0 Ie UD = Ie Therefore any member of Ie induces a base-isomorphism 000 of onto 18 , as desired. The next few results 1.3.7-1.3.10, and Remark 1.3.11, are due to and and address the question concerning possible improvements of 1.3.6. 1.3.7. Let and 18 be base-isomorphic Crs 's via F• 01 If x E A is regular, then so is Fx. Lemma 1.3.8. Let be the full Cs with base U. Let x 01 be a regular element of not in the minimal subalgebra of , with Then there is a base-automorphism F of such that Proof. For each equivalence relation E on let For each s E OiU ,let Es {(IC,,,) : K,fc E sK s,,1 • Then we put Thus Y is in the minimal subalgebra of Clearly x y , so there is atE y _ x Say t E sEx Then for all K,fc E we have tIC tfc iff sK = s". Hence there is a permutation f of 42 Change of base 1.3.9 U such that fsK = tK for all KE ex. Thus "'"f is a base-automor- phism of Since sEx, we have f 0 s E fx Now 1 (f 0 s ) = f;.x1 t, t x , and fx is regular (cf , 1.3.7), so f 0 s f. x. Thus x r fx , as desired. Let a w , let E Cs ,and suppose that 1.3.9. a x E A, x is regular, < w ,and x is not in the minimal sub- algebra of Then there is a homomorphism f of onto a Cs 18 Q' such that fx is not regular. Let U be the base of , and let r;;. be tre full Cs R!:£2l. Q' with base U By Lemma 1.3.8, let F be a base-automorphism of such that Fx .,. x • Let q E Q'u be arbitrary. For any c E C let Gc = (c n Q'U(q)) U (Fc n (Q'u-Q'u(q» Because 0 it is easy to verify that G is an endomorphism of Let f = A 1 G and 18 = f*m It remains only to check that Gx is not regular. Since Fx .,. x , say s E Fx""x. Let z E "u - Q'U(q) be arbitrary (note that lui> 1 since Now we set Thus z' E Q'U"'" Q'U(q) and q' E Q'U(q) • Furthermore, since x and Fx are regular (cf. 1.3.7), we have z',q' E Fx"'" x. Hence z' E Gx and q' Gx. But ex1 z' = Ax1 q' and Thus Gx is not regular. Theorem 1.3.10. Let Q' w. Then every non-minimal locally fin ite di mensional regular Cs is isomorphic to a non-regular Cs Q' Q' 1.3.11 Change of base 43 Proof. By 1.5.2 (i) below, is simple (the proof is easy and direct). Hence the theorem follows from Lemma 1.3.9. Remark 1.3.11. From Lemma 1.3.7 and Theorem 1.3.10 it follows that for each a w there exist locally finite-dimensional isomorphic Cs 's with finite bases, regular, which are not base-iso- 01 morphLc , Thus regularity cannot be dropped, for or m , in Theorem 1.3.6. Andreka and Nerneti have also proved the following: (1) In 1.3.6 we cannot replace E Lf by E De see a Q! [AN3] , Prop. 3.5(iv). (2) In 1.3.6 we cannot replace E Cs by E Gs • a a (3) The condition that one of the bases is finite cannot be removed. They also noted that a w Cs cannot be replaced by a Ws (or Gws) ; we give the simple example. Let p = (0 : K: < a) a a and q = (0 : K: even < 01) U (l : K: odd < 01) Let and m be the minimal Ws 's with unit elements a 2(p) and respectively, a and let be the minimal Cs with base 2 Q! we have > , and since is simple (cf. 1.5.2), it follows that Similarly, Clearly and are not base-iso- morphic. Remark 1.3.12. To complete the discussion of change of base when one base is finite, consider the case of two Cs 's and Q! with bases U,W respectively, where Q! lUi < w Then it is possi- ble to have W even though lui f \W\ • For example, let and m be minimal Cs 's with bases U and W respectively, subject a 44 Change of base 1.3.13 only to the condition a luI ,\w! ; then by 2.5.30. But also the following simple result shows that not all possibilities can be realized (also recall Remark 1.3.2). Theorem 1.3.13. Assume that is the Cs with base U gener• a atedby {{(u:l«a)}:uEU} Then if is any Crs with base a W such that , we have lUi \W\ • Proof. Let f be the given isomorphism from m onto , and for each u E U let x = {( u : K < a)} Then (fx : u E U) is a u u system of pairwise disjoint elements of B Furthermore, for any u EU and < a we have Xu ' and so fxu Hence for every u E U there is awE W such that (w: K < a) E fx Hence u Remark 1.3.14. By Cor. 1.4 of Andrcka , [AN3] , the algebra m of 1.3.13 is regular. The construction can be modified to give a Ws with the same conclusion as 1.3.13. Namely, fix U EU and let rY. o p = (u :K < rY.) Let m be the Ws with unit element rY.U(p) gener O rY. ated by {p o :u E U} . u Theorem 1.3.13 and Remark 1.3.14 show that in general the size of a base cannot be reduced. We now prove a theorem giving important special cases in which it is possible to decrease the base; the theorem is due to Monk, and Nemeti. It is an algebraic version of the downward LowenheimSkolem theorem, and is proved as that theorem is proved in Tarski, Vaught [TV] • It generalizes Lemma 5 in Henkin, Monk [HM] • First we need a definition and a lemma. 1.3.15 Change of base 45 Definition 1.3.15. Let and m be Crs 's with unit ele• 01 ments Vo and VI respectively. If the mapping (X n VO:X E B) is an isomorphism of m onto then we say that is subisomorphic to m , and m, is extisomorphic to • This definition gives algebraic versions of the notions of elementary substructures and elementary extensions. Note that if is sub isomorphic to m then the unit element and base of are contained in those of m • 1.3.16. Let u and m be Crs 's with unit elements 01 Vo and VI respectively, and assume that is subisomorphic to m Then if X E B is regular in m , so is X n Vo in • Rema rk 1. 3 •17. For each 01 O!: W there is a regular Cs sub 01 isomorphic to a nonregular Cs (thus 1.3.16 holds only in the direc 01 tion given); the example is due to Andreka and Nemeti. Let , resp. , be the full Cs with base w + 1 , resp. w Set p Cl' o (w : K < 01) q = (0 : K < 01) (thus qO = wand qK = 0 for all w K 1 0). Set X {u E O!(w + l)(P) : uO = w} U {u E CY(w + 1) rv O!(w + l)(P) : uO O}. lQ 0' } Let m and = 6rs; {X n W·. 11 p = 11 q , p EX, but q l X Hence X is not regular in m • Clearly, however, X n CYw {u E O!w : uO = O} is regular in and nO!w) = 1 ; hence by 1.4.1 below, is regular. Finally, is subiso morphic to m. This follows from the following three facts: 46 Change a f base 1.3.17 (1) If and are simple CAa'S generated by txo1 and {xl} respectively, with = = 1 , and if for every K < wand E < 2 we have o c(K)[d(K X K)·n s x 1 o and A then there is an isomorphism f of lB' onto such that fX X l o (2) '11 and 18 satisfy (1) , with = , = lB' , X = Xl , X n Cl w Xo (3) If f is an isomorphism of lB onto such that fX X n Clw , then f = (y n Cl'w : Y E B) It is straightforward to check (2) , except possibly that and lB are simple. is simple by 1.5.2 below. That both and lB are simple is seen by 2.2.24. For (3) , note that all K < CI , and hence that both f and (y r\ aw : Y E B) are homo- morphisms from -2lb8 into 18Ll9 agreeing on which generates 18118 by 2.2.24 and (2) Hence (3) holds. Finally, (1) is a special case of Theorem 17 of Monk [MIl We sketch the proof of (1) for completeness. It follows from the fol- lowing statement that there is an isomorphism f of 18')8' onto lBl.l)1' soc h that d for all K,A. < Oi: KA (4) For all finite R,S a x a and all finite r,6 Oi , 1.3.18 Change of base 47 o iff o • Since for all y and K, Y = 0 iff cK;Y = 0 , one sees tha t (4) is true by using the hypothesis of (1) and 2.2.22, proceeding by induc- tion on \FdR U FdS U r U Given such an isomorphism f, that f preserves c is also easily seen by 2.2.22. K Theorem 1.3.18. Let m be a Crs with unit element V and Cl' base U Let K be an infinite cardinal such that \A\ s K s lUi Assume S U and lSI s K. Then there is a W with SWU such that \W\ K and: (i) Each of the following conditions a) - c) implies that m is ext-isomorphic to a Crs with unit element V n O'W: Cl' a) 91 E Ws Cl' b) K == K \0'1 ; c) is a regular , and is m Gs0' K = L KJ..I. , where the least infinite cardinal such that < A for all x E A Cl'U(p) (ii) If is a Ws0' with unit element then m is ext-isomorphic to a Ws with unit element O'w(p) or (iii) If m is a Cs and K = K\0'\ , then m is ext-isomor- a! phic to a Cs with base W . rt (iv) If is a. regular GSa! (resp. Cs ) and (i) (c) holds, Cl' then m is ext-isomorphic to a regular Gs0' (resp. Cs0' ) with base W (v) If m is a Gws then m is ext-isomorphic to a Gws a! a! with base W 48 Change of base 1.3.18 (vi) If a K , then is ext-isomorphic to a Crs with a base W. Proof. We assume given well-orderings of U and V. (i) (a) and (ii): Note that \a\ \A\ • There is a subset TO of U such that = K , S Rgp ,;; , and X naT l' 0 whenever 0 XEA \Tol U TO 0 l' Now suppose that 0<[3 < K and T has been defined for all 'i < [3 . 'i Let M = and let T = M U (a E U: there is an X E A, a I-' < a, and a S u E aM(p) such that a is the first e Lement; of U with the property that E X} • J1a Let W = T = UT By induction it is easily seen that K 'i easy to check. (i) (b) and (iii): We make the same construction, begin- ning with TO U such that \Tol = K whenever o l' XEA; to construct T we replace above by 8 a The condition Ki \ = K is used to check that \T[3\ = K for all [3 ,,; K . To check that (X n O!w n V : X E A) preserves C it is I-' la\ enough to note that Ci < cfK because K = K , and hence any p E aW is in CiT for some 8 < K (i) (c) and (iv) : If is discrete then B the conclusion is clear. Now suppose is non-discrete. Then \a\ \A\ K. Furthermore, we may assume that a 2 , since the case a 1 is treated by (i) (a) and (ii) above (cf. 1.1.8). Now we proceed as in (i) (b), except that TS is defined as follows: T M U (a E U there is an X EA ,a I-' E kX , a S 1.3.18 Change of base 49 such that a is the first element of U with the property that v E X for some v E V with The condition in (i) (c) is used to check that for all e K, and that u E rW with \f\ < A implies for sone e < K• To check that h = (X nO'w n V : X E A) preserves C IJ. we need to prove (*) for X EA. If IJ. , then (*) is clear. Assume E • Then the inclusion in (*) is clear. Suppose p E C[V]X n n V IJ. So p E V , and plJ. EX for some a. Choose \I E 0'''' tl1} and let a u = .... tlJ.}) U t\l})1p. Then u E for some < K, so by the definition of , v E X for some v with Then V\I = U\l = P\l , so it follows since '21 E Gs and 0' 2 that pj.1 EV ,while obviously 0' plJ."IJ. E Thus _.../:iX 1 v = /:iX 1vj.1 , so by regularity of X, PvlJ.".,. EX Thus p E C[O'wnv] (X n aW n V) , as desired. Far the regularity required IJ. in (iv) , see Lemma 1.3.16. (v): Let the unit element of m be U O'T (pi) iEI i ' h Q'T(pi) n Q'T(pj) were i j o for i f j Choose JI with \J\ s K and X n U Q'T(pj) 4 0 for all non-zero such tba t S c U j EJTj jEJ j T X EA. Now take any j E J. The set tX n E A} is the J a: ( 0) universe of a Ws , as is easily checked (since ToPJ may not be a Q' ] member of A , we cannot use the notation R{, y = )• We Y J 50 Change of base 1.3.19 denote this by If IT.\ < K , we set W0 = T. If J JJ IT.I ;;:: K, by (ii), is ext-isomorphic to a Ws with unit element J J where rgp. U (8 n T.) W. T. and \WJo\ = K• With Vi = J JJJJ the desired conclusion is easily checked. (vi): Let Zo be a subset of V such that \zol s K, S c UZEzoRgZ, I UZEzoRgzl = K, and X n Zo r 0 whenever 0 r X EA. If nEw and zn has been defined, let zn+l = Zn U (z EV: there is a q E zn ' an XE A, and a < a such that z is the least element of V with z EX and Le t V I = UZ and It is easily checked that m nEw n is ext-isomorphic to a Crs with unit element Vi and base W satis- 0'. fying the desired conditions. Remark 1.3.19. The conditions in 1.3.18 are necessary for the truth of the theorem. First, Andreka and Nemeti have noticed that for each a wand each infinite K with K\O'.\ 1 K there is a local1y- finite dimensional Cs m of power \0'.\ ' with base of power Kla\ , 0'. such that m is not ext-isomorphic to any Cs with base of power K 0'. This shows that Ws cannot be replaced by Cs in (i) a), the con- 0'. 0'. dition K = cannot be weakened in (i) b) or (iii) , regularity cannot be dropped for Cs I s in (iv), and "Gws" cannot be replaced a a by "cs .. in (v) • To construct this algebra, let U = O'.K Let 0'. R = (i,g) : f,g E: aU and I(K < a : fK 1 gKl\ < w} 1.3.19 Change of base 51 Let F be a function from aU into U such that for all f,g E aU we have Ff = Fg iff (f ,g) E R• Let T = (I< + 1 : I< < w) U (I< : I< Ea,...w) . Ct • c[ Ctu} X = Ct Set x= {q E U• F(q 0 T)} • Clearly u and = to} qo ° t:.X Let 18 be the full Cs with base U and let m= 6iJ(18){ X} Thus a m is locally finite-dimensional, \A\ = a , and the base of m has power I< \a\ . Suppose W U and \W! = I< • To show that m is not ext-isomorphic to any Cs with base W it suffices to show that a n CtW) 1 CtW • Note that each equivalence class under R n (Ctwx aW) has cardinality la\ U I< < I Fourth, the hypothesis that I< is infinite is essential by Remark 1.3.2. Finally, Andreka and Nemeti have shown that in (i) (c) one cannot re- place "Gs II by "Gws ". We describe their interesting example: for a r:J r eg each Ct to we construct an E Gws n Lf with a base U such a Ol that \A\ \Oll s Iu\ and having the property that for all Wk U, if \wl \Ol\ then (a n etw: a E A) is not an isomorphism. This provides the desired counterexample, with I< = let\. Let It is easy to define p E such that for all y,O , if Y < ° < then 52 Change of base 1.3.19 01 (po) Py wand such that for every infinite r wand every y < there is a 0 < e with y s 0 and Po E 01r For each y < e let (p ) OiU Y ,and let X = (f E V : fO l w} y Let III be the Gws with unit element V generated by (X} . We 01 claim that III is the desired algebra. Note that III has base a , ard that U U whenever ys:II<8 Since LX = 1 , by 2.1.15 (i) we y o . have u E Lf Now we show that if and , then 01 \wl = \Oi\ is not ext-isomorphic to any Crs with unit element V n otw Let 01 Iwnw\+l We now consider two cases. I< E w Now C[V] (d(K X K) • n sOX) = V (I<) z. C[Z] (d(K XK)'rr - sOX) (K) z. But then E W , by our choice of 0 , contradicting the fact that (po) E Z otw • I.3.l9 Change of base 53 It remains to show that !I.l is regular. For this purpose we need the following fact about regularity. It is a part of Lemma 1.3.4 of Nemeti [AN 31, but we Lnc Iude its short proof for completeness. (*) Let !I.l be a Gws with unit element V, XEA, and r a finite Ci subset of Ci. Suppose that for all f,g, if f EX, g E V and U r) 1 f = U f) 1 g , then g EX. Then X is regular. To prove (*), assume its hypothesis, and suppose that f EX, g E V , and (6X U 1)1 f = (6X U 1) 1 g ; we are to show that g EX - ... Let ® = r .... (6X... U 1) , and for each k E V let k ' = (Q'....®) 1 k U (8 X {fO}). Since fO = gO and 8 is finite, we have f' ,g' EV Since {K: fK l' f'K} is a finite subset of tY.""6X, we have f' EX. Clearly U r) 1 f' = U r) 1 g' , so by the hypothesis of (*) , g' EX. Finally, {K: gK 'I g'K} is a finite subset of Ci""p;X, so g EX, as desired. Now we prove that !I.l is regular. Let YEA. Since ...t:,X = 1 , we can apply 2.2.24. Note that for any K E w we have for Z = X or Z = V""X • Thus by 2.2.24 we can write where \r\ < w, Ry,Sy \; Ci XCi and \Ry\, \Sy\ < w for y E r , \®y\,\Ey\ < W for y E r , and lJoyo,vye: E sx for y E r, {) E ®y , e: E Ey. Let 54 Change of base 1.3.20 () = UyEr(FdRy U FdSy U : 0 E By} U (\)yo : 0 E Sy}) • Note that O. Thus to prove that Y is regular it suffices by (*) to suppose that fEY, g E V ,and 01f = 01g , and show that g E Y. Since fEY, choose y E r so that f En(K,A)ERy DKA nn(K'AlESy"'DKA n n089y s n nI5ESy s (V'" X) o Since spX = (h E V : hp w} , for any p < C'i it is now easy to see that g EY• We shall consider the question of increasing bases in section 1.7, since we need ultraproducts to establish these results; see 1.7.19- 1.7.30. We wish to conclude this section by considering a question related to the changing base question: when is a Ws with unit ele- C'i ment isomorphic to one with unit element C'iW(q)? The following theorem is a generalization of Lemma 6 of [HM] due to Andreka and Theorem 1.3.20. Let (resp. and la (resp. la' ) be (the full) Ws ' s with unit elements V and bases U a o VI ' and o and U , respectively. Consider the following conditions: l (i) and la are base-isomorphic; -1 t here .,p E Vo and qr E VI such tha t P'\ P ,-I q' \q' and \ UO",Rgp' \ = \Ul",Rgq'\ (iii) la' ; (iv) is base-isomorphic to JB' • 1.3.21 Change of base 55 Then (i) => (ii), while (ii) (iii) (iv) Proof. Say V = O'U(p) = O'U(q) (i) => (iv): trivial. o 0 and VI 1 (iv) => (iii): trivial. (iii) => (ii): Let f be an isomorphism from a,,.. ' onto ""\U' • Choose q ' EV1 so that f(p} (q '} If pK = pA. , then (p} DKA. ' so (q'} DKA. and q'K = q'A. By symmetry, -1 ,-1 pip = q'\q • Also, it is easy to check that \U_Rgp\ = \(d E A' : d is an atom S; I(d E B' : d is an atom:S: CO(q'} n Thus (ii) holds. (ii) => (iv): this is proved in [HMJ , but we sketch the proof here. Let f = (P'K,q'K) : K< O'} . Then f is a one- to-one function from a subset of U onto a subset of U ' and it can o 1 be extended to a one-to-one function f' from U onto U By o 1 Theorem 1.3.1, is an isomorphism from onto the fuli Ws ill" 0' Q' (f'''p) with unit element U If rand 6 are finite sets such that 1 (O'-r) 1 p = (O'-r) 1 p ' and (0'-6) 1 q = (0'-6) 1 q ' , clearly (O'-(r U 6»1 (flop) (0'''' (r U 6» 1 q. Thus O'u (f' 0 p) = O'u (q ) , as 1 1 desired. Remark 1.3.21. It is easy to see that in 1.3.20, (ii) does not imply (i) in general. The condition of base-isomorphism in (i) cannot be replaced by isomorphism. This follows from the following theorem of and Nemeti. 56 Suba 1gebras 1.3.22 Theorem 1.3.22. If is a locally finite-dimensional WS m a with base U and q E aU , then is isomorphic to a Ws with a unit element aU(q) • Proof. Let m have unit element aU(p) • For any XE A let fX [u E aU(q) : there is a v E X with kX1 u = ....6X1 v} Using the regularity of m (1.1.16) , it is easy to see tl:a t f is an isomorphism from the with unit 18tm into \sUB , l8 full Wsa element ctu(q) . Now let K < Oi and XEA . If u E fCKX , choose v E CKX so that /.t.C u = 6C X1 v Define wE aU be setting for any A < Q' KX1 -K u A WA { VA 6C X1 w . Hence by regularity wE C Choose a E U such that "" K t? K K w EX. Now 6X 1 w = 6X 1 uK , so uK E fX • Thus u E CKfX The a VM a ,.... a a converse is straightforward. Some results related to 1.3.20 and 1.3.22 are given in 1.7.27- 1.7.30. Our various classes of set algebras are clearly closed under the formation of subalgebras, and we shall not formulate a theorem to this effect. The following theorem gives an important method for forming regular set algebras. The proof is due to and Nemeti. I.4.l Suba 1gebras 57 Theorem 1.4.1. If is a Cs generated by a set of regular a elements with finite dimension sets, then m is regular. Proof. We shall use (*) from 1.3.19. Let B be the set of all finite dimensional regular elements of m; it suffices to show that B E Su m • Clearly DEB for all K,A < Ci , and clearly B KA is closed under -, since kX = k(-X) Now let X,Y E B ; we show that X n YE B. In fact, we shall verify (*) with r = kX U n Y). Suppose r 1 f = r 1 g , f EX n Y and g E aU. Then... 6X1 f = ,...6X1 g so g E X since X is regular. Similarly, g E Y, as desired. Finally, suppose XE Band K < a ; we show that CKX EB• To this end we verify (*) with r = 6 X 6C X• So, suppose ,... .. K 6X 1 f = 6X 1 g , f E CKX , and g E aU Then for some u E U we '" - K have E X• Thus t,X 1 fK = t,X 1 gK and E O!u , so by the lu __ u ... u gu K regularity of X, gu EX. Thus g E CX , as desired. K Remarks. 1.4.2. As mentioned in the introduction to this paper, regular cylindric set algebras arise naturally from relational struc- tures and the notion of satisfaction in an associated first-order language. Using 1.4.1 we can express this construction of regular Cs 's without recourse to an auxiliary language. Namely, let m= O! (A,Ri,C) j)iEI,jEJ be a relational structure, and let O! w. Let Pi be the rank of R and a the rank of OJ for each i E I and i j j E J. Set X = {(x E OIA :p.1x E R.}:i E 1} U ({x E O!A : 0.(x : K < 0 .) = x }: j E J} • J K J OJ 58 Subalgebras 1.4.2 Clearly each memb.er of X is regular and finite dimensional. by 1.4.1, the Cs of subsets of aA generated by X is regular. a This is the same Cs described in the introduction to this paper in a terms of a language for j)j • r eg Conversely, given any E Cs n Lf , there is a relational a a structure j)j such that is obtained from j)j in the way just des- cribed. We shall prove this, which is rather easy, in a later paper where we discuss this correspondence in detail. The assumption that the dimension sets are finite in 1.4.1 is essential, and cannot even be replaced by the assumption that j)j is dimension complemented, or by the weaker assumption that the regular elements mentioned have dimension sets with infinite complements. To see this, let a;:: w , let be full Cs with base w , and a a let X = { x E w : for every odd K {I( < a : K is odd} U 1 , and hence j)j = E Dc Furthermore, a X is clearly regular. But COX = [x E aw : there is a "- COX is not regular. Andreka and Nemeti have shown that "Cs" cannot be replaced by O! "Gws" in 1.4.1. They also established the following interesting O! facts about Gws 's (where is the minimal suba1gebra of a (1) For any Gws , O! ;:: W, is regular iff for every two O! subbases Y and W of j)j, \Y\ = \wl < w or \Y! ,\w\ ;:: w ; (2) There is a Gws j)j, a;:: w, having elements X,Y such that a kX = = 1 and both 6.1{X} and are regular but 6,{X,Y} is not; 1.4.3 Suba 1gebras 59 (3) There is a Gws Ct W, such that is the largest Ct regular subalgebra of and there is an element X E A such that = 1 ,X is regular, and COX is not regular. (4) For every Gws the following two conditions are equivalent: Ct (a) m is normal (see 1.2.6); (b) if is the full Gws such that , then every sub• Ct set X of consisting of regular finitedimensional elements is such that is regular. We also mention the following useful and obvious property of regular Cs's: Theorem 1.4.3. is a regular Cs ,then Zd9j {O, I} . Ct For the rest of this section we consider the problem of the num ber of generators of set algebras, in particular, conditions under which a set algebra has a single generator. This question was considered in 2.1.11, 2.3.22, 2.3.23, and 2.6.25. In particular, following 2.1.11 the following result was stated, the proof being easily obtained from the proof of 2.1.11: (*) If 2 Ct < wand K < w , then the full Cs with base is Ct generated by a single element. By generalizing the proof of 2.1.11 further we obtain the following generalization of (*) , due to Monk. Theorem 1.4.4. Let Ct < W, and let m be the full Gs with tx o for distinct i,j EI 60 Subalgebras 1.4.4 2·\I\s;.a,and l<\Ui\ Proof. The theorem is trivial if a s; 1, so assume that Q' 2 • We may assume that I = e < w with 2e S; Q' For each I<: < e let be a one-one function mapping UI<: onto some < w Our single generator is x {u E V : if I<: is the element of e such that u E Q'UI<:' then t Ku2K < tl<:u2K+l} • Now for each I<: < e we define a sequence Y E wA by recursim : K while for 0 s; < W we set Then it is easily seen by induction on that for all < w Hence if K < 6 and v E UI<: we have {u E Q'U" : u "+1 = v} = Y "'" Y rv2 '" I<: ,t v 1<:, tl<: v+l K Thus for K < e and u E Cl'U we have I<: Thus X generates m, since A is finite. 1.4.5 Suba 1gebras 61 Remark 1.4.5. Note that the assumption a < w in 1.4.4 is in- essential if we replace "generated" by "completely generated". Theorem 1.4.4 has been generalized by and who showed in [AN4] that if for each a with 2 a < w we let the smallest such that there is a systElll (U : y < \3) y of pairwise disjoint sets with w > IU I > 1 for all y y < S such that the full GSa with unit element aU is not generated by a single element, Uy Also note that in Theorem 1.4.4 one may assume that lUi! = 0 is pos- sible or even replace the assumption that 1 < is a Gs implies the existence of pairwise disjoint non-empty a a j E J , such tba t V = U jEJ Wj , and an easy argunent shows that IJ\ \1\ • Closely related to (*) above is the following theorem of Stephen Comer. Theorem 1.4.6. Assume that a < w , and that is a Cs with 01 a base U such that lUI a. Then can be generated by a single element. 62 Subalgebras 1.4.6 We may assume that a 2 and UfO. For each x E aU let x' = n(x: x E X E A} Thus x' EA, since A is finite. We shall show that if x maps a onto U then x' generates To prove this we need four preliminary results. (1) If x,y E au, K,A < a, K fA, and x 0 [K/A] = y , then s' = CKx' n D ,so s' E Sg{x'} KA For, x Ex' , and (a""'(K}) 1 x = (a"'(K}) 1 y , so y E CKx' n D • KA Hence y' CKx' n D • Now let X EA be arbitrary such that KA y E X Clearly x E CK(X n D ) , so x' CK(X n D ) Thus KA KA Hence CKx' n D y' ,and (1) is established. KA a (2) If x,y EU, are distinct ordinals = y ,and x then y , E Sgfx'} K For, under the hypotheses of (2) we have y = x 0 [KIA] o [A/!J.] ,so (2) follows from (1). (3) If x,y E aU, K,A,!J.,\I are distinct ordinals < a, x ° [K/A,A/KJ = y and XiJ. X\l then y' In fact, y = XO[!J./K] o[Kh] ° [A/!J.] o [!J./\I] , so (3) follows easily from (1). (4) If x,y E au, Rgy Rgx ,and \Rgx! < a , then y' E sg{x'} • For, write y = X°O' with a E aa ; express a as a product of transpositions and replacements, and use (1)-(3). Now let x map a onto U; we show Sg(x'} =A. If lUI < a , 1.4.7 Subalgebras 63 then by (4) y' E Sg{x'} for each y E C¥u , and for any X E A we have X = U y' E Sg{x'} • Assume that = c¥. For each K < c¥ yEX lUI let KY = x ° [K/O] ; thus (KY)' E Sg{x'} by (1) Also let z '"' x o [0/1]; again z ' E Sg{x'} by (1) • Now if wE C¥u and \Rgw\ < C¥ , then either Rgw RgKy for some K E cy"" 1 , and hence by (4) w' E Sg{( y)'} sg{x'} , or Rgw Rgz and w' E Sg{z'} sg{x'} K C¥ Hence it suffices to take w E U with \Rgw\ = C¥ and show that w' E Sg{x'} • Let t '" w o [0/1] • Then t' E Sg[x'} by the situation just discussed. Thus the proof is finished as soon as we prove Since w E cot' D ' the inclusion c is clear. Now n nO so in (5) follows. 1.4.7. In contrast to 1.4.6 we now show that for there is a Cs with base U such that ex c¥ luI '" . and cannot be generated by fewer than elements. The exam- ple is due to Henkin, with some simplifications by and Let (t be the full Cs with base a : 13. For each < 13 let C¥ X "" {q E C¥(ex '13) : q is one-to-one and \' q = 13)} , l. L »ax 'J Clearly (X : t. < e) is a partition of d(ex X ex) into pairwise dis- l. 64 Subalgebras 1.4.7 joint non-empty sets, and C X = C d(a X a) for all < Band K l, K K < Q'. Let Y = {Xl, : z, < I"} and let = 6W-(CS-\. We shall show that cannot be generated by less than 10gZl" elements. First let be the minimal subalgebra of , and let \8 = (Y U M) Note that Atl8 = Y U {x: x E AtM, x f d(o: X a)} Now we claim (1) for all b E B and all K < a , C b EM. K In fact, since B is finite it is enough to prove (1) for an atom b , and by virtue of the mentioned description of Atl8 this is clear, since CX = C d(a X ct) for all l, < B• K z, K Because of (1) we have, first of all, B = A Now suppose that GkA and = 6 (s/91,)G we show that B zlG\ • Let Then by (1) = 6rf(\8) (G U M) = rscf) (E U M) Since X k d(ct X ct) E for all z, < B' it follows that Y c {z n d (ct X o ) : z EA , and so , as desired. It is perhaps interesting that the algebra just constructed be generated by Y elements, where y is the least integer 108213. In fact, let f E B(Y2) be one-to-one, with fO = (1 : < y). For each < Y, let Z = U{X : z, < 13, f \I = l} \I Then, as is easily checked, X- z d(ct X a) o n\I X = (d(Q' X a) '" U{Z : f = O}) n n{z : f \I I}. l, \I t. v z, Thus {Z : < y} generates , as desired. \I r.4.8 Subalgebras 65 Remarks 1.4.8. G. Bergman independently obtained examples of csZ's which require a large number of generators. In connection with 1.4.6 and I.4.? it is natural to define the following function. For Z < W let q(a,l3) the smallest y < w such that every Cs with C\' base can be generated by y elements. Thus the example in 1.4.? shows tha t q (ct ,C\' • 1.4.6 says that = 1 for ct. and Nemeti have established many properties of this function q. For example, q(ct,ct+ 1) = 1, q(2,13) = least integer;:;, logZ (B - 1) for > Z, and q(a,a least integer generalizing 1.4.6 and 1.4.7. Note that these results on the function q are relevant to our discussion in 1.3 of change of base. For example 1.4.6 implies that if a CS cannot be generated by a single element, then it is not even a isomorphic to a Cs with base of power a J. Larson has shown that for 2 < w there are 2 isomor- phism types of one-generated Cs "s , P. Erdos, V. Faber and J. Larson ct [EFL] show that there is a countable CS not embeddable in any finitely 2 generated CS • 2 In I.?lO and I.?ll we show that IGs and IGws (for arbitrary C\' Q' r eg a ), ICs and IWs (for 01 < w ) ,and ICs n Lf are closed Q' Q' Q' Q' under directed unions. Andreka and Nemeti have shown that ICs and Q' r eg ICs are not closed under directed unions for Q' We also should mention that H. and I. have solved Problem 2.3 of [HMT] by showing that for each a > 0 there is a simple finitely generated Cs not generated by a single element; see [AN2j. a 5. Homomorehisms The following result about CA's in general will be useful in what follows. Theorem 1.5.I. Let be a CA and Then a {x EEl d : x EI and d E o}} Proof. This is clear since is a minimal CA a Turning now to set algebras, we begin with a result concerning simple algebras. Theo,rem 1.5.2. (i) Any regular locally finite-dimensional Cs a with non-empty base is simple. (ii) Any locally finite-dimensional Ws is simple. a (iii) For a < w any CSa with non-empty base is simple. Proof. Trivial, using 2.3.14. Corollary 1.5.3. Let E CSa WS with base U, 0 U a luI> Then the minimal suba1gebra of is simple. The characteristic of is lUI if \UI < a n w , and it is 0 if lu\ a nw • 1.5.4 Homomorphisms 67 Corollary 1.5.4. Let m be a Gws with subbases (U : i E I) , CI i each UiTa, I T a an d CI:i!: 1 Then the minimal subalgebra of is simple iff either (1) for all i,j E I we lave < m \ Uil \Uj \ CI nw in which case m has characteristic Iuil (for any i EI ), or (2) for all i E I we have Iuil CI nw, in which case m has characteristic a Remarks 1.5.5. Theorem 1.5.Z (iii) does not extend to Gs ' s CI and Gws ' s . In fact, if 1 ,,; a , m is a full Gs with unit element CI CI CI , where U. n U. = a for i T j , and if 1 > 1 and each U iEI Ui L J 1 \ subbase U is non-empty, then m is not simple. For, choose i and let fX = X n ClU• for all XEA Then it is easy to verify La that f is a homomorphism of m into a Cs 58 with base U (and a i a hence that IBI > 1 ), but f is not one-to-one. So m is not simple. A similar construction works for Gws 's. This phenomenon is more fully CI explained by the fact tffit Gs 's and Gws 's are often subdirectly a CI decomposable; see 1.6.3, 1.6.4. For CI:i!: W , I.5.Z (i) does not extend to arbitrary locally finite-dimensional Cs 's For, let 58 be the full Cs with base Ci . Ci Z , let p = (0 : < CI), and let m= ){ClZ(p)} . Now ...t/I'Z(p)) a so m E Lf by 2.15 (i) , while m fails to be simple by Z.3.l4. CI Also, for CI:i!:W 1.5.Z (ii) does not extend to arbitrary loIs 's ; in CI fact, one cannot even replace "locally finite dimensional" by "dimension complemented". In fact, let CI:i!: ill , and choose f CI with rand Ci- f infinite. Let p (0 : K: < CI) and let 58 be the full loIs CI with unit element Ci2(P). Let X = {f E ClZ(P) :f1 f p) , and let m = (X} Then clearly mEWs n Dc ,while by 2.3.14 m is CI 0: 68 Homomorphisms 1.5.6 not simple, since for every finite and Nemeti have shown that 1.5.2 (ii) does not extend to regular dimension-complemented Cs ' s for 0' 2: w They also noted 0' that 1.5.3 does not extend to Gs 's or Gws 's for 0' 2: 2 In fact, a 0' let V = (o} and V = P,2} , and put p = (0 :K < 0') , q = (1 : K < 0') o l Let u be any GSa (resp. Gws ) with unit element 0' U O'uiq) ); in particular, could be minimal. Then is not simple. To see this, observe that some CSO' with base Uo is a homomorphic image of , but m is not isomorphic to ill since Finally, concerning 1.5.4 it is worth remarking that for any Gws m with subbases 0' (Ui: i E I) we have V"!l = (\Uil : 0 < \uil < a n w} (see 2.5.25). Remarks 1.5.6. We now discuss possible closure under homomorphisms of our various classes of set algebras, where not all natural questions have been answered. The situation is summarized in Figure 1.5.7, which we now discuss; cf. Figure 1.1.4 (1) It will be shown in 1.7.15 that HGs 1Gs RGws for a a (J/ all 0' r eg (2) For o < 0' < W we have 1Ws U (l} = ICs (where 1 is 0''''' 0' "'" r e g the one-element Cs ), and hence IWs U {l} = ICs ICs RCs 0' a" 0' 0' 0' r eg = HWs = RCs QI 0' For the remainder of these remarks assume that 0' 2: W• (3) Andreka and Nemeti have shown that there is an E Ws 0' with infinite base such that lim et IWs U (l} QI - (4) In 1.7.21 we show that any homomorphic image of a Cs or 0' a Ws with an infinite base is isomorphic to a Cs This is not 0' 0' 1.5.6 Homomorphisms 69 r eg r eg r eg IGws reg = IGws IGs = HGs HGws = HGs = HGws ex ex ex 0/ ex ex 0: Figure 1.5.7 r eg", true if we replace "Cs" by "Cs as is shown by the following 0: ex example of Andreka and Nemeti. Let U be the set of finite subsets of Of, and let H be a one-one function from ex onto U For each K Of let uK = Of X {HK} and let Y = C(HK)u Set x = UK regular, let yEA We may assume that \0:- 6Y\ ill. We show then that y E (and hence y is regular). Since = , there is a d E such that y ffi d E (using 1.5.1). Let z = y ffi d ; we show that z = 0 (hence Y = d , as desired). Suppose z 1 0 Since z E Sg{y} ,we have Moreover, z E Ig{x} implies that for some finite f c ex • Now fix q E z Then K < 0: Thus q E C(f)YK for some [ex - (HK U r)] 1 q I;;; U Since \0: - 6Y\ w and \llz - llY\ < ill , we K A. can choose AE ex - (6z U HK U f) • Fix f.L E ex - {K} • Then ql\J. E z '" A. but c(f)x , as is easily checked, contradicting 70 Homomorphisms 1.5.6 Next we construct 18 E m.l with \Zd > 2 (hence 18 I/:. , by 1.4.3). Let I = Ig cKx • -x :K -< a} , and let 18 = Clearly x/I E Zd18 , so we only need to show that x I/:. I and -x I/:. I Now if Y EI then there exist finite subsets r,6 of a such that Suppose x E I , with rand 6 as in (*) where y = x. Choose }. < a so that 6 = H}. Then u}. Ex, but u}. is not in the right side of (*), a contradiction. Suppose -x EI, with r and as in (*) where y = -x . Then H E -x , but R is not in the right side of (*) , a contradiction. (5) In contrast to (3) and (4) we now show that RCs rt a ICs HWs rt rcs , and HWs rt IWs u ' where 1 is the one- a a a a Ct OJ element Cs Of course our example will have a finite base. The 01 example is due to Monk; in Demaree [Del there is an error in the con- struction of such an example. Let 2 K < w. Our construction is based on the following obvious fact: (*) is an atomless Cs or Ws of characteristic > 0 , then Ct 01 m has no element nK,}. Let m be the full Cs (resp. Ws ) with base K (resp. unit a a element aK(p) , where p = K X {OJ Let I={X:XEA, IX! Clearly I is a proper ideal of We claim that m/l is not iso- morphic to a Cs or Ws Suppose to the contrary that m/l is a Ct is omorphic to a Cs or Ws 18 • By 1.5.3 , m and 18 have i somor- a a phic minimal subalgebras m', respectively. The two formulas r .5.6 Homomorphisms 71 d«K + 1) X (K + 1» o hold in and hence in m'. Hence the base of m has power K• Noting that W/I is atomless, we can obtain a contradiction to (*) by exhibiting an element X of A such tin tOT X/I S; d(W/I) for AJ.!. all A,J.!. < a Set X = {x E a K : XA T 0 for exactly one A < a} It is easily checked that X satisfies the above conditions. (6) Andreka and Nemeti have modified the above construction to reg show that HCs ¢ ICs a a r eg (7) and Nemeti have shown that ICs ¢ HWs • CI CI (8) It is clear that Gs ¢ HCs , since the minimal subalgebra CI CI of any W E HCs is simple or of power 1 by 1.5.3, while there are a clearly W E GSa without this property. r eg (9) The inclusion IWs rCs will be established in 1.7.13. a a It implies that IWs HCS ' and this inclusion is easy to establish. a a In fact, clearly Ws c SHCs since the full Ws with unit element a Ci Ci is a homomorphic image of the full Cs with base U. Hence a IWs SHCs G HSCs HCs a CI a CI r eg (10) It remains open whether ICs c HCs or HCs a CI O! (11) Andreka and Nemeti have shown that H(Cs n Lf ) ¢ ICs Ci a a By 1.5.1, the inclusion holds trivially if "Cs" is replaced by CI "Csreg" or ''tvs''. Ci CI (12) From the definition of characteristic we know that if W has characteristic K, m> m , and \B\ > 1 , then m has character- istic K • The meaning of characteristic for set algebras, described in 1.5,.3 and 1.5.4, is further elucidated by a result of Andreka and r eg Nemeti according to which for each cardinal K 2 there is a Cs W Ci 72 Homomorphisms 1.5.6 with base of power K, having a homomorphic image 18 such that every Gws isomorphic to 18 has base of power > K• O! r eg (13) Recall from 1.3.9 that for every E Cs , if there is a in an element x, not in the minimal algebra of , with r eg \b.x\ < OJ , then Hll n Cs ct Cs . ... O! O! (14) In contrast to (13), Andreka and Nemeti have constructed r eg a such that Hll ICsreg , mr n Cs >: Cs , and is not a O! 01 simple. The construction is simple: let 18 be tle full Cs with O! base OJ , and let = : x E 01OJ} Note that for any YE ((x} :x E OIw} we have (*) for every finite r 01 and every K E 01 ,." r we have 0 cKco(r)Y - Now if f E and E Cs ,then each YE {f{x) : x E aw) 01 r eg satisfies (*), and so by Theorem l.3 of [AN3] we have IS: E Cs a r eg Thus Hll Cs Cs By (4) above, it follows that n a a Finally, by 2.3.14 it is clear that is not simple. (15) Generalizing the construction given in (4) above, Andreka and Nemeti have shown that for any K 2 there is an E with base K and some 18 E mr with \Zd18 \ > 2 • (16) Contrasting to (15) , Andreka and Nemeti have shown that re g for any K 2 there is an E Cs with base K such that is 01 not simple, but \Zd18 \ :s;; 2 for all 18 E mr (17) Figure 1.5.7 simplifies considerably if we restrict our- selves to set algebras with bases and subbases infinite and to a OJ Let us denote by Cs, Gs ,etc. the corresponding classes. Then we ao O! ao 01 1.5.8 Products 73 obtain Figure 1.5.8; see (3), (4), (9) Here "'? means that we do not know whether equality holds in the two indicated cases. =? Figure 1.5 .8 6. r.Productsc. In terms of products, we can express a simple relationship between Gs 's and Cs 's , and between Gws 's and Ws 's We first express a a a this relationship more generally for Crs 's For this purpose it a is convenient to introduce the following special notation. Definition 1.6.1. Let be a Crs with unit element v and m a m let W (;; V Then rtw is the function with domain A such that for any a E A , rtwa=wna.m Theorem 1.6.2. Let be a full Crs with unit element m a for i,j EI and i T j , and for all i E I. Assume that m E CA For each i E I let m be a i 74 Products 1.6.3 the full with unit element Vi' Then m P In fact, i EI m there is a unique f E such that rtv. pj. 0 f for each i E I Proof. Clearly there is a unique f mapping B into PiEIAi and satisfying the final condition. By 2.3.26, E for i each i EI, so by 0.3.6 (ii). Clearly f is one-to-one and onto. The assumption m E CA is not actually needed in 1.6.2. Ci reg Corollary 1.6.3. For Ci 2 we have IGs SPCs and IGg Ci Ci Ci SR;sreg Q' Proof. First suppose that E Gs say has unit element Ci a U U , where U. n u. 0 for distinct i,j E I , and U. T- O i EI i 1 ] for all i E I . Let V. = Q'U. for each i EI and let m, , f i. be as in Theorem 1.6.2 ; clearly r:.mv. = 0 for all i E I since a 2 '" 1 and m E CA Clearly C 1 f is an is amorphism of onto a subdirect Ci product of Cs 's , as desired. Ci Second, suppose ,each a Cs with base Ci We may assume that U n U = 0 for distinct i,j EI i j Let V. for each i E I , and again let m, and f be as in Theorem 1.6.2. -1 Clearly C1 f is an isomorphism of IS: onto a Gs Ci The second part of 1.6.3 is handled by 1.1.15. In an entirely analogous way we obtain 1.6.4 Products 75 Corollary 1.6.4. For 0'.:2: 2 we have IGws SPWs 0'. 0'. Corollary 1.6.5. Let E IGs \A \ > 1, 0'. < W• Then the 0'. following conditions are equivalent: (i) E ICs 0'. (ii) is simple. Proof. By 1.6.3 and 1.5.2 (iii) (treating a < 2 separately). r e g Corollary 1.6.6. For 0'.:2: 2 we have PGs IGsO'. PGs a 0'. r eg r eg r e g 1Gs ,PGws = 1Gws ,and PGws IGws a a a 0'. 0'. Remark 1.6.7. Nore of 1.6.3, 1.6.4, 1.6.6 extend to 0' s: 1 • Remarks 1.6.8. Closure properties under H, S, P of our classes of set algebras are summarized in Figure 1.6.9 for 0':2: 2 ; see also Figures 1.1.4, 1.5.7, and 1.5.8. We now discuss this figure. IGws HSPGws IGs HSPGs SPCs SPWs 0' 0'. Of 0' 0' 10'. ______HPCs HCs )?------PCs 10'.______0'. ICs 0'. :: H\:_PC 1 HWs01_____ 10'. PWsO'. IWs 0' Figure 1.6.9 (0' :2: 2) 76 Products 1.6.9 (1) For a 1 the diagram is different; then the classes are just five in number, increasing under inclusion: (a) IWs ECA is simple} ; a a (b) ICs ECA : [j is simple or = I} ; a a \AI (c) PCs CA is a product of simple CA r s} a E a a (d) RPCs a ; (e) RSPCs CA = SPWs a a a (2 ) The example [j in 1.5.6 (5) also shows tha t HWs PCs in a¢ a general, for a OJ In fact, (*) continues to hold for all E PCs a r eg r eg (3) Andreka and Nemeti have shown that Cs rt PWs and Cs rt PCs a a a a for a OJ • (4) To show that PWs RCs for a OJ , Iet and l8 be Ws r S a'* a a with bases K,A respectively, where 1 < K < A < OJ since each non-trivial homomorphic image of a Cs has a well-defined a characteristic, while [j X l8 does not (c f , 2.4.61 for the definition of characteristic). (5 ) For any a we have SPCs RPCs (for a OJ this was shown by a '* Cli 1.6.9 Products 77 and For a = 0 this is well-known (SPCs = BA , tx while if E HPCS and IAl w , then /A\W JA\ by S. Koppelberg a = [Ko]). For 0 < (i < W let be the full GSa with base W, and let 18 be the subalgebra of u.m generated by w zero-dimensional elements. Suppose 18 E HPCs ; say P ;;" 18 each a non- a iEI i r, i trivial Cs By the above result of Koppelberg, since Izdllli = w we a must have \1\ < w Since each is simple by 1.5.2 (iii), it follows that 18 ';:; P for some J f;; I. But has only finitely many 0 - dimensional elements, a contradiction. Now suppose (i w. For each E w 2 let be the minimal Cs with base to , let 18 = P III a and let be the minimal sub- algebra of 18. Then 'Dl fI. HPCS For, assume that hE Ho(PiEIG:i,'Dl), a where E CSa for all i E I. Let = For each E w 2 let a be the term z. Thus a 1 holds in a Cs G: iff G: has base of cardinality • a It follows that f 0 for all E w N 2 , so for all E w 2 there is an i E I such that G:i has base of power z, • Now define ) f E PiEICi by defining fi = 1 if the base of has ele- ) ments for some fi = 0 otherwise. Thus a (':tl) s f if to t. E w ..... 2 is even, and a(.D)· f = 0 if t. E w ..... 2 is odd. Hence and 'hf = 0 in these two cases. But = 0 , , so by 2.1.17 (iii), hf E Sg(18(''Dl){a('ffi): z,Ew ..... 2} , t: which is clearly impossible. (6) From (4) it follows, of course, that PCsO'rt ICs even for a 78 Products 1.6.9 But we show in 1.7.21 that a product of Cs 's with infinite 01 bases is isomorphic to a Cs for 01 r e g (7) and Nemeti have noted that Cs rt SPDc and Ws rt SPDc 01 01 01 O! for all 01 > 0 In fact, let III be tre full Cs with base K 2 , 01 and let m be the subalgebra of III generated by the atoms of III Then by Cor. 1.4 of [AN3] , But the statement (*) for all x, if c'-' x o for all A < 01 then x o A holds in every De , and hence in every (S: E SPDc Since every atom 01 O! x of falsifies (*) , we have III I/:. SPDc The case of Ws 1 S is 01 01 similar: any full Ws has an atom. O! (8) In connection with (7) we should mention the following general fact about Lf 's and DC's not found in [HMT]: SPDc rt SPLf O! O! O! 01 for O! IJ) In fact, write O! = rub. with r n b. = 0 and \r\,\6\ IJ). Then the statement (**) for all x, if o for all AE r then x o holds in every Lf hence in every SPLf but fails in some Dc O! 01 01 ( 9 ) From 1.6.13 it follows that PWs IWs • 01 01 (10) Among the questions about Figure 1.6.9 which are open the most important seems to be whether ICsreg HPWs 01 01 (11) If we restrict ourselves to 01 IJ) and to set algebras with bases and subbases infinite, 1.6.9 simplifies as in Figure 1.6.10, where we use the notation of 1.5.7 (17) • 1.6.11 Products 79 HSP Gws 00 ct Figure 1.6.10 Again =? means that equality of the classes in question is not known. Some of the theorems needed to check this figure are in [AN3]G.2. Now we discuss direct indecomposability, subdirect indecomposa- bility, and weak subdirect indecomposability. We give some simple results about these notions and then we discuss some examples and problems. Theorem 1.6.11. Every full Ws is subdirectly indecomposable. a Proof. Let be a full Ws ,with unit element ctU(p) • Given tx o yEA ,choose fEy Then there is a finite r r:t. such that (ct- 01 f = (a-r)1 p. Thus (p} c(f)Y So is subdirectly indecomposable by 2.4.44. Corollary 1.6.12. Any subdirect1y indecomposab}e Cs is isomor- ct phic to a Ws Q' 80 Products 1.6.13 Proof. By 1.1.11 and 1.6.4. Corollary 1.6.13. Every Ws is weakly subdirectly indecomposable. 01 Proof. By 0.3.58 (ii), 2.4.47 (i), and 1.6.11. Corollary 1.6.14. Let 'll E IGws Then the following two condi- 01 tions are equivalent: (i) 'll E IWsa' ; (ii) 'll m for some subdirectly indecomposable mE IGws 01 Proof. (i) implies (ii) by 1.6.11. (ii) implies (i) by 1.6.2. Corollary 1.6.15. Any regular Cs with non-empty base is directly 01 indecomposable. Proof. By 1.4.3 and 2.4.14. Remarks 1.6.16. Throughout these remarks let 01 W • (1) Examples (I) and (II) in 2.4.50 are Ws 's which are res- 01 pectively subdirectly indecomposable but not simple, and weakly sub- directly indecomposable but not subdirectly indecomposable. (2) To supplement our discussion of homomorphisms we shall now show that for any K 2 there is a Ws with base K having a homo- a morphic image not isomorphic to a Ws The first such example was a' due to Monk; the present simpler example is due to and Let p (0 :K < 01) and let 'll be the full Ws with unit element 01 OI (p) • K Let x = {f E V: wl f<: p or the greatest K fK f 0 is even} • Let 1.6.13 Products 81 We claim that > 2 , hence mil Ws by 1.6.13 and 0.3.58. It In fact, clearly xiI E , so it suffices to show that x,-x I If I' c wand I' KE w , then c (Ox. -x {f EV: (w.... K) 1 f 0= (w"""K) 1 p) Thus for every y E I there is a K < w such that for all fEy we have (w"'K) 1 f = (w",K) 1 p Hence, x,-x I• (3) In contrast to the situation for homomorphic images (see 1.5.6(5)), we show in 1.7.17 that any direct factor of a Cs is ex isomorphic to a Cs Note that a CA is a direct factor of some ex ex Cs iff is isomorphic to a compressed Gws (cf , 1.2.6). ex ex (4) The full Cs with base of any cardinality 2 is directly ex decomposabIe. r eg (5) A Cs which is directly indecomposable (by 1.6.15) but ex not weakly subdirect1y indecomposable can be obtained by modifying Example (III) of 2.4.50. Namely, let p = (0: K < ex) , q = (1 : K < ex) , and let be the Cs of subsets of ex2 generated by {{p},{q)) . ex By Cor 1.4 of [AN3], is regular, and it is clearly not weakly subdirect1y indecomposable. (6) Example (I) in 2.4.50 can be similarly modified to yield r eg a Cs which is subdirect1y indecomposable but not simple: is ex the Cs of subsets of ex2 generated by {{p}} , with p as above. ex reg Clearly is not simple, while E Cs by Cor. 1.4 of [AN3] . ex Now let a 'f x E A By 1.5.1 write x = y ffi d ,where y E and d E sg(m){O} • Clearly then there is a finite f ex such that {p} c(f)x. Hence is subdirect1y indecomposable. 82 Products 1.6.13 (7) Example (II) in 2.4.50 can be modified to yield a Cs 01 which is weakly subdirectly indecomposable but not subdirectly indecom- posable. This was noted by and who also constructed r eg a Cs with these properties. To construct such a Cs ,let be 01 01 the full CS with base 2 . Let (f :K < w) be a system of pair- OI K wise disjoint infinite subsets of 01. Let p = (0 :K < 01) • and for each K < 01 let Let I Ig(\8){X : K < w} . Thus by K 1.5.1 we have (*) B={x$d:xEI and dESg(\8){O}} Now \8 is not subdirectly indecomposable, by the same argument as in 2.4.50 (II). Now suppose that 0 t y E B ; by (*) write y = x $ d with x E I and d E Sg OO){O} Then we can choose KE wand a finite 0,;;; 01 such that x c UA P E c(@)y for some finite @,;;; 01. Since this is then true for every non-zero y E B , it then easily follows from 2.4.46 that \8 is weakly subdirectly indecomposable. Since y to, we have two possibilities. Case 1. x· -d to. For any f Ex. -d we have f E 0I2 (p) , and hence p E c(@)(X.-d) for some finite @ c 01 • Case 2. d· -x '!' 0 Fix fEd· -x. For all A < K choose E fA (0 U Ad) Let \l! {IJoA :A < K} , h = [(M)-- 1 f] U [(01"" - 1 p] k = [(01 \l! ) 1 h] U (1 : \I E \l! > Then h E d since d is regular, kEd similarly, k x ,but k E 0I2( p) • Hence the desired conclusion follows 1.6.13 Products 83 as in Case 1. (8) Andreka and Nemeti have shown that the converse of Theorem 1.6.15 fails: there is a directly indecomposable Cs not isomorphic 01 reg to a Cs • Let K 2 be arbitrary, and let 18 be the full Cs 01 01 with base K. Choose r 01 with 0 E r , and both r, OI .... r infinite. Let p = (0 : K < 01) Choose qO,q1 E OIK so that r 1 qo = r 1 q1 = r 1 p and q '\(ql) For each z, < 2 let x = (f E OIK(qz, ) .r : f = r1 p} 0 t. Let _ «(S:)( } We claim that is the desired algebra. - 6f y,xO,x1 To check that is directly indecomposable, we need an auxiliary con- struction. Let 18 = Thus 18 is a regular Cs ,by Theorem 01 1.4.1. Now let C (a E A: a or -a is for some z, < 2 and some finite 8 c Q'} • Note that C is closed under Let D = (IT /\I'az, : K < ci, a E K(CUB)j and E-( '\ a: A < w, a E AD} • Now we claim: LZ, (*) A = E• To prove (*) , since y,x E , it suffices to show that EE sum • O,x1 E Clearly E is closed under , and for all K,,, IT a=rr d·b, IJ. (cdn d). c z (cdn d)· c b (IT cdd). c b K 1J. and hence Thus we conclude d (**) (c n d). c z E E • K 1J. On the other hand, _cdrr d·c z =, [c (-d) .cKZ] , K 1J. d Thus - cKTI d· c z E E, so by (**), cKz EE Hence we have 1J. Z E {O,l} by 1.6.15. By (*) we can write Z=, d.b +e+g, L \I where 0 and g are finite subsets of a. Choose distinct E r "" (U """lib U U.;'Vel U (l U S) • \I ....- \I \I ....." \I Note that for each \I < K• Hence z cz=' b +ce+cg L \1« \I iJ. iJ. as desired. r eg It remains to show that ICs Suppose F E Is (91,G:) a . where G! is a Cs with base U Let y' = Fy and x' = Fx for a . z, z, 0 z, < 2 Since cel 1 and Y• slY' -dOl = 0 , similar equations hold for s', and so there is a u E U such that fO u for all f E r' Also, X d for all K E r ,anq X s y ; it follows that fK = u o OK o for all K E r , for all f E Xo Similarly for xl' Since and Xl • Xl = 0 , cannot be regular. o 1 (9) It follows from 1.7.13 that any subdirectly indecomposable Cs is isomorphic to a regular Cs a a (10) The following problems about these notions remain open: (a) Is every weakly subdirectly indecomposable Cs isomorphic to a a regular Cs ? a (b) Is every weakly subdirectly indecomposable Gws (or a isomorphic to a Ws ? a 86 Ultraproducts 1.7.1 7. We shall begin with a general lemma (due to Monk) about ultra- products of Crs 's • To formulate this lemma it is convenient to intro- at duce some special notation. Definition 1.7.1. Let F be an ultrafilter on a set I, U (U i E I) a system of sets, and at an ordinal. i: (i) By an (F ,U,at) - choice function we mean a function c mapping such that for all Ie < at and all + (ii) If c is an (F ,U,at) - choice function, then we define c mapping at(P - into P exU by setting, for all i E1Ui/F) i E1 i q Eex (P - and all i E I i E1Ui/F) + (c q)l.' = (c(K,q ). :K < at) • K 1. (iii) Let A = (Ai : i E I) be a sys tem of sets such that A Sb(atU•) for all i E I , and let c be an (F,U ,at) - choice func- i 1. t Lon , Then there is a unique function Rep(F,U,at,A,c) (usually abbrevi- ated by omitting one or more of the five arguments) mapping PiEIA/F into such that, for any a E P ' i E1Ai Rep (a/F) Lemma 1.7.2. Let F be an ultrafilter on a set I, U=(U.:iE1) 1. a system of non-empty sets, and at an ordinal. Let c be an (F,U,at)- I choice function. Further, let u E Crs ' where each !lli has base U at i 1.7.2 Ultraproducts 87 and unit element Vi ' and set V = (Vi: i E I) • Then Rep(c) is a homomorphism from PiE1'JJ/F into a ers et Furthermore, for every o ¥ a/F E PiE1Ai/F , if ZE F, s E PiE1Vi ' si E a for all i E Z and w;: «s.1< : i E I) : I< < a) it follows i that if c' is any (F ,U ,et) - choice function such that c' (I< ,wI< iF) ;: wK for all K < Q' , then Rep(c') (a/F) ¥ O. Proof. Let f ;: Rep(c) , X = PiEIU/F T f(V/F) Clearly f preserves +. Now let K,A. < et ; we show that f preserves d KA. Note that fd k; T since f preserves + • Now let q E T Then I< A. + {i EI: (c q). E V.} EF , and so + [v.l iff {iEI:(cq).ED L}EF iff q E fdKA. KA. + = (c q).\} EF iff (i EI: c(K,qK). ;: c(K,qA.).} EF iff L iff q E D[T) KA. Now let a E PiE1A • We show that f preserves Clearly i f(-a/F) ,; T'"" f(a/F) Now let q E T'"" f(aii) Thus (i EI: (c+q)i E Vi} E F and (i EI: (c+q)i E a i} F, i.e. , (i E I: (c+q). E v. '"" ail EF Therefore q E f(-aiF) Next, let also K < Q' ; we show that f preserves c K • First + [Vi] suppose that q E f(cKa/F) Let M;: (i EI: (c q\ EC ail thus K K ME F. Then there is an s E PiE1U such that [(c+q)i)si E a for i i all i EM. We show that qK E f(a/F). Let Z = s/F (i E I: s L c(K,s/F)1} Then ZEF since c(K,s/F) E s/F For any i E Z n M we have 88 Ultraproducts 1. 7 ,3 + I( I< (c q -)i (c(i\,(q h).: A. < a) s/F siF + I< I< [(cq),l [(cq).l,Ea.+ c(l<:,siF)i I< - Thus, indeed, q _ E f(a!F) Hence q E crT] f(a/F) Second, suppose s!F K that q E C[Tl f(a/F) Thus q E T and (a {K}) 1 q for K some p E f(a/F) . Let M E I : (c+p\ E a ; thus ME F. Also = ji i} + since q ET the set Z = {i E I:(cq)iEVi} is in F. Now let + + + iEMnz. Then (c q)i E Vi and (a"" (K}) 1 (c q), >;; (c p). E a, , + eVil proving that (c E c a Thus q E f(CKa!F) ,since M n Z EF q\ K i We have now verified that f is a homomorphism. For the second part of the conclusion of the lemma, assume its additional hypotheses. Let q = (WK/F:I< < a) and fl = Rep(c') • We show that q E f(a!F) (hence f(a!F) 0 , as desired). In fact, for any i E Z we have «wK), :K < a) So q E f(a!F) , as desired • Next, we derive some specialized versions of Lemma 1.7.2 which are more easily applicable; they are due to and Nemeti. 1.7.3. Assume the hypotheses of Lemma 1.7.2. Also suppose that F is \a\+ -complete (which holds, e.g., if Ci < w). Then (i) for any (F,U,a) - choice function c' we have Rep(c) = Rep(c') ; (ii) Rep(c) is an isomor phism; (iii) For any a E PiEIAi we have 1.7.4 Ultraproducts 89 (Rep(c) HalF) that qK = WK!F for all K < a and {i E I : pJ. 0 w Ea.} E F} 1 1 Proof. It suffices to prove (iii), since the fact that c is arbitrary then implies (i), and thus (ii) follows from Lemma 1.7.2. a - So, let a E PiEIA i and q E (PiEIUi!F). We want to show that the following two conditions are equivalent: (1) {i E I: (c+q). Ea.} EF 1 1 (2) there is awE a(PiEIU such that qK = wK!F for all K < 01 i) and (i E I : pJ. 0 w Ea.} EF• 1 1 If (1) holds, we let w = suppose that (2) holds, and let H = (i E I: pj. 0 w Ea.} • Then for 1 1 any K < a ,both wK and c(K,qK) are members of qK , and hence the set ZK = (i E I : (wK)i = C(K,qK\} is in F• Therefore the set Y = H n Z is also in F. For any i EY we have nK (wK). : K < < 1 a> and hence (1) holds. Lemma 1.7.4. Assume the hypotheses of Lemma 1.7.2. Let Rep(c)*- Then: (i) EI K implies E K for K Gws or K = Cs in the Q' Q' latter case, has base PiEIUi!F. (ii) If in addition F is \a\+ - complete, then III EI K implies EK for K = Gs or K = Ws a Q' 90 Ultraproducts 1, 7.4 R!££!. Let f = Rep(c). First suppose E l Cs Thus V. Q' 1 Q'U. for each i E 1 We need to prove that f(V/F) = Q'x , where 1 x PiEIUi/F. Let q E Q'X. Then (c+q)i E Q'U Vi far all i E 1 , = i = so q E f(V/F) • The converse being obvious, we thus have E Cs m Q' with base X. Next, suppose that E l ews • Then for each i E 1 we can write Q' V. = : j E J.} , 1 1J 1 where n = 0 whenever j k • Let 1J 1 Now for each j E PiEIJi we set W' = (q EQ'x :{ i E I : (c+q). E S. ..}E F}, J 1 1,J1 Now we claim For, let q E f(V/F). Let M = (i E 1: (c+q). E V.} so that MEF 1 1 + i E M Thus Choose j E Pi EIJi so that (c q)i E Si,ji for all q E W , as desired. Clearly each Wi f(V/F) , so (1) holds. j For assume the hypothesis of (2) and let q E W.. Let Z = J (i E 1 : (c+q)i E Si,ji} and let H = (i E 1 : (c+q)i E Si,ki}. Then Z n H (i E 1 : ji = ki} F ,while Z E F , so H F. Thus q l Wk ' as desired. 1.7.4 Ultraproducts 91 (3) For any we have W. U O'Q • qEW J J j For, first let q E w.. Let K < 0' Now (i EI: (c+q) . E s ...} EF J and hence (i EI: c(K,qK\ E Yi,jJ EF, and further, since c(K,qK) E qK, qK EQ.• Thus in (3) holds. For the other direc- J tion, it suffices to take q E W K < 0' , u E Q. , and show that j, J qK EW.• Let M = (i E I: (c+q). E s...} and Z = fi EI: c(K,u). u J E Y. ..} Then M E F since q E W. and ZEF since and J c(K,u) E u So M n ZE F. Let i E M n Z. Then (c+qK). u (c(r..,qr..)i : r.. < + K [(c q)ilC(K,U)i E Si,ji Thus qK E W. , as desired. u J Now (1), (2), (3) inunediately yield that 18 E Gws ,upon noting 0' that W = W if j/F = kIF. j k + Now we turn to the proof of (ii). So, assume that F is \ 0'\ - complete. Let EI Gs Since Gs Gws , we can assume the above 0' 0' 0' notation, where in addition for each i E I and j,k E J we have i Y Y or Y Y 0 (that is, in the terminology of 1.2.6, i j = i k i j n i k = is a normal GwsQl). We claim Clearly In fact, assume y E PiE1Yi,ji Let H=(iEI:y. EYi ,k.} H {i EI: ji = ki} F , so H F and hence y Qk. So (4) holds. 92 Ultraproducts 1. 7.4 By (1), (2), (3) and (4) it now suffices to show (5) For any OiQ • f(V IF) J a Oi To prove (5) , let q E Q Say and qK = yK!F j y E (Pl.. ElY' l.,J'i) for all K a Then for all i E I we have pji 0 y < a , l. l. Hence by Lemma 1.7.3 we conclude that q E f (Vif!') . It remains to check (ii) for K = Ws So, we suppose that a I E Ws Since Ws ,;; Gws ,we still have the above situation, with Oi Oi Oi each J a singleton {t ; we write Y for Y, t' , for i i} i i., l. Pi,ti' Wand Q for W. and Q. where j is the unique member of JJ P J • Thus by (1) f(V!F) W , and by (3) , i E1 i = Let r = «p.K : i E 1)!F :K < Oi) . Thus r E OiX Now we claim: l. (6) For any q E OiX, q E W iff {K < Oi : qK 1 rK} finite. (Hence W = OiQ(r) , as desired.) To prove (6), let q E OiX • Choose y E 0i(P Y ) so that qK YK!F for all K < Oi i E1 i = Let G = {K < Oi : qK 1 rK} • For each K < ix let ZK Hence the set is in F (by \ Oil + - completeness). Now for all i E Z we have {K = pl.,K} = Oi G. Let M = {i E I: c(K,qK), = y i} Thus '" l. K • MEF also. Now if q E W , then the set N = {i EI: (c+q\ E is in F. For i E M n N n Z we have, for all K < Oi, (c+q)iK = c(K,qK)i = YK i and hence (by the definition of N), G = {K : YK i 1 PiK} is finite. 1. 7.5 Ultraproducts 93 On the other hand, if G is finite then for any i E M n Z and any KE a_ G we have (c+q).K = c(K,qK). = YKi = P.K and so, since M n Z E F, {i EI: (c+q). EEF and q E W. This completes the proof. Lemma 1.7.5. Assume the hypotheses of Lemma 1.7.2. Let K = Ws a or GSa Then for every non-zero a E P. lA./F there is an (F,U,a)- choice function c' such that Rep(c')a t 0 and + Proof. If F is \a\ - complete, the desired conclusion follows from Lemma 1.7.3 and Lemma 1.7.4 (ii). So we assume henceforth that F + is not \a\ - complete. In particular, a w. We let sand w be as in the last part of the proof of Lemma 1.7.2. Let X = P i E1Ui/F Assume now E Let N = {i E I: IUil > I} . Then it is easily seen that there is an (F ,U,a) - choice function c' s.vtLs fyd.ng the following two conditions: (1) c ' (K ,wK/F) = wK for all K < a (2) c'(K,y)i t wKi whenever K < a, y EX, y t WK/F ,and i EN. Again let f = Rep(c') • By Lemma 1.7.2, fa to. Now let q = If N rt. F ,then [x] = 1 and hence 0 t f(V/F) (;; aX = {q} ax(q) , so f(V/F) = OIX(q) Assume that NE F. Let p E OIX , i EN, and K < a. Then + (c' p) K = sKiff y/F = w /F , and hence (C'+P)i E Vi iff i i K p E OIX(q). Since NEF it follows that p E f(V/F) iff p E OIX(q) , as desired. 94 Ultraproducts 1. 7.6 1 + Next, assume WE GSa Since F is not Ia\ - complete, there l)] is an h E 1a such that [1!(h\h- n F o (see, e.g., Chang-Keisler [CK], p. 180). Since E 1GS for each i E 1 there is a subbase m a Now there clearly is an (F,U,a)- choice function c' satisfying the following three conditions: (3) c'(hi,y!F). E Y for all i I and all y EX; i E (4) for all and all c'(K,y!F) E PiETYi K Let Again let f = Rep(c') By Lemma. 1.7.2, fa f- 0 Now we sha11 show that f (V!F) = a W By (4) we have aWs;; f(V!F) Now let q E aX such that qK W for some K < a. Thus the set H (i I: C(K,qK)i Y F If i EH, then by (3) we have = E i i) E C(K,q ). '1- Y. hence K Since HE F, it follows that q i f(V!F). This completes the proof. Our final version of 1.7.2 concerns regularity. 1.7.6. Assume the hypotheses of Lemma 1.7.2. Let f Rep(c) Also suppose a PiEIA and for each i E I a is regular E i i in m and s;; • Then f(aiF) is regular. i ... -- Proof. We assume all the hypotheses. Let r = 1 U tf(a!F) , and assume that p E f(aiF), q E f(V!F) , and r 1 p = r 1 q. We want to show that q E f(a!F) Let + + H = {i E I: (c p ) i E a i and (c q)i E Vi} • 1.7.7 Ultraproducts 95 + Thus H E F. By the definition of c ,for all i E I we have + + r 1 (c =f1 (c q). Thus for any i E H , the regularity of a p\ 1 i + implies that (c q)i E a Since HE F, it follows that q E f(a/F) i Remarks 1.7.7. The above lemmas are algebraic forms of the lemma for ultraproducts. The exact relationships between them and lemma will be discussed in a later article. and Nemeti have shown that various hypotheses in 1.7.2-1.7.6 are essential, and that these lemmas do not generalize to arbitrary reduced products. Now we shall use the above lemmas to prove various closure prop- erties of our classes of set algebras. Theorem 1.7.8. UpK IK for KE {Gws ,Gs } a (:i Proof. By Lemmas 1.7.2, 1.7.4 (i), and 1.7.5, each member of UpK is a subdirect product of memebers of K If a 'f 1 , then IK = SPK ; if ex = 1 ,then by 1.7.3 and 1.7.4 we have UpK = IK the proof is complete. Theorem 1.7.9. UpCs = ICS for ex < w • ex a Proof. By Lemmas 1.7.3 and 1.7.4 (ii) • Theorem 1.7.10. UL E K whenever L is a non-empty subset of K directed by , for K E {IGws ,IGs }U {ICs : ex < w} U ex ex ex {I(Ws n Lf ):ex an ordinal} • a a Proof. This is immediate from 1.7.8 and 1.7.9, since SK = K, for all choices of K except the last. Let K = Ws Lf ,and let exn ex 96 Ultraproducts 1.7.11 L be as indicated. Then UL E Gws by what was already proved, while a UL is simple by 1.5.2 (ii), 2.4.43, and 0.3.51. Hence UL E IWs by a 1.6.4. The following generalization of a part of Theorem 1.7.10 is due to and [ANI]: r eg Theorem 1.7.11, If 0 f L ICs Lf is a set directed by an a r eg then UL E ICs an Lf a Proof. By Theorem 1.7.10 we may assume that a w. For each E L let be an isomorphism onto a regular CSa with base ; further, let = E Let F be an ultrafilter on L such that E F for each E L There is an isomorphism g of onto P such tba t g(b!F) = : IS E L)/F for all IS EL ll.\a/F b E PlRELB Let c be an (F ,U,a) - choice function, and let f = Rep(c) By Lemmas 1.7.2 and 1.7.4 (i), f is a homomorphism from Now for each b E let h 'b = (b : b E B, IS E L) U (0 (l8) : b B E L), and let hb = h'b/F. Then h is an isomorphism of UL into m/F (see the proof of 0.3.71). Now let = (f 0 g 0 h)* UL re g We claim that fog 0 h is an isomorphism, and IS: E Cs Lf an a First note that each member of L is simple, by 1.5.2 (i); hence UL is simple, by 2.3.16 (ii). Thus fog 0 h is an isomorphism. By 2.1.13, IS: E Lf To show that IS: is regular we apply 1.7.6. Let a b E Then for each E L, (h 'b)m is regular in , and 1. 7.12 Ultraproducts 97 (h = J;,b since fog 0 h is an isomorphism. Therefore by 1.7.6, fghb is regular, as desired. The following lemma is due to and 1.7.12. Let be a Crs with base U and unit element 01 v , and let F be an ultrafilter on a set I Let c be an (F,(U: i E 1),01) - choice function, and let f Rep(F,(U: i E 1),01, : i E I) ,c). Define 6 E A(IA/F) and e: E U(IU/F) by 6 «a: i E I)/F : a E A) e: «u : i E I)IF : uE U) We assume also that for every rEV there is a Z E F such that, for all K < 01, c(K,e:(rK»;l (rK : i E Z). Finally, let g = f 00 and = Then (i) g is an isomorphism from onto (ii) is sub-isomorphic to (recall the definition of E from 1.3.5, and subisomorphism from 1.3.15); (iii) o g (recall the definition of r-t, from 1.6.1). Proof. Clearly g is a homomorphism from onto By 1.3.1, is an isomorphism from onto , so if we establish (iii), then (i) and (ii) will follow. To establish (iii), let a E A ; we want to show (1) ga n 'Ell (e: 0 s : sEa} . 98 Ultraproducts 1.7.13 First suppose q E ga n7v Since q E 7v , there is an s E V such that q = 0 s By the hypothesis of the lemma choose ZEF such that c (I( ,( sK: )) (Sl( : i E Z) for all I( < 0' • Let H (iEI:(c+q).Ea} ; thus HE F . Since H n Z EF , we can choose i E H n Z . Then s = (sIC: K: < 0') = : IC < 0') + (c(IC,qK:)i : II: < 0') (c q\ Ea. Thus q = E 0 s implies that q is in the right side of (1). Second suppose q = 0 s "With sEa. Since a l;; V , we have q Again by the hypothesis of the lemma let Z E F be such that + for all IC < Ci • Then (c q)i = sEa for all i EZ ,so q E ga n 7.v r eg We now use this lemma to establish that Ws ICs , also due 0' 0' to Andreka and Nemeti. It generalizes the fact that WSO' ICs ' a established in Henkin, Monk [HM] • r eg Theorem 1.7.13. Ws ICs 0' 0' Let be a Ws with unit element V = O'u(p) • Let 0' \1\ \0'1 Uw (in a later proof we shall choose I in a special way; for now we could take I = a). Let F be a \1\ - regular ultrafilter over I (for the notion of K: - regular ultrafilter see Chang, Keisler [ CK], p. 201). Then, as is easily seen, there is a function h E I!:f a : Ir\ < w} such that !: i E I : II: E hi} EF for all K: Now let 6 and E be as in 1.7.12, and set X = IuiF Let c be an (F, (U : i E I) ,0') - choice function sat isfying the following condition: 1.7.14 99 (1) For all K < i E I , and all y EX, if K l hi then C(K,y)i = pK ; if K E hi and y = EU with u E U then c(K'Y)i u • Let f = Rep(c) We shall show that f 015 is the desired isomorphism. By 1.7.2, is a homomorphism onto a Crs 18 • Now we show that a fl)V = Q'x so that 18 is a Cs Since fl)V trivially, we ix show the other inclusion. Let q E It suffices to show that + + (c q)i E V for all i E I So, let i E I Note that (c q\ E + If K l hi , then by (1) we have (c q)iK = C(K,qK)i = pK . Since + hi is finite, it follows tba t (c q)i E Q'U(p) , as desired. To show that f 0 I) is an isomorphism we apply 1.7.12. We have only one hypothesis of 1.7.12 left to check. Let rEV Then there is a finite r such that r) 1 r = r) 1 p Let Z £i EI: r c hi} By the choice of h we have Z E F Let K < and i E Z ; we show that c(K,ErK)i rK (as desired). If KE r then KE hi and so c(K,ErK)i = rK by (1). If K l r then rK = pK and c(K,ErK)i = rK by either clause of (1) Thus tre hypotheses of 1.7.12 hold, and hence f 0 I) is an isomorphism. It remains to show that 18 is regular; but since f 0 I) is an isomorphism, this is immediate from 1.7.6 and 1.1.16. re g Theorem 1.7.14. = IGwsQ' >;; SPCsQ' Proof. Using 1.1.10, 1.1.11, 1.1.18, 1.1.19, 1.6.4, 1.6.6, and 1. 7. 13 we have 100 Ultraproducts 1. 7 .15 r e g where K = Gs or Gws The theorem follows. at at The following result, HGwS IGs ' was known to the authors for a at at long time using representation theory. The present direct proof is due to and Theorem 1.7.15. HGws IGs at at Proof. The at < 2 is trivial, so assume at 2 Let E Gws ' and let L be an ideal of We want to show that at E IGS at Case 1. at < w. For each z E A let kz = -c(at)z. Note that kz is zero-dimensional and E GS Let ;a = : z E L) . at Let F be an ultrafilter over L such that Iv E L : v ;2: z} E F for all z E L Now define h mapping A into PZELBz/F by setting, far any a E A , ha (a. kz : z E L) IF Now (a ·kz: a E A) E ) for each z E L by 2.3.26, so z «a .kz: z E L):a E A) E by 0.3.6(ii). Hence by 0.3.61 we infer that h E . Now we claim that h * ; to show this it suffices to show that If a E L , then a. -c(at)v = 0 for all v;2: a ,so Iv E L: v ;2: a} (v E L : a . kv = O} ; since (v E L : v ;2: a} E F , it follows that ha O. On the other hand, if a E A L and z E L then a :f c(at)z and hence a • kz '! 0 ; thus ha '! 0 , as desired. Thus h* , so E SUpGs By 1. 7.8 , E IGs at at 1. 7 .15 Ultraproducts 101 2. a w. Using 1.6.3, it is enough to take any a E A - L and find a homomorphism h of onto a Cs such that a ha 1 0 and h* L = (O} • Let I Lx(rs;;a:\rl 6 r} EF for all (z,r) E I. Let have base U and unit ele- ment a (Pj) v U( Y. : j E J} , J where for distinct i,j E J. Let X = lufF Since a f L, there is a function r E IV such that r(z,r) Ea· for all (z,r) EI. Then there is a function such a (p .. ) E y .. forall i EI Next we let L J Q (kiF: k E PiEIYji} , s = «r.K:i E 1):K Let c be an (F,(U: i E I) .o)- choice function such that the following conditions hold. (1) If K (2) If K Let f Rep(F ,(U : i E I) ,a,(A : i E I) ,c) . Also let 5 be as in 1. 7 .12 Then we claim that f 0 /) is the desired homomorphism. First 102 Ultraproducts 1.7.16 we note tha t by the second part of 1. 7.2 and by (2) , we have f6a 1- 0 Next we show that (foO)*L = (O} Let z E L Set Z = (v,r) E I: v ;;>; z} thus ZE F Let q E aX We show, for any i E Z that + (c q\ z (thus q f6z , as desired). Let i E Z say i (v,r) where v z. By (3) we have c(K,qK)i = riK for all K E a - r ; + Since it follows that thus (a - r)1 (c q)i = (a - r)1 r i r i c(r)v + + (c q)i c(r)v and, since v;;>; z (c q) i z It remains only to show that (f 0 0)* is a Cs we show that a fOV = aQ Lff (4) For any qE ax an dany . E I , (c+q). EV L (+)C q iYjiE o., For, if q E aX and i E I , then with i = (z,r) we have by (3) + (a - r)1 (c q\ = (a-r)1ri An easy argument yields (4) + a Now suppose q E aQ By (1) , (c q\ E Yji for all i E I , + so by (4) (c q)i E V for all i E I and hence q E fOV On the + other hand, let q E fOV Then the set Z = (i E I : (c q)i E V} is + in F By (4) (c q)i E a y .. for each i E Z , i.e. , This C(K,qK)i E Yj i for all K < a and all i E Z Thus q E "o completes the proof. Theorem 1.7.16. For a > 1 we have 1Gs HSPGs = HSPGws a a a r eg HSPWs HSPCs HSPCs a a a Proof. HSPGwS = HGws 1Gs 1Gws using 1.7.15; the other a a a a parts of the theorem are easily established by using 1.7.13. The following result is due to Monk. 1.7.17 Ultraproducts 103 Theorem 1.7.17. Any direct factor of a is isomorphic to Csa a Cs a Proof. We may assume that et w . We use the notion of com- pre ssed Gws in 1.2.6. In fact, we first prove the fo l l owirg inde- a pendently interesting result noticed by and (1) ICs is isomorphic to a compressed GWs } in fact, any 0: a compressed Gws is subisomorphic to a Cs a a The theorem follows immediately from (1), using 2.4.8, since any zero-dimensional element of a compressed Gws is clearly the unit a element of a compressed Gws Now let be a compressed Gws C( C( Say the unit element of is V and its base is U. Let F be a let\ -regular ultrafilter on et Let E and 6 be as in 1.7.12. Let X = etui"F Let c be an (F, (u : 'A < a) ,a) - choice function such that (2) for all K < a and all u E u, C(K,EU) Let f = Rep(F,(U: f... < a),Oi,(A: f... < a),c) . Then the hypotheses of 1.7.12 are met, and hence f °6 is an isomorphism. Now by I.7.4(i), * Actually if we examine the proof of I.7.4(i) we C( see that = Q for all j,k E in its with I = a Qj k PiEIJi and hence (f06)a1* too. Also recall from is a compressed Gwset = that proof that the base of (f 0 6)* is X Let W etx ,.. f6v If W = 0 we are finished, so assume that W 1 O. Note that W it- self is the unit element of some Gws Now we claim a 104 Ultraproducts 1. 7.18 (3) 21 '" for some Gws with unit element W. m O! In fact, applying 2.3.26(ii) to the full Gws with unit element V O! and then restricting to 21 we find that 21 '" for some Ws with O! base U. By 1.7.13, is isomorphic to a Cs looking at the O! proof of 1.7.13 we see that we may assume that the base of is X Since W O!x is zero-dimensional, we get a Gws m with unit element O! W such that '" m ,as desired in (3) ; let g be a homomorphism from 21 onto m . By 0.3.6(ii), 21 is isomorphic to a subalgebra of mx (f 00) *21 and by 1.6.2 mx (f 00)*21 is isomorphic to a Cs 21' in fact, the O! function h = (ga U foa : a E A) is an isomorphism from 91 into It is easily checked that ha n 7v = Ea for all a E A ,so (1) holds. We shall establish a few more results about closure properties and relationships between our classes of set algebras in 1.7.28-1.7.30, after discussing again change of base. First we make some remarks about the results already established in this section. Remarks 1.7.18. Many of the results above cannot be improved in the obvious ways; we now make several specific arguments to this effect. (1) If O! ;;, w and is a Cs with base U such that O! 2 S; [u] (see, e.g., Chang, Keisler [CK] p. 202). Now'll has characteristic and hence so does . But any Cs of characteristic \ui , ex Ivl has base of cardinality IU\ ' and hence has at most elements. I- Thus 'll/F ICs . This same remark and proof apply to Ws 's a a (2) For any a W, any non-discrete 'll E WS ' and any non- a prinicpal ultrafilter F on W we have w'll/F IWs For, let a x = (dK,K+l :K < w) . Then 0 < x/F < 1 and O. By 1.6.13, w'll/F f IWs ct Now we again discuss change of base (see section 1.3), using ultraproducts. First we prove a sharper form of part of 1.7.13, due to And reka a nd ti, Theorem 1.7.19. Assume 0':<:2. Let 'll bea Ws with infinite ct base V. Let Y be a cardinal such that IA\ . \u\ !" f1<1e where Ie is the least infinite cardinal such that \6x\ < Ie for all r eg x EA Then 'll is sub-isomorphic to a Cs with base of power a Proof. Let 'll have unit element aV(p) Let I = rnax(a,y) , and let F be a \ II - regular ultrafilter on I Introducing the notation in the proof of 1.7.13, we see from that proof tha t f 06 r e g ,.,,* is an isomorphism of 'll onto a Cs and E 'll is sub-isomorphic 01 to Also note that has base X. By Proposition 4.3.7 of Hence we may apply I.3.18(iv) to get a subset W of X such that E *U I;;; w, IW\ y , and such that r e g is ext-isomorphic to a Cs with base W Thus E 'll is sub- ex - isomorphic to Choose a set WI U and a one-one function E -1 -* from WI onto W such that E I;;; E Let 19 = (E I ) (c f , 1.3.5). 106 Ultraproducts 1.7.Z0 Thus E by 1.3.1 and 1.3.7, and it is easily checked that m is sub-isomorphic to Using this theorem we can prove one of the basic results about set algebras, due to Henkin and Monk. First we formally give a defini- tion used in 1.5.6. Definition 1.7.20. For K E rcs ,Gs ,Gws ,Ws } we denote by Ci Ci Ci Ci the class of all mE K having all subbases infinite. Theorem 1.7.21. For Ci w we have m be a ' say with unit element UiEIV i ' where and U is infinite for all i E I , and V.nV.=o i 1. J For each a EA let 18 be the full Ws with for all a E A - {OJ • a 0: m unit element set h (recall 1.6.1). Thus Via' and a rtV. 1.a h E Hom(m,18 for all a E A, and h a if Let a a) a " 0 a " 0 K=UiEI\Uilu \A \ u \0:\ Now let X = 0:(2K) (za ) , and let X = O:(ZK) -U {Xa:a E A"" {OJ} For a o every a E A let ill be the full Crso: with unit element X and a a set ga = rt 0 ja 0 h Thus ga E Hom(m,iJ for all a E A , and xa a a) gaa "0 for a" O. Hence m \ • By 1.6.2 we have 1.7.22 Ultraproducts 107 Remarks 1.7.22. By 1.7.16 and 1.7.21, I Gs is an algebraically 0> a closed class for a w Thus for many purposes, cylindric set algebras with infinite bases are the simplest GA's with which to isomorphically represent abstract GA's and have improved 1.7.21 by showing that any Gws is sub-isomorphic to a G.g 0> a a Before turning to results concerning change of base by increasing the cardinality of the base, algebraic analogs of the upward Skolem-Tarski theorem, we give an example supplementing those of section 1.3. The example, due to and shows that if 1 < a < wand K < w , then there is a GSa ij with base Ii: such that if ij 58 E CS a then the base of 58 has power Ii: (whether or not Ii: < a ); because of this example we assume in most of our theorems that the base is infinite. For each A < Ii: let a :: {(A: < a)} • Let ij be the full Cs with A 0' base Ii: Thus aO, ••• ,a are the distinct atoms Now suppose K_ l ij E CS ' say f E Is(ij,58) • Then faO, •.• ,fali:_l are distinct atoms a so the base U of 58 has at least Ii: elements. Now suppose dBa , > Ii: Since ::IA«fa ' it follows that jfa > 1 for some \ul A A\ A < Ii: • Now (c a a rI c(QNl)a = 0 , so (cOfa .... faA) n O A A) A A o , contradiction. The following general lemma and theorem about increasing a base are due to AndrekaI and Nemeti;I their parts dealing with WS and Csreg0' a are due to Henkin and Monk. 1.7.23. Let be a Gws with base U and unit e1e ment a V. Let F be a \0'\ - regular ultrafilter on some set I. Then there is an (F, (U : i E I) ,0') - choice function c such that, letting 108 Ultraproducts 1.7.23 f = Rep(F,(U: i E I),a,(A: i E I),c) and letting 6 and E: be as in 1.7.12, we have: (i) f 0 0 is an isomorphism from onto (f 06)* ,.Jk (ii) E: is sub-isomorphic to (f 06)* ; (iii) -E = rt.... o f 0 0 ; E:V .... Ci (iv) E:V = (E *U) n f5v (v) the base of * is lufF ; (vi) EK implies (foO)* E K for all Proof. Assume the hypotheses. Say V = , where o n for distinct j,k E J Let Y = (Yj : j E J) and X = IuiF. Let Thus R is an equivalence relation on J Let K be a subset of J having exactly one elenent in common with each equivalence class under R. Let LI K have exactly one element in common with each equivalence class under F«K: iEI») ,with (j: i E I) EL for each j E K• Now we define functions w E X(I and v E X(IJ) Let Y EX. Choose k U) y = k E I K such that y n PiEIYki f 0 ,and k constant if y = E:U for some u E U. Let vy be the unique element of L n kiF. Thus y n PiE1Y(vy)i f 0 , so we can pick wy E y n PiEIY(vy)i with wy = (u : i E I) if y = EU. Now wand v have the following properties: (1) wy E y for all y EX; (2) WE:U = (u : i E I) for all u E U 1.7 .23 Ultraproducts 109 (3) for all y E X and i E I we have (wy)i E Y( ). ; vy 1 (4) if k,q E Rgv and {i E I :Y Y E F, then k q • ki = qi} Since F is \a\ - regular, choose h E I{r k a : \f\ < w} such that {i EI: K: E hi} E F for all K: < 0'. Now let c be an (F, (U : i E I) ,a) - choice function such that for all K: < 0', Y EX, and i E I , and y E* U , _{p (vy)iK: if K: hi (wy)i otherwise. Note that c(K:,y) E PiEIY(vy)i for all K: < a and y EX. Let f,6,E be as in the statement of the lemma.. Byr.7.12, (i), (ii) and (iii) hold. For (iv), use (2) and the definition of c. To prove (v), let Z be the base of (f 0 6)*flI ; we are to show that Z = X , and it is obvious that Z X. Suppose y E E* U ; say y = EU with u E U o in fact say u E Y. with j E J• Now let q = (EP.K: : K: < 0') Thus J oj (p.) y + q E O'x , and f or any i E I we have (c q)i = (p.) E O'y. J k V using J u J (2) and the definition of c It follows that q E f6V , and hence y E Z• Now let y EX"" E*U. Let q = (y : K: < 0') • Then for any i EI, (c+q)i = (c(K:,y\: K: < ci) = (0' "" hi)1 p(vy)i U «wy)i: K: E hi) (p(vy)i) c V E Y(vy)i - Hence again q E f6v and y E Z. So (v) holds. Now we turn to the parts of (vi) • The cases K = Gws and 0' K = Cs a re taken care of by 1.7.4. If flI is regular, then so is 0' (f 0 6) *flI by 1.7.6, since f 0 6 is an isomorphism. Next suppose 110 Ultraproducts 1.7.23 K =' Gs Ci For each r E Rgv we set Now (5) if r,r' E Rgv and r 1 r' , then Q Q , =' 0 • r n r E P Y d ri EI z.} EF• Z iEI r'i ' an t L Then [i E 1 : Y. =' Y I.} E F , so by (4) r = r' • r For, let q E fOV • Thus the set M = [i EI: (c+q). E V} is in F We claim that For each i E M choose ji E J so that Now for any i E M and K < Ci we have by the note following the defini- tion of So for any i EM and K < Ci we have c , so Yj i = Y(vqK)i • Thus for any K < 0' , C(K,qK)i E Y(vqK)i = Yj i = Y(vqO)i . qK E QvqO as desired. (7) If r E Rgv, K < Ci , and y E Qr ' then c(K,y) E PiEIYri • For, choose z E y n PiEIYri' By the remark after the definition of c we also have c(K,y) E y n PiEIY(vy)i Thus the set M = (i E I: zi = C(K,y)i} is in F. Since M <;;; (i EI:Yr i = Y(vy)i} we infer from (4) that vy r, so (7) follows. (8) For all r E Rgv we have CiQ <;;; f5V • r In fact, let q E O'Q By (7) we have, for all K < Ci , r + c(K,qK) E PiEIYri Hence for all i E I , (c q)i = :K < Ci) E !Y.y • <;;; V. Hence q E f5 V • L 1. 7.24 Ultraproducts 111 By (5), (6), (8) we have (f 06)* E Gs Q' In the case K = Ws we redefine c Let E Ws , say 01 01 We may assume that luI > 1 For each u E U and 1«01 let c(l(,eu) = (u:iEI) Now suppose y E X e*U and I( <01 Choose k E y and let Z {i EI: k i i PK} . Thus Y Y Y Z E F since y Epl< Let c(1< ,y) Z 1k U (u : i E 01 Z) , where y i y y y u U {PK) Thus c is an (F, (U : i E I) ,01) - choice function, y E with the additional property (9) for all K < 01 and all y E X- {Epl<} we have pI( Rg c(K,y) • Now the hypotheses of 1.7.12 clearly hold, so (i) - (iv) are true. To establish (v) and (f 0 5)* E Ws it suffices to shew that 01 O f5v = OIX(E p) First suppose q E OIX(EO p) Let i E I Then + 01 (c q)i E U, obviously. Let r = {I< < 01 : ql< r epK} . Then r is + finite, and for I( E 01 - I' we have (c q\K = c(l(,qK\ = C(K,EPK)i pK. Thus (c+q)i E OIU(P) = V. Hence q E fOV • Conversely, suppose q E OIX _ Q'X(EOp) ; we show that q f fOV • Let r = {K < 01 : qK r EpK} ; thus I' is infinite. For any I( E rand i E I we have C(I<,qK)i i pI< by (9) Therefore (c+q)i OIU(P) = V for all i E I , and consequently q t fOV • This completes tbe proof of 1.7.23. This lemma immediately gives Theorem 1. 7.24. Let be a Gws with an infinite base, and 01 let K be any ca r'df.na L, Then is sub-isomorphic to a Gws with 01 base of cardinality K, such that =0 n OIW for some W • 112 Ultraproducts 1.7.25 r reg reg reg Moreover, if EKE tWS ,Cs ,Gs ,Gws ,Gws ,Cs ,Gs } then 0'.0'.0'.0'.0'.010'. also m EK• Using 1.3.18 we easily obtain the following more specific result. TheorEm 1.7.25. Let m be a Gws with an infinite base U, 01 and suppose that \A\ U \U\ K Then is sub-isomorphic to a 91 Gws m with base of cardinality K, such that 1 = liB n OIW for 01 some W. Moreover: reg reg (i) if rWs ,Gws ,Gs ,Gws ,Gs } then m EK 0'. 0'. 0'. 0'. 0'. r eg) (ii) if E Cs (resp. E Cs and K K\ 01\ , then 0'. 0'. reg m E Cs (resp. m E Cs ) 0'. 0'. r eg r eg Proof. The parts concerning Ws , Gws ,Gws Cs and Cs 0'. 0'. 0'. ' ex ex are immediate from 1.3.16, 1.3.18 and 1.7.24. For the other parts reg dealing with Gs and Gs we use a direct construction, not Ci O'. involving ultraproducts (which actually works for some other classes). Suppose is a Gws ,say with base U and unit element v = Ci CL. (pj) U ,where n = 0 for distinct j,k E J• Let (Ze: e < K) be a system of pairwise disjoint sets, with for all a < K, and let fa be a one-to-one function mapping u onto for each e < K further, we assume that Zo = U and U11d. Then for distinct (a,j),(y,k) EK X J we have Opj) OI(f*y.)(fe n O'.(f*Y )(fyOpk) = 0 Let m be the full Gws with e J y k 0'. unit element • For each x E A let gx 0'. -1 E Z8: f 0 YE x}. It is easily checked that g is an iso- 8 -1 morphism of onto a subalgebra of m, and in fact g c v n c for each c E C ,i.e., m is sub-isomorphic to Moreover, 1. 7 .26 Ultraproducts 113 the base of is of power K In case E GSa ' clearly E GSa • Finally, suppose Suppose x E A, Y E gx , z E gV , a -1 and .. U 1)1y = ... U 1)1z Say y E Za and fa 0 y EX, -1 while z E aZ and f 0 z EV Since yO = zO we have a = y y Y -1 Thus since AX = , because g is an isomorphism, we get f a 0 z E x by regularity of x and hence z E gx • For GWsa's in general it seems to be more interesting to increase the size of various subbases rather than to merely increase the size of the base; that is the purpose of our next theorem. U and unit Theorem 1.7.26. Let be a Gwsa with base element V = U ay (pj) where ay(pj) n ay(pk) = 0 for distinct jEJ j' j k j,k E J. Let K be a cardinal-number-valued function with domain \a\U\Y.\ J such that Kj (\A\ n 2 J) Uwand Kj \Yj\ w whenever j E J and Kj f \Yj\ • Then is sub-isomorphic to a Gws with unit element a ,with n = 0 for distinct j ,k E J, where Y W for all j E J for which K \Y ' and W Y with j = j j = j\ j j \W.\ =K. forall j E J such that K. > \ Y. \ • Furthermore, JJ JJ m V = ClJ n 1 • Proof. By 1.6.2 we have ,where isa WSa with unit element for each j E J ; in fact, the isomorphism JI h is given by (ha)j = a n ayjpj) for all a E A and j E J. Note that \Y. \ wand J 114 Ultraproducts 1.7.27 for each j E J for which Hence by 1.7.25, \8. is sub- J isomorphic to a Ws with unit element such that \8. = a J j JJ and hence Y W if K '" IY ,while \W '" K for all j E J• j = j j j\ j\ j We may assume that Wnw = Y n Y and hence n cL(pk) = 0 j k j k jWk for distinct j,k E J• For each j E J let f '" (c n : c E C.) j J J thus f. Finally, let for each a E A JJJ then g is an isomorphism from onto a Gws with unit element a (as is easily checked), and g-ld '" V n d for all d ED, as desired. We now return to a question discussed in 1.3.20-1.3.22: changing the function p in the unit element aU(p) of a Ws The a following theorem of and strengthens results of Henkin and Monk. Theorem 1.7.27. Suppose a til • Let be a Ws with unit 01 element aU(p) , and let q E aU. Then is homomorphic to a Ws a with unit element ay(q) with UY, where if \UI < til or IU"" Rgq\ = lUI \A\ then one may take Y'" U Proof. We may assume that \U\ > 1. Let 1 = {r c a: \r\ < til} , and let F be an ultrafilter on 1 such that {r E 1 : r} EF for all E 1 • Let X = lUfF Choose k E aU so that kK 1 pK for all K < a. Let be as in 1.7.12 Then there is an (F, (U : i E 1) ,01) - choice function c such that for all y EX, r E 1 and K E 01 "" r , 1.7.28 Ultraproducts 115 if y = e:qK: , c(K:,y)r "'" = {.:: otherwise • Now let f = Rep(F,(U: i E I),Q',(A: i E I),c) • Thus by 1.7.2, f is a homomorphism from onto some Crs 18 Since the functim 0 Of . of 1.7.12 is an isomorphism of into , f 00 E Horn ,18 ) + Now if h E Q'x and rEI , then {K: E Q' ,.., r : (c h)rK: 'f PK:} = {K: E Q''''' r : c(K:,hK:)r 'f PK:} {K: E Q''''' r :hK: 'f e:qK:} ,so (C+h)r E Q'U(p) iff h E Q'X(e:oq). Thus 18 is a Ws with unit element OfX(e:oq) Q' Choose Y;:l U together with a function t mapping Y one-to-one -1 onto X such that e: t. By 1.3.1, t induces an isomorphism t from 18 onto a WSQ' with unit element oty(q). Thus t 0 f 0 0 is the desired homomorphism. If \U\ < w , then \U\ = IX\ and hence Y = U. Assume now \U,.., Rgq\ = \u\ \A\ • By I.3.l8(ii), t*** f 0 is ext-isomorphic to a WsQ' with unit element OW(q) for some W with lUi = Iw\ ' UW• Thus there is a one-to-one function s mapping W onto U such that So q = q. By 1.3.1, s induces an isomorphism u from t*** f 0 onto a Ws with unit element Q'U(q) , and hence u 0 t 0 f 0 0 is Of the desired homomorphism for the last part of the theorem. Our next theorem, due to Henkin and Monk, is related to 1.7.21. Given a GwsQ' With all subbases finite, it is not always isomorphic to a Cs see 1.6.8(5). It is natural, however, to try to reduce the Q' number of subbases. Theorem 1.7.28. Let K: < UJ s; Q' • Suppose that is a GwsQ' with unit element V such that every subbase of is of power K: • 116 Ultraproducts 1.7.28 Let A be the least cardinal such that for each equivalence relation R on a having all equivalence classes infinite the following inequality holds: a \{q E V: q Iq-1 = R} I S; A• I{q E K: q1-1 q = R} I• Then is isomorphic to a Gs with subbases. a Proof. Let have unit element V = U aU (pi) , where u iEI i aU(pi) n aU(pj) = 0 i j for distinct i,j E I . Now set R = -1 cr:> {ql q : q E a K} and It = {R E It : all equivalence classes of Rare infinite} Now we define a relation - on It by setting R =R' -1 -1 iff there exist q,q' E aK such that R=qlq ,R/=q/\q' , and I{s: s < a and qS f q/S}\ < w (1) is an equivalence relation on It • Indeed, only the transitivity of = is questionable. Suppose R = -1 I -1 R' == &" , say q,q' ,r' ,r" and R = q\q , R = (q '\ q I ) = R" = rII \ r 11-1 and q E aK(q') r" E a K(r') Then there is a one-to-one function k from K onto K such that k ° q I r ' since K R == R" • Similarly we have /, -1 (2) if R= R lUI = K, q E aU , and R qlq ,then there is a q' E aU(q) such that R ' = q,\,-1 q The following statement is clear. ce (3) If RER lUI = K ,and q E aU , then there is at most one 1. 7 .28 Ultraproducts 117 ,-I such that q'\q = R (4) For every R ER there is an R' E Roo such that R = R' • -1 01 For, say R = q \q ,where q E K: • Le t r = U{ 13 IR : 13 < 01 and \I3/R\ < w} • Since K: < W we have IOI/R\ < wand hence \rl < w Choose 13 E 01'" r , and let q' = (01'" r)1q U (ql3 : 13 E r) q' is as desired. Now by (1) - (4) there is a function R E IRoo such that p.1 = R. for all i EI, and LLL Now let (YI3:13 < A) be a system of pairwise disjaLnt sets, each of power K:. Set W = Lb equivalence relation on W , and the ,., - classes are weak spaces. Let K = W1-. Next, for each i E I let Li = {S EK: there is a q E S with q\q-1 R } Then by the choice of R, = i (5) if i, j EI and R. 'f R., then L. n L. o· L J L J ) -1 (6) if i EI, then \i/(R\R )\ $ ILil • In fact, using finally the hypothesis of the theorem, and (3), I \i/(R\R- )\ I{q EV :q\q-l = R i}\ -1 } s; \ {q EW: q \ q = R \ i \Lil • By (6) we can choose a one-to-one SEI K such that Si EL for i all i E I. Then (7) for all i E I there is a q E Si 118 Ultraproducts 1. 7.28 This is true since Now let m map K onto I such that mSi = i for all i E I is a Ws with By I.6.2 we have m \ c PiEI 58 i where unit element for each i E I For each TEK let be i the full WSQ' with unit element T. By 1.6.2 PTEK (S:T is isomorphic to a Gs with unit element W, which has A subbases. Thus Of to prove the theorem it suffices to show that P 58 c P • i EI i I TEK First we note: (8) for eyery TEK there is a homomorphism from j{\mT into Itr For, say T = • Let b E UmTy be one-to-one and onto, and set -1 r = b 0 q By I• 7. 27 let h be a homomorphism from j{\mT onto a Ws j{\ I with unit element Then 0 h is as desired in (8) , Q' b where b is defined in 1.3.5. (9) For every i E 1 there is an isomorphism from j{\i into -1 1-1 Thus W = For, by (7) choose k E 5i such that k\k = Pi Pi Si ay(k) for some < A• There is a one-to-one function b from U. 1 onto such that bop. = k • Thus '"b is the desired isomorphism, 1 by 1.3.5 and 1.3.1. By (8) and (9) there is a function h with domain K such that E Hom(l8mT'(S:T) for any TEK and hSi E Ism(j{\i for each i E I. It follows easily that is the desired is omorphism from PiEl 18 i into 1.7.29 Ultraproducts 119 Corollary 1.7.29. Let K < W S; a. Suppose that is a Gws with unit element V = ,where n 0 a 1 1 1 J for distinct i,j E I • Assume that Iuil = K for all i E I , and \II s; K• Then is isomorphic to a Cs a Proof. We may assume that K > 1 Let R be an equivalence relation on a having every equivalence class infinite. Then I{q E V :qlq-l = Rl\ :s; \1\ ' since if q\q-l = R = q'\q,-l with q,q' E V ,and q T q' ,then Of : T is infinite and so CiU(pi) q E i ' for distinct i,j EI Thus by 1.7.28 it suffices to check (*) if R is an equivalence relation on a and \a/RI S; K, then the set {aK(q): q\q-l = R} has at least K elements. To prove (*) , first choose q E a K such that qlq-l = R. Let f be the pe rmutiatidon of K such that fA. = A. + 1 for all A. < K- 1 , while f(K - 1) = O. Then for all distinct < K we clearly have fA. 0 q ,while (fA. 0 q)\ (fA. 0 q)-l = R ,so (*) follows. Remarks 1.7.30. (These remarks are due to and (a) The special hypotheses in 1.7.19 are necessary. Namely, if S; \ then there is a Ws with base U such that w U\ :s; Ci a \A \ s; a and any 18 E Cs n either has only one element or else has base Ci of power > 0:. To construct such a WS ,let = K• Let a lui w E UK be one-to-one and onto. Let p = (0 : < (1) , and let be the full Ws with unit element V = OfU(P). We now construct Ci a E KC• For every A. < K let kA. be a one-to-one function mappirg 120 Ultraproducts 1. 7 . 30 K into {v E U: A. < wv} • Then for each A. < K let Now if 0 < < a, A. < K, and q E aA. ' then wq s; max {wq : 0 < \) < a} < w(k max{wq : 0 < QI})A. = wqO ' \) \) Thus (1) if 0 < < a and A. < K, then aA. • We also clearly have (2) co a A = 1 for all A. < K; (3) a n a = 0 for distinct A.,II. < K• A e- Let = : A. < K} • Clearly \A\ s; a. Now we claim (4) if 18 E WSQl n with unit element ay(q) , then \Y'" Rgq\ K• For, suppose that f is a homomorphism from onto 18. Thus the conditions (1) - (3) hold with (a},. : A. < K) replaced by (faA. : A. < K) . 0 For each A. < K choose YA EY so that qYA E faA. ' using (2) for faA • Thus Y is one-to-one by (3) for faA and Furthermore, YA Rgq for 0 < A. < K by (1) for faA. ' if YA. 'I qO. So (4) holds. Now suppose h for some CS (S; with base W 0 , E a 'I and assume that \W\ s; QI. Let q map a onto W, and set V = aw(q) • Then 0 h E , where 18 is a WsQI with unit element V, contradicting (4). 1.7.30 Ultraproducts 121 (b) In 1.7.27 one cannot replace ''homomorphic'' by "isomorphic". In fact, let be the Ws with unit elerrent aU(p) generated by a {p} ,where U '" a + 01 and p '" (0 : I< < a) , and let 18 be the full Ws with unit element aU(q) ,where q = (I< : I< < a) Suppose a f E Since {p} s for all I<,A < a , there is an . diO . r E aU(q) r E f{p} with r E dl (c) By the argument for (a)(4) above, the condition \U Rgq\ IU\ in 1.7.27 is necessary. (d) It is not known if the condition lUI \A\ in 1.7.27 is needed. (e) The hypothesis on A in 1.7.28 is in a sense best possible. Namely, suppose that is a Gws with unit element V such that a every subbase of is of power I< , and that is isomorphic to a Gs 18 with A subbases; assume in addition that {q} E A for all a q E V ; we show that the indicated inequality holds. Let h E , and Ie t W be the unit element of 18. Let R be an equivalence relation on a with all equivalence classes infinite. If q E V and q\q-l = R, then (1<,1..) ER implies that {q} and hence PI< = PI.. for each p E h{q} ; and similarly pI< 1 pt.. whenever -1 (1<,1..) Rand p E h{q} • So q E V and q\q = R implY that -1 -1 pIp = R for all p E h{q} • Thus (h{q}:q E V,R = qlq ) is a system of pairwise disjoint non-empty subsets of {q E W: qlq-l '" R} Hence -1 -1 \{q E V : q Iq R} \{q E W: q \q = R} \ '" A'\ {q E al< : q Iq-1 = R} \• 122 Reducts 1.8.1 (f) The assumption \1\ SKin 1.7.29 cannot be improved, by (e) upon considering R a x a • (g) and have proved the following algebraic version of the various logical theorems to the effect that elementarily equiva- lent structures have isomorphic elementary extensions: Let E Cs 00 a and j8 • Then and j8 are sub-isomorphic to Cs I S and 113' a respectively such that and j8' are base-isomorphic. 8. Reducts • _.ww"..;PW We restrict ourselves in this section to the most basic results ,, about reducts. A more detailed study is found in Andreka, Nemeti [AN3] to which we also refer for the statement of various open questions. 1.8.1. Let be a crsB with base U and unit element V Let a be an ordinal and let p E a B be one-to-one. Fix x E X EA. For each y E aU set + -1 y «B- Rgp)1x) U (y s p ) thus y+ E Bu. For all YEA let fY = {y E aU : y+ E Y) • Then f is a homomorphism of into a Crs and fX 1 0 • a, Proof. Let W fV. Clearly f preserves + and Sin: e + (x e p) = x we have x s p E fX, Le., fX'I O. It is routine to check that f preserves d for K,),. < a. Now suppose that YEA , KA 1.8.2 Reducts 123 I<: < Q' , and yEW I For brevity set 18 = To prove that fc 08)y C[W]fY , let y E fc(18)y Thus y E fC[V]y , i.e., KKK pI<: y+ E Thus y+ E V and E Y for some u E U. It is easily checked that = ; hence E Y, SO E fY and so y E The other inclusion is established similarly. Theorem 1.8.2. If Q' and B are ordinals with Q' 2 and PE Q'B is one-to-one, then I Gws co Q' First we take any E WS Q and show that E IGws I-' Q' Q' To this end, by 2.4.39 and 1.6.4 it suffices to take any non-zero X E A and find a homomorphism f of into some Ws such that fX 0 Q' 01 Say has unit element 13U(p) , and x E X For each y E OIU define y+ as in 1.8.1; then define f as there also. Applying 1.8.1, we see that f is a homomorphism of into a Crs 18 ,and fX 0 • Q' Now it is easily checked that f(13 U(P» = {y E Q'u : \ry\ < ill} , where r = {K E Rgp :YP-lK pK} for all y E Q'U. Clearly iff y -1* -1* \ pry\ < W, and pry = tK < Q' : yK " ppK} Thus Q'u(pop) , so "'. a WsQ" as d' d The theorem itself now follows easily from O.5.l3(iv) and 1.6.4, the final statement being c le ar from the above. Remarks 1.8.3. Under the hypothesis of 1.8.2 and using 1.8.2 (p)Gsr eg _c IGs reg by I 7 14 we also have Rd(P)GS IGs and Rd01 13 Q' •• , 01 13 Q' I I and Rd(P) Cs I Cs if 01 W by 1.7.21 But Andreka and Nemeti Q' co 13 co Q' have shown that if Rgp 'I 13 then generalizing examples of Monk. 124 Reducts 1.8.4 Theorem 1.8.4. Let a and be ordinals and let p E be reg, one-to-one and onto. Then '" IK for KE tCrs,Gws,Gs, Gws a Cs,Ws} • Proof. The arguments being very easy, we restrict ourselves to r eg• two representative cases, K '" Crs and K'" Gws First suppose m is a ' say wi th base U and unit element V Now the function y+ in 1.8.1 does not depend on any elerrent x ; we have simply y+ -1 yop The hypotheses of 1.8.1 hold, so the function f defined there is a homomorphism of into a Crs 58 with unit element fV, a and fX r 0 for every non-zero XEA ,i.e., f is one-to-one. Thus E ICrs ,as desired. a Now suppose that m E It is easy to check that 58 E Gws Assume that YEA, y E fY , z EW, ani U 1)1y a ;"" '" - U 1)1 z ; we want to show that z E fY Now Iet (S; '" !Rb (p )m For any K < a we have c(58)fY fY iff c«(S;)Y '" Y iff c(m>y '" y K K pK Thus '" First suppose that 0 E • Then p-lO E , and so yp-lO '" zp-lO , i.e., y+O z+O Hence U 1)1y+ '" u 1)1z+ Clearly y+ E Y and z+ E V , so ... - z+ E Y by the assumed regularity, so z E fY as desired. Second, assume that 0 Now y+ E Y, so E V (using m E , and hence EY Clearly also E V , and U 1)1 '" u 1)1 ,so E Y. Hence z+ E Y and z E fY , as desired. We have shown in our two representative cases. For the other inclusion, it suffices to note that pEa-1 is one-to- -1 one and onto, )K IKe by what was already shown, and clearly a 1.8.5 Reducts 125 =K 01 Now we turn to neat embeddings, for which we also require a technical lelIllIla. LelIllIla 1.8.5. Le t [( be a Crs with base U and unit element 01 v • Assume that 01 S and W We also assume the following conditions: (i) V = {x: x = 0I1y for some yEW} ; (ii) for all xEW, K: < 01 , and u E U, if then For any XEA let fX = {x E W: 011 x E X} • Then there is a crs !8 wi th base U and unit e lene nt W such 13 that f EI sm([(,!Rb 18) and c (l8)fX = fX for all XEA and K: E 13 ,.,. 01 01 K: Proof. Let !8 be tre full with unit element W• Clearly f preserves +. Now let x E Wand X E A By (i) we have 011 x E V. Hence x E f(V,., X) iff 0I1x E V'" X iff 011 x X iff x E W,., fX. Hence f preserves If 0 X E A ,choose x EX. Then by (i) there is ayE W such that x y. Thus y E fX This shows that f is one-to-one. Clearly f preserves for < 01 (again using (i». Now suppose that XE A, K: < 01, and x E C[V]X . we want to show that x E C[W]fX By the definition of K: ' K: f we have x E Wand 0I1x E Hence 0I1x E V and EX for some u E U. Since (0I1x)K: E V, our assumption (ii) yields u that xK: E W. Hence x E C[W]fX as desired. The converse is similar. uK:' Finally, suppose that XEA, K: E 13 ,.,. 01 , and x E C[W]fX K Thus x E Wand xK: E fX for some u E U Hence xK: E Wand u u 126 Problems 1.8.6 Since K a , this means that a1x EX, and hence x E fx , as desired. From this lemma it is easy to prove Theorem 1.8.6. Assume that a and K E {Ws,Cs,Gws,Gs} • Then K c ISNr K.. • a 0: Corollary 1.8.7. If 2 a then IGws a Proof. By 1.7.14, 1.8.2, and 1.8.6. 1.8.8. It follows from 2.6.48 and 1.8.6 that for any result of the representation theory of CA 's , to appear in a later 0: paper, is that if 0: 2 then IGws a 9. Problems JUQ. 4C We begin by indicating the status of the problems listed in [ HMI' ] as of January 1981. In Problem 0.6 one should assume that 0: is less than the first uncountable measurable cardinal (see Chang, Keisler [CK]). Under this corrected formulation, the consistency of a positive answer relative to the consistency of ZFC plus certain other axioms has been shown by Magidor [Ma] and Laver [L] Problem 1.2 has been solved I affirmatively by B. Sobociriski [ S] • Andreka and Nemeti solved Problem 1 Problems 127 Problem 2.3 affirmatively; see [AN2] and [N]. Problem 2.4 has been solved affirmatively by J. Ketonen [K] for Boolean algebras, and hence for discrete GA'S. Problem 2.8 was solved affirmatively by D. Myers [My] and Problem 2.9 negatively by W. Hanf [H]. 2 was solved negatively (except for a < 2) by and see [AN2] and [N]. Problem 2.12 was solved negatively by R. Maddux [Md]. Now we shall list some problems left open concerning set algebras. Problem 1. Let a w. Given a normal Gws with base U, a £ is there a CSa with same base U such that E (Cf. 1.2.6-1.2.13) • Problem 2. Let q be the function defined in 1.4.8. For every + aEw..-2 let q a be the largest e E w such that 1 Give a simple arithmetic description of q, or at least of q+ Problem 3. Is IWB closed under directed unions for a w ? a (Cf. 1.4.8 and 1.7.11.) reg r eg Problem 4. Is ICs HCs or HCs()' = HCs ? (Cf. 1.5.6.) a ()' ()' Problem 5. Does I Cs = H Ws ? (Cf. 1.5.6(17) and 1.5.8.) a> ()' a> a reg Problem 6. Is ICs !=: HPWs or ICs c HPWs ? (Cf. 1.6.8.) ()' a ()' ()' Problem 7. Is H Ws HP Ws ? a> a a> ()' Problem 8. Is HP Ws I Cs ? a> a a> ()' For these two questions cf. 1.6.8 aa d 1.6.10. 128 References Problem 9 Problem 9. Is every weakly subdirectly indecomposable Cs iso- a morphic to a regular Cs ? a Problem 10. Is every weakly subdirectly indecomposable a r eg) (or Cs isomorphic to a Ws ? a a For these two questions cf. 1.6.16. Problem 11. Is the condition lUI \A\ in 11.7.27 needed? (Cf. here also 1.7.30.) REFERENCES [ANI] H. and I., A simple, purely algebraic proof of the completeness of some first order logics, Alg. Univ. 5(1975), 8-15. [AN2] H. and I., On problems in cylindric algebra theory, Abstracts Amer. Math. Soc. 1(1980), 588. [AN3] H. and I., On cylindric-relativized set algebras, this vol., [AN4] H. and N4meti, I., Finite cylindric algebras generated by a single element, Finite algebra and multivalved logic (Proc. ColI. Szeged), II eds. B. Csakany, I. Rosenberg, Colloq. Math. Soc. J. Bolyai vol. 28, North-Holland, to appear. [CK] Chang, C.C. and Keisler, H.J., Model theory (second edition), North-Holland 1978, xii + 554 pp. [D] Daigneault, A., On automorphisms of polyadic algebras, Trans. Amer. Math. Soc. 112(1964), 84-130. [De] Demaree, D., Studies in algebraic logic, Doctoral Dissertation, Univ. of Calif., Berkeley 1970, 96pp. [EFL] Erdos, P., Faber, V. and Larson, J., Sets of natural numbers of positive density and cylindric set algebras of dimension 2, to appear,Alg. Univ. References 129 [H] Hanf, W., The Boolean algebra of logic, Bull. Amer. Math. Soc. 8(1975), 587-589. [HM] Henkin, L. and Monk; J.D., Cylindric set algebras and related structures, Proc. of the Tarski Symposium, Proc. Symp. Pure Math. 25(1974), Amer. Math. Soc., 105-121. [HMT] Henkin, L., Monk, J.D., and Tarski, A., Cylindric Algebras, Part I, North-Holland (1971), 508pp. [HR] Henkin, L. and Resek, D., Relativization of cylindric algebras, Fund. Math. 82(1975), 363-383. [HT] Henkin, L. and Tarski, A., Cylindric algebras, Lattice theory, Proc. Symp. pure math. 2(1961), Amer. Math. Soc., 83-113. [K] Ketonen, J., The of countable Boolean algebras, Ann. Math. 108(1978), 41-89. [Ko] Koppe1berg, S., Homomorphic images of complete Boolean algebras, Proc. Amer. Math. Soc. 51(1975), 171-175. [L] Laver, R., Saturated ideals and nonregular ultrafilters, to appear. [Ma] Magidor, M., On the existence of non regular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc.249 (1979),97-111 . [M1] Monk, J.D., Singulary cylindric and polyadic equality algebras, Trans. Amer. Math. Soc. 112(1964), 185-205. [M2] Monk, J.D., Model-theoretic methods and results in the theory of cylindric algebras, The Theory of Models, Proc. 1963 Symp., North- Holland, 238-250. [Md] Maddux, R., Relation algebras and neat embeddings of cylindric algebras, Notices Amer. Math. Soc. 24(1977), A-298. [My] Myers, D., Cylindric algebras of first-order languages, Trans. Amer. Math. Soc. 216(1976), 189-202. I [N] Nemeti, I., Connections between cylindric algebras and initial algebra semantics of CF languages, Mathematical logic in computer science, eds. B Domolki, T. Gergely, Colloq. Math. Soc. J. Bolyai, vol. 26 North-Holland (1981), 561-606 [S] B., Solution to the problem concerning the Boolean bases for cylindric algebras, Notre Dame J. Formal Logic 13(1972), 529-545. [TV] Tarski, A. and Vaught, R.L., Arithmetical extensions of relational systems, Compos. Math. 13(1957), 81-102. On cylindric-relativized set algebras by H. Andreka and I. Nemeti This work is based on the book [HMTJ and the paper [HMTIJ. The abstract theory of cylindric algebras (CA-s) is extensively developed in the book [HMTJ. Most of the motivating examples for the abstract theory of CA-s are cylindric-relativized set algebras (Crs-s). The present work is de- voted to the study of Crs-s, more precisely to certain distinguished classes of Crs-s introduced in [HMTIJ. Such a distinguished class is r e g. r e g Gs The role played by Gs in CA-theory is similar to the role played by Boolean set algebras in Boolean algebra theory. For example, r e g, the fundamental link between model theory and CA-theory is Gs see the introduction of [HMTIJ. It was proved in [NJ that the class connec- ting classical finitary model theory to CA-theory is exactly GsregnLf, in a sense at least. Recently much attention was given to the meta-struc- ture consisting of all first order theories and all interpretations be- tween them. It was proved in CGJ that this structure can be represented r e g isomorphically by Gs achieving considerable insight and simplifica- tion this way. Following these motivations we shall give special atten- r e g. tion to Gs We shall use the notations introduced in CHMTJ without recalling them. The present paper is a continuation of CHMTIJ and is organized parallel to CHMTIJ. We refer to CHMTIJ for an introductory discussion of the contents of the individual sections; we have practically the same section-titles as [HMTIJ. The items in [HMTIJ are numbered by three figures like 1.2.2. The first figure is always I and therefore we omit it, e.g. the reference [HMTIJ2.2 in this paper means item 1.2.2 of [HMTIJ. We refer to items of the present paper by strings of figures, e.g. 0.5.1 refers to item 0.5.1 of this paper, moreover this item is found in section 0, and it is a sub- item of item 0.5. In general, the figures when read from left to right 132 correspond to the subdivisions in which the item referred to is found. We shall be glad to send full proofs of statements claimed but not proved (or not proved in detail) in the present work, whenever requested. Acknowledgement. We are most grateful to Professor J.D. Monk for guiding us iri this work as well as in our research concerning algebraic logic in general. o. Basic conce£ts and notations We use the notations and definitions of [HMTIJ and [HMTJ without recalling them. Especially we use [HMTIJ1.l where the classes Cs , a r e g, r e g, r e g, r e g Ws , Gs , GWs , Crs , Cs Gs GWS Crs of cylindric- N N N a a a a a -relativized set algebras were introduced. All these algebras are normal Bo -so a Notations: Let be an algebra similar to CA -so Then 1{)L m a exists since 1 is a constant symbol of CAa-s. We define Mn( OL) £! Sg ( vt) {1ot J and '!'!"W,( {It) £! {}\, ) {1Ot }. Let V be a Crs -unit. Then (Sf) V denotes the full Crs with a a unit V. This notation is ambiguous if v=o but we hope context will help. Let Then and Let H be any set. Then Sb H denotes the set of all finite subsets w of Hand GC H denotes that GESb H. -w w As a generalization of the notation introduced in [HMTIJ, the following notation will be very useful. Let f,k be two functions and let H be a set. Then f[H/kJ (Dof H)1f u H1k. The notations f: A - B, f A -- B, f: A >- B, and f : A >-- B mean that A1f is a function mapping A into (onto, one-one into, one-one onto respectively) B. In accordance with [HMTJ, 0.1 133 means that and similarly for Hom etc. We shall use the notations Zd, etc. introduced for CA -s in a [HMTJ to Crs -s as well, despite of the fact that a Crs need a a not be a CA E.g. let mECrs , and let xEA. Then a '1 g (eJ!,)x g til. { iEa : c.x f. x}, Zd(J1, g {xEA : (OL) x = O} and l (ZdOL, +,',-,0,1), cf. [HMTJ 1.6.1 and 1.6.18. By [HMTJ 2.2.3 we have that the axioms (C (C and (C O)-(C3), 5) 7) of [HMTJ 1.1.1 are valid in Crs. Therefore [HMTJ 1.2.1-1.2.12 a are true for Crsa-s, although they are stated for CA -s only, a because in their proofs the only axioms used are (C (as it O)-(C3) is explicitly noted on p.177 of [HMTJ). Also [HMTJ 1.6.2, 1.6.5-1.6.7 are true for Crs -s, since their proofs use only (C and a O)-(C3) 1.2.1-1.2.12 of [HMTJ. Because of the above, in the proofs we shall apply [HMTJ 1.2.1-1.2.12 and 1.6.2, 1.6.5-1.6.7 to crs -so By a 1. 6.2, 1. 6.5-1. 6.7 we have that for every dECrsa Let Then A=B implies eJl=15. Therefore we let Zd A Zdm. for mECrs In general, notions applicable to Cr -s g a a will be applied to cylindric-relativized fields of sets. The above argument holds for Boolean set algebras too. Particularly for any Boolean set algebra £s we let At B g At.f6. Definition 0. 1. Let U be any set and let vcaU . By a subunit of V we understand an atom of the Boolean field Zd Sb V of sets. Subu(V) denotes the set of all subunits of V. I.e. Subu(V) = At Zd Sb V We define base(V) g U{Rgp pEV}. We say that Y is a subbase of V iff y = base(W) for some WE Subu(V). Subb(V) denotes the set of all subbases of V. Let -mECrs Then base (01.) base (iOl.), Subu ({[) Subu ( 1'!l) and a Subb({}l,) = Subb(lO(,). W is said to be a subunit of a iff W is a subunit of itt, and Y is said to be a subbase of a iff Y is a subbase of 1« 134 0.2 The above definition of subbase a0rees with [HMTIJ 1.1 (vii). Note that a subbase might be empty iff a=O. Notation: Let KCCrs and H be a cardinal. Then - a K W1EK (VUESubb (a) ) IU I=H} and H K {elEK co (VUESubb(U)) The above notation agrees with [mUIJ 5.6(17) and [HMTIJ 7.20. Note that the one-element Crs is in Crs n Crs for all H since it a }{ a co a has no subbases. In this connection we recall the following: Let Then Gs {fJ1.ECA H a a That is HGSa is a variety definable by a finite scheme of equations consisting of (C and the above one. This is immed- O)-(C7) iate by [HMTJ 2.6.54 and [Ht1TIJ 8.8. Lemma 0.2. Let Then (i)-(iii) below hold. a (i) is the disjoint union of all subunits of (ii) Let WESubu(OL). Then wcay(p) for some YESubb(OL) and for some pE10L ( iii) VCEGws iff every subunit of {}(, is a weak space. a Moreover, let and let V = be such l that (Vi,jEI)[Hj = OJ . Then Subu(V) = i J : iEI} . l The proof of Lemma 0.2 depends on the followin0 lemma. Lemma 0.2.1. Let a be a complete SCra Let zd d n 1a} = < [{cio.•.cinx : nEw, iE + xEA ) . Then (i)-(ii) below hold. (if zd: A- Zd01, , z d : At'{l[ At Zd{)(" and (VxEA) zd(x) = 0.2.1. 135 = n{yEZdOL : xsy}, i.e. zd(x) is the "zero-dimensional closure" of x. (ii) If ex, is atomic then lot LAt7JU. Proof. Let be a complete SCr(J, and let the function zd be defined as in the statement of 0.2.1. Proof of 0.2.1(i): Using [HMTJ 1.2.6(i) it is clear that zd A- Next we show zd(x) = n{y E zdOL : XSy} for any xEA Let y E zdOL and suppose xSy Then for every nEw C i O·· .cinxSy n+l and i E (J, by [HMTJ 1. 2.7 and therefore zd(x)Sy (by (Co» By xszd (x) E Zda then zd(x) n{y E ZdOL : XSy} Suppose x E Ata. We show that zd(x) E AtU Let xSzd(x), y E ZdO(,. Then zd(x)-y E zdOL, by [HMTJ 1.6.6 and 1.6.7. If xSzd(x)-y then zd(x) S zd(x)-y, and therefore y=O by YS zd(x). Suppose x i zd(x)-y. Then x i -y by x S zd(x) and therefore xSy since XE AtU. Then zd(x)Sy by xsy E ZdeJ(, and by zd(x) n{y E ZdfJl, : XSy} Therefore y = zd(x) by ySzd(x). We have seen that zd (x) E At 7J a. Therefore zd: AtOL- At 7/.m. Proof of 0.2.1(ii): Suppose VL is atomic and complete. Then lor, = L Ate;{, follows from the theory of Boolean algebras. L AtflL S Uv L At 'fJ a by zd: AtOL - Atya and by (V'xEA)xSzd(x). Then l = = LAta S L At'ffJ!, proves that let = L At ?/eJt. QED (Lemma 0.2.1.) Now we turn to the proof of Lemma 0.2. Let or, ecrs (J, and let Proof of 0.2(i) : J:. is a complete and atomic SCr(J, hence let = lJ:' = L At Zd.r = L Subu(Q) by 0.2.1. Proof of 0.2(ii) : Let WE Subu(tt) and let pEW be arbitrary. We have W = L{ciO"'Cin{p} nEw, 1., E by.. 0 2 1(')1., since pEW E At Zd Sb lor,. Therefore We (J,base(W)(p). 136 0.3 Proof of 0.2(iii): If every subunit of {t is a weak space then dE Gws by 0.2(i) and by the definition of Gws . If a:S1 then a a the other direction holds too, since every nonempty subset of a Gwsa -unit is an a -dimensional weak space then. Suppose Let V = iEI} be such that (Vi, kEI)[i*k l Then by [HMTIJ 1.12 we have At Zd Sb V : iEI}. l QED(Lemma 0.2.) Proposition 0.3. Let V be a Crs -unit. Then statements (i)- a -(iv) below are equivalent. (i) V is a Gwsa -unit. (ii) The full Crs with unit V is a CA a a ( iii) {{ q J qEV} is a The Crsa with unit V and generated by CA a (iv) Every Crs with unit V is a CA a a Proof. Since Gws C CA by [HMTIJ 1.9(i), it is clear that (i) a a (iv) (ii) (iii). So, it is enough to prove that (iii) implies (i). We shall need the following lemma. Lemma 0.3.1. Let tlECA bEA and assume a Then b :S holds in f][, for all i, jEa . Proof. Assume the hypotheses of Lemma 0.3.1. Then the minimal sub- CA and hence nbeJt F c , d , ,=1 Then a l lJ b s c . (d,, 'b) =b holds in VL, by the definition of tlO.. VL. 1.e. l lJ '1) b :S holds in {}{,. QED(Lemma 0.3.1.) 0.3. 137 Now we return to the proof of Prop.0.3. Let V be a Crs -unit a and let be the Crs with unit V and generated by {{q} : qEV}. a a Assume 0.3(iii), i.e. assume (tECA Let U = base (V) and let a = Then and hence V holds in J: by 0.3.1. i In particular, (VqEV) (Vi,jEa) qq(j) EV V $ c.(d .. ·V). J l lJ Now we show that a i . j Case 1 Assume . By [HHTIJ 2.2 and by V $ sjVsiV , it is enough to prove that $ V holds in J:. for all i,jEa , J l Then there are such that l lJ J lJ lJ i _ j q fq(i) - hq(j) Let k E Let u g q (k) . Then i u = h(k) = f(k) = q(k), by k {i,j} By V $ skV and {f ,h} c V we have i i { f}EA f u = ff(k) EV and Note that since tt is generated by { {p} pEV} , and fEV . Then h E E f I since and i h , c V• Since l J l (X tit CA by our assumption, we have by (C ) that h E cjc ( f L, Then mE a 4 i E j (:Ip ci{f})h Ph(j) Then p(i)= h(i)= q(i) by Hj and i ) By P E ciU} we have p=fp(i) and therefore qEV by i i pEV and p = q We have proved siv.sjv $ V fp(i) fq(i) J l for all i, jEa Case 2 Assume a$2 a$l [HMTIJ 1.8 If then Crsa= Gws a by and therefore we are done. Let a=2 \\Te define u - v iff u,vEU and (u,v) EV It suffices to show that is an equivalence relation on U Suppose (u,v) EV; we prove that (v,u )EV In fact, (u,u ) EV by V s;v (Lemma 0.3.1), and similarly C v ,» ) EV Clearly (u,U)EC«.« so (u,u)Ecora 1CO{(v,v)}, Oc1{(v,v)}, and hence (v,u ) EV Now suppose ( u, v ) , t v , w ) EV I ... we show that (u,w ) EV We have ( w,v ) EV by what was just proved, so ( u,v )E ECet Ol ) EV (u,v )E Oc 1{(w,w ( w,w by Lemma 0.3.1 again). Hence EcfJI. c «{( w w and therefore (u,w ) EV By these we have proved 1 0 ' that V is a GS2 -unit. 138 0.4. QED (proposition 0.3.) Remark 0.4. In Proposition 0.3, condition (iii) cannot be replaced with the condition that some crs with unit V be a CA since a a, IGws CI Crs n CA by CHMTIJ 2.14 we have a a a Problem 0.4.1. Let V c aU. What are the sufficient and necessary conditions on V for 'l1IaC (5Rt V) E CA ? a Note that by CHMTJ 2.6.57 we have E CA iff m.w(M V) E a E IGs Similarly, by Prop. 0.3, V E CA iff VE I GSa a rsff a By the above we have the following discussion of CHMTJ 2.2.10 (stating (V(1ECA ) (VbEA)C('Vi, =>-1?Q vtECA J) • The a J 1- b a condition b:O: is necessary for 't?R.bOLECA , but the J l a condition b;:: is not necessary in general by CHMTIJ 2.1, J l 2.14), though it cannot be omitted completely, as Prop.a.3 shows. E g b;:: is necessary if ()(, is a Crs and {{ q J: ••J 1- a : gEb} ::. A . Definition 0.5. Let mECrs Then eJl. is said to be normal iff a Subb(fJL) is a partition of base (a) , Le. eJL is normal iff CVY, ZESubb(fJI.,»CY=Z or ynZ=OJ. is said to be compressed iff I Subb (elL) 1:$ 1. a is said to be widely distributed iff (VW, vesusu WI,) ) CW=V or base (W)nbase (V) =0 J . wd Let KC Crs Then Kno r m, Kc omp, and K denote the a classes of all normal, compressed and widely distributed members of K, respectively. E.g. {fJl.EGws eJl,is compressed}. a The above notions were introduced in CHMTIJ 2.6. In CHMTIJ 2,1, an elementary characterization is given for those 0.6. 139 Gws -units which are members of a CS ' for a23. It is shown a a in [HMTIJ 2.3 that for a=2 there is no abstract characterization of Gws -units as members of a given Cs At the end of [HMTIJ a a 2.3, Theorem 0.6 below is quoted. Theorem 0.6 says that there is no abstract characterization of those GSa -units which are members of a given Cs a, for a2w . Theorem 0.6. (noncharacterizability of GSa -units) Let and let U be an arbitrary set. Let xcau be any nondiscrete Gs a rt D(U) -unit, i.e. let X ];., 01 . Then there is an automorphism no rm such that t (X) is ------anot a Gws -unit (hence t(X) is not a GSa -unit). We need the following lemma. Lemma 0.6.1. Let a2w and let J: = G'fi "u'. Let f : U >-- U be a permutation of U Let p E aU. Then there is t E Is (or,[) such that t{q}={foq} for every qEau(p) and t{q}={q} for every . Proof. Let everything be as in the hypotheses of 0.6.1. Let d 'f@'¥J:: and p@ p(;lUdet : d E At Z ) Let r = « xnd : d E At Z): : xEC Then r E Is (.t',p) by [HMTIJ 6.2 and by [HHTJ 0.3.6 (ii) , -1 since l- = L At Z We have r ( URgk : kEP ) , since d ('\>'d, bEAt Z)bnd=O Let a = au(p) and b @au(fop) Then a,b E E At Z. Let z : At Z >-- At Z be the permutation of At Z defined as z (At U{(a,b), (b,a)} . For every d E At Z let if d=a h = f d dr-1 if d=b and d*a Id if d${a,b} 140 0.6.1. d (For the notation r see [HMTIJ 3.S(i).) Let g = « hd(kd) : d E At Z): kEp . . Now g E Is CP, P('J(Pz (dl : d E At Z » by [HMTJ 0.3.6 (iii), since for every d E At Z we have hd E Is (Wd£ , 'R£Z(d/)' by [HMTIJ 3.1. By [HMTIJ 6.2 (and by [HMTJ 0.3.6 (ii» -1 -1 again we have r EIs (P ('b(f,z(d).t : d E At Z) ,.t), since r (U Rgk k E P(Rld£ dE At Z» (U Rgk k E P(RIZ(d).t d -1 d E At Z » Let t = r ogor. Then t E Is (£,1:) . Let xEC be arbitrary. Then by the definitions of rand g we have t(x) -1 -1 -1 r gr(x) = r g( xnd : d E At Z) = r (hd(xnd) : dE At Z a U{hd(xnd) : d E At Z} == r(xna)Uf Therefore if then t(x) rex) = {foq : qEx} and if then t(x)= x QED(Lemma 0.6.1.) Now we turn to the proof of Theorem 0.6. Let and let U g a Xeau be any set. Let J: 15ff u Let be any nondiscrete Gs a -unit. Then there is a subbase Y of X such that IYI>l and YjU (and aye X ) Let m,nEY and be such that mjn Such m,n,w exist by IYI>l and by YjU Let f d = 1Id U {( m,w ) , (vl,m )} Then f is a permutation of U. Let p (m : iEa ) By Lemma 0.6.1 there is t E Is(£,J::) such that ('Vq E au(p»t{q} {foq} and ('Vq E U U au(fop»))t{q} = {q} We shall show that t(X) is not a no r m Gws -unit. Let (n : iEa) and g (w : iEa i , Then a n w hence tF t(X) But E t(X) . nOw Then t(X) is not normal since the ranges of are not nOw disjoint and one of them is in t(X) while the other is not. QED(Theorem 0.6.) Proposition 0.7 below says that neither the WS -units nor the a wd Gwsa -units have any abstract characterizations as members of Cs a -so 0.7. 141 Moreover, there is a VI- such that those members of A which wd are Wsa-units (or GWS -units) have no abstract characterization a even if fJL is fixed, I. e. even relative to 'U. Indeed, let a:?:w and ){>2 Let «,X and t be as in Prop.0.7. Then E n o rm wd G norm E Ws and Gws . Let K E {Ws GWs wS ' a a a, a, a c omp} no r m Gws . By Ws eKe GWs then E K but K. a a - a Thus the K -units have no abstract characterization in it. Proposition 0.7. (noncharacterizabilityof Ws -units) Let a and ){>1. Then there are 'OLE Cs , t E Is ( Proof. Let a:?:w, ){>1 and S (s : iEa for any set s. d a (0) Then X is a Ws -unit. Let X = ){ a Let and Let rI6 = rl ,', VI- and rl (-X) Then .(5,l1EMn X 11. "'a . a and by [HMTJ 2.5.25 cI5-:;;n since Vdf, = V 11, is finite. Since XEZd A, {J(, -:;; d's ,,11,. Clearly f = « znx, zn-X > zEA >E Is (01, ,.ex1[) and f (X) = ( 1,0 > By [HMTJ 0.3.6, there is hE Is (£,,11., J:f, x lD with h( 1,0> = (0,1 > and hoh C Id Then a a (0) Observing that -X= ){ is not a WS -unit completes the proof of the first statement. a To prove the second statement assume ){>2 . Let u ){ . For every n<3 define v au(n) n X a{2,0}(2). 2 is a Ws -unit and a for some t E Is(a,U) . For every n<3 let JSn U £1 (i5frv n ) {x and let tSli(5lr(V 1 V2 )) {x . For every n} 1ux2} i gEls (r!:s0,J::.) such that g(x Let Q U{V : n<3} . Now O)=x1Ux 2. n we show that Q, and Gd = {x Ux satisfy the conditions of O,x 1 2} Prop. 4.7. Clearly, It is easy to check that every element of G is Q-wsmall in the sense of Def.4.5 (actually, see Def.l.2) and that G satisfies condition (i). Condi- tion (ii) is satisfied since uGcQ Therefore IX is regular by 4.7(1) since every element 9 of G is regular by Also, rlQEls a by 4.7(11) since is simple by [HMTIJ 5.3. Let 'R rl Then R = sg{x Ux . Let lS4-6"oX.[){( xO,O ), Q -o: O'x 1 2} (0,x Ux )} By [HMTIJ 6.2 there is such that 1 2 d -1 Let f = «g z,gy : (y,z )EBOXC). By gEls (Jf ' .( ) and by general algebra then O fEls (,zOxr,.eox[) By f « X ' 0 ) ) = ( 0,X UX and f«0,x1Ux2) O 1 2) d -1-1 Now = (Xo'O ) then fEls(n,n) Let t = riO oh ofohorlQ tEls({X"Q) QED(Proposition 0.7.) Remark: All the conditions are needed in Prop.O.7 because of the following. Let OLE Cs and XEA If x=l then x a X is a Ws -unit iff X=l iff t(X) is a Ws -unit. If x=O a a then X is not a Ws -unit. Let x=2 , base (fJL) =2 and XEA be a norm a Ws -unit. Assume t(X) is not a Gws -unit. Then a a Subb(teX))={{u},2} for some uE2 Thus and t(X) '1. d .. for every i Remark 0.8. By Theorem 0.6 , Prop.0.7 and by [HMTIJ we have a complete description of abstract characterizability of the distin- guished types of units defined in [HMTIJ 1.1. Recall that in [HMTIJ comp seven kinds of units were introduced: Cs GSa' Ws GwS ' a, a, a 0.8. 143 wd, no r m, GWS aGws a Gwsa -units. af these 7 kinds of units exactly one has abstract characterization, and this one is the class of Gws -units. a By [HMTIJ 2.1 exactly those members x of Cs -s are Gws -units a a for which This is an abstract characterization J l of Gws -units as members of Cs -so We call this characterization a a abstract because the property of x is preserved J l under isomorphisms. By Theorem a.6 and Prop.a.7 none of the remaining 6 types of units has any kind of abstract characterizations. Moreover, anyone of these classes of units is even destroyed by automorphisms. Thus they cannot be characterized even if the Cs a in which they are characterized as its members is fixed. (Theorem comp 0.6 shows this for Gs , Gws and Gwsnorm_units and Prop. a a a c omp, wd no r m a.7 shows this for GWS Gws and Gws -units.) a a a The csa-units and the GSa -units have an even stronger negative property not shared by anyone of the remaining 5 kinds of units. By Theorem 0.6, in any full Cs for any nondiscrete Gs -unit a r a there is an automorphism taking x to a GWS a -unit which is not a GSa-unit; hence the same holds for Cs -units. Next we a show that Ws -units do have an abstract characterization as a members of full Csa-s. Let £ be any full Cs and let XEC a Then x is a WS -unit iff [(3yEAt Zd C)X::S:y and (l;Ii,jEa) a since by [HMTIJ x is a Gws -unit iff (\;Ii,jEa) J l a and since At Zd C = {aba s e(.[) (q) : gEl}. Define the J l set of formulas ¢(X) g {(3y)[ A c.y=y A \;Iz( A c.z=z .. (z=oVz<=y» A xsy • A i i,jEa} . By the above, for any full Cs J:. an element aEC is a Ws- a a -unit iff J: 1= ¢[aJ. This is an abstract characterization of Wsa-units in full Csa-s. However, this characterization is not 144 0.9. elementary since the set of formulas contains infinitary formulas. Since Ws C GwscomPnGwswdnGwsnorm we have that Theorem a a a a 0.6 does not extend to any of the remaining 5 kinds of units (from CS - and GSa -units) in its full strength. a Proposition 0.9. Let . (i) ICrs is a variety, i.e. I Crs = HSPCrs . a a a r e g r e g ( ii) I Crs is a quasiequational class, i.e. I Crs a a = SPUpCrsr e g . a r eg r e g (iii) ICrs * HCrs if a>3. a a r e g (iv) ICrs *I Crsr e g iff a>2 iff Crs * Crs a a a a The detailed proof can be found in the preprint [AN6J. A brief outline of proof appeared in [NJ. To save space we omit the proof. Lemma 0.10. Let .e!tES Cr Let x,yEA. Then l::.(x Y) (l::.x) (l::.y). a Proof. Let i E (l::.y) We may suppose i E since is symmetric. Then c.x=x and i.e. 1- Therefore either x'ciY'-y*O or (-x) '(ciY'-yHO . d Case 1 Assume x'CiY'-y*O . Let Z = x'ciY'-Y . Ci(CiX'Y) = ci(x-y) Ci(-(x Y» by x=cix and by (C3). Z x'-Y x y. Therefore 0 * Z (x y),ci(-(x y» This implies ci(x Y) * x Y, since ci(x y) = x Y implies Ci(X $ Y)'ci(-(x $ Y» (C and (C ) 3) 1 Case 2 Assume -x'ciy'-Y * 0 . Let x ciY = cix $ ciY c (x $ Y) by X=C.X and by Eo a 1= i 1- 1= (CiX ciY c i (x $ Y». Therefore 0 * Z s c i (x $ Y) --(x $ Y) shows that iEl::.(x $ Y) 1.1. 145 QED (Lenuna 0.10.) We note that Lenuna 0.10 above becomes false if we replace SCr a by "normal Bo By the above proof, in every Bo satisfying a a (C 1), (C 3) the conclusion of Lenuna 0.10 holds. None of these two conditions CRn be dropped. Actually, in Rny Bo the conrti, tion a a (ciX=X - ci-x=-x) : iEa} is equivalent with the conclusion of Lenuna 0.10. 1. Resrular c:(lindric set al Definition 1.1. Let V be a Crs -unit. Let xCV and HC Then a a x is H-regular in V iff => kExJ. x is regular in V iff [either a=O or x is l-regular in VJ. Let f1. ECrs a Then x is H-regular in a iff it is H-regular in 10.. ()(, is said to be H-regular iff every element of A is H- regular in a and et is regular iff every element of A is regular in Instead of "x is regular in V" we shall say "x is regular" when V is understood from context. The above definition of regularity agrees with [HMTIJ 1.1(ix). A natural question about Gwsa-s is: do regular elements generate regular ones? [HMTIJ 4.1, 4.2 and section 4 here deal with this question. By [HMTIJ 4.1, 4.2 the question arises: which collections of not necessarily finite dimensional elements do generate regular subalgebras. Theorernsl.3, 4.6, 4.7 and 4.9 concern this question. They will be frequently used in constructing regular algebras. Definition 1.2. Let Then xEA is said to be small in 146 1, 3. .eJ(, iff for every infinite KC 6ftx we have a (Vr C 0:) (,38 C = 0 -iU -w K)c(8)c(r)x SnPL g {xEA : x is small in m i . Theorem 1.3. Let be generated by Assume every 0: element of X is regular. Then is regular. Before proving Theorem 1.3, we shall prove some lemmas which might be of interest in themselves. Definition 1.3.1. Let HCo: and fj(,ECA Then 0: a(8)c g {xEA : (lfr ::'W 0:) (.30 ::'W 0:-H)C U)x=O} and g {xEA : . By [HMTJ 2.1.4, .a denotes the subalgebra of a with universe fJI, Notation: The superscript will be dropped if there is no danger of confusion. Sometimes we shall write or Sm(OL) instead of smm. . EI C(6)x H follow immediately by the definition of It remains x,yEI Let to show that X+yEIH whenever Let H a rC 0: -w be arbitrary. We have to show that (x+y)=O a implies that implies that Let 8 g 8 U8 Clearly, 8C 1 2 -w 1. 3.3. 147 a a a a a c(G)c(r) (x+y) c(G )c(G ) (c(f)x + c(f)Y) :s; c(G ) (c(G )C(r)x + 2 1 2 1 a a a + c(G )c(r)Y) = C(G2)C(G = 0 , by CA 1= ci(x+Y) :s; C.X + ciy· 1 1Ur)Y a l We have seen that x+yEI E H The above proves that I H 11ft, by [HMTJ 2.3.7. Proof of (ii): Let Y E IHnDmH be arbitrary. We have to show that y=O YE Dm means that fd = is finite. By H a a have that . But C(G)C(f)Y = C(f)Y since and Therefore c(f)Y=O, i.e. y=O Proof of (iii): Let be arbitrary. Then by Let fC a be fixed. Then by -w xESm arid by . Then BC , i.e. we have seen that -w QED(Lemma 1.3.2.) tlL Lemma 1.3.3. Let Ol.ECA be generated by XCSm Let HCa Then a Dm-eJl.= sg(a) (XnDm ) H H Proof. First we state a fact about ideals and generator sets of CA -s which is an easy consequence of the definition of a ( [HMT ] 2 . 3 . 5) . Fact(*): Let aECAa be generated by XCA Let BE Sua and (Fact(*) is true because is generated by {b/I : and the function (b/I : bEB > is one-one.) Now we return to the proof of Lemma 1.3.3. Let be generated by XCSm and let HC be fixed. Then I E 1101" Dm E Sua, a H H and CDm by Lemma 1.3.2. Hence by Fact (,',) H- H and thus Dm = sg(xnDmH) , by Dm E. H H QED(Lemma 1.3.3.) 148 1. 3.4. Lemma 1.3.4. Let a>O, and xEA. (i) Statements a.-e. below are equivalent. 01') a. x is regular (in 1 . b. x is { i}-regular for some iEa c. x is H-regular for some HC a -w HC a d. x is H-regular for all nonempty -w e. (VqEx) (VkE1U) (CRgqnRgk * 0 and L1X1Q2:.k J => kEx) (ii) Let Hand G be nonempty subsets of a. suppose that the symmetric difference H @ G of Hand G is finite. Then x is H-regular iff x is G -regular. Proof. Proof of (ii): It is enough to prove the following: is H -regular x is HUF -regular). The direction "H-regular (HUF) -regular" is obvi.ous by observing x R -regular] • Let FC a that is L -regular is -w and O*Hca and suppose that x is HUF -regular. We prove that x is H -regular. Let kEV and (HUL1x)1k 2:. qEx Since H*O there Let this b be fixed. Let g Notation: g Since P is finite, we have and by • By 2:. and by HUF -regularity of x we have Then kEx since This proves that x is H -regular. Note that in this direction the assumption H*O was essential (namely: every 0 -regular element is F -regular, but while w1 is 1 -regular in W1Uw{ 1 ) , it is not 0 -regular). Proof of (i): c. => d. holds by (ii) of the present lemma and by the fact that if GCHca then G -regularity of x implies H- -regularity of x Then a. - d. are equivalent since a. => b. => c. and d. => a. hold by a>O e. => d. holds by the definition of H- -regularity. If L1x*O then d. e. can be seen by choosing H = L1x 1. 3.5. 149 Suppose Let qEx and kEV. Then RgqnRgk*O implies the existence of an element b such that oEx kOEV and kOE .• qb b b x kEx. The above prove that e. d .. QED(Lemma 1.3.4.) Lemma 1.3.5. Let and let HC a be nonempty. a (i) H regular elements generate H regular ones in ",/ 0 Ofu,OL • • (ii) If VL is normal then H regular elements generate H regular ones in Proof. Let and let O*HCa. By definition, if an element xEA is H regular then x is H regular too, in any . Let x,yEDm both be H regular. Let G Now G (jl H H Then by Lemma 1.3.4 (ii) and by it is finite by x,yEDmH is enough to show G regularity instead of H regularity. Both x and yare G regular by Let qE(xny) and kEV be such that . Since both x and yare G regular, we have kE(xny). Thus xny is G regular. By this, (i) is proved. Obviously, if (JLEGws then d .. is H regular, for every i, jEa . a l] Suppose a is normal and iEa Let kEV and Then i i for some Further, kiEV because RgknRgq*O b b i i (by H*O), qbEV , and V is normal. Then kbEx by G regularity of x , and hence kEc .x Therefore c.x is G regular. l l QED(Lemma 1.3.5.) Lemma 1.3.6. Let be generated by XCA Suppose a d (¥HCa ) Dm = Sg (xnom ) Then is regular if every element of -H «o H Proof. Assume the hypotheses. Let yESg WI,) X be arbitrary. Let H=lU",y . Clearly yEDm Then yESg ({)I,) (XnDmH), by our assumption. H Every element of XnDmH is H regular, by Lemma 1.3.4 and by H*O. 150 1. 4. Then y is H -regular by Lemma 1.3.5. Since this means that y is regular. QED(Lemma 1.3.6.) Now Theorem 1.3 follows immediately from Lemmas 1.3.3, 1.3.6. QED (Theorem 1.3.) Let HC a, OLECA and xEA. Recall from [HMTJ that x is an a H -atom of a if it is an atom of the Boolean algebra rtHvt of H- closed elements of vt. Corollary 1.4. (i) Every normal Gws g8nerated by its atoms is regular. (There- a fore every GSa generated by its atoms is regular.) no rm (ii) Let OlEGws and let HC a Suppose A=Sg At C1HOL . a Then a.-b. below hold. a. If H is f inite then oOL is regular. b. Let be a set of regular elements. Then Proof. Since A = Cloa we have that (i) is a special case of (ii). Hence it is enough to prove (ii). By Theorem 1.3 it is enough to prove the following claim. Claim 1.4.1. (i) At C1Hm SmQ for every HC a and OLECA • a ( ii) Let HC a and Then every H-atom of a is -w a regular. Proof. Proof of (i) : Let y be an H -atom of d. Let be infinite and let rc a Then is a HUr -atom by [HMTJ -w ccnY lolO.3(i). I I;:::w and by IKI;:::w Then by [HMTJ 1. 5. 151 a a 1.10.5Ci), yicodol which implies cCf)yicodol and therefore n6 6CC Cf)y) = . Thus Kn6CCCf)y)*0 Let i EK Cc Cr) y) a a Then cCf)y > cicCf)Y E which implies c iCC r) y=O, by (II, cCf)yEAt CICHUf)Q We have seen that yESm CNote that [HMTJ 1.10.3, 1.10.5 hold for H -atoms where H is infinite, too. This is true because in their proofs no condition implying IHI Proof of CE) : If a a:2:w Let He a Let y be an H -atom. Then either 6y=0 or -w a else ; and if 6y=0 then c , by [HMTJ 1. 10.5 (i) y s od0 1 and by a:2:w If then y is regular by IHI Remark 1.5. Cl) Lemma 1.3.5 becomes false if we omit the condition a{2} H*O Namely, let be the full Gs wi.t.h unit V a 2u .r: a a2 Let x { fE : f(O)=O}. Clearly, x is 0 -regular in V but a a2)=0 cOx = 2 is not 0 -regular in V , since 6C C2) None of 1. 3, 1. 3.5 Cii) , 1. 3.6, 1.4 and 1. 4.1 (ii) is true for Gws in general. This follows from the following. CSee also a a g prop. 4.2. ) Let a:2:w , I a,<{1} and V g 1Ua2CI) Let -m a • Then l E At1l1, is not regular in V . (3) The condition in Cor. 1.4 Cii)b that the elements of Y be regular cannot be omitted (even if we suppose that is infinite): there exists a CS with some non-regular H -atoms. let a:2:w a aX{OI clearly b is an H -atom of ,[ . Now b is regular iff H is finite. 152 1.6. Remark 1.6. (Notions of regularity) Definition 1.6.1. Let KCCrs Then a o r e g K { CJLEK Of, is O-regular} . Kzdreg { OlEK A) a is o-regular}. i r e g K { {tEK (iaEA) (3iEa) a is {i)-regular} . c r e g K { OlEK (iaEA) C'ViEa) a is {i}-regular} . It seems that among the above notions of regularity the most interesting ones are cregularity and zdregularity. Proposition 1.6.2. Let • o r e g (i) Gwsr e g Gwszdreg GWs i r e g Gwsc r e g Gws a a a a a Gwscomp reg Gwscomp oreg a a i r eg c r e g o r e g ( iii) ICrs ICrs ICrsr e g ICrs ICrs and a a a a a Crs crszdreg Crsc r e g a a a c r e g (iv) ICrs ICrszdreg SPCrso r eg. a a a Proof. Let Proof of (i): GwS r e g Gwsi r e g Gwsc r e g a a a follows from 1.3.4(i). Now we show GwSzdreg C GWS i r eg. Let a a i r eg. EGws -Gws We show By there is xEA a a a a such that (ViEa) x is not {i}-regular. Then and x is not 1-regular. Then there are such that fEx and kE-x. By f*k and we have that there is WESubu(U), I base (W) I 2 such that either fEW or kEW. Since and -x is not 0- -regular, we may assume fEW. Let be such that fEd. Then By aEGws we have and by a Then proves that d'x is not O-regular. Hence by 6(d'x)*0. Crsc r e g C Crszdreg can be seen as follows. a a c r e g Let O(.ECrs and let x E A. Let iEl\.x. Then x is {i}- a -regular by Thus x is O-regular by We show g 1. Let a a X1,1 a x { 1} , v g {O}U{l} and.r ;ffv. 2.1. 153 o r e g ('Vf , kE l.e ) [fO=kO => f=k J. I:. tf. Gws since Then J: is regular by a {O}EC is not O-regular. c omp reg proof of (ii) : Let !J1,EGws Vie show oftE Gwsoreg . Let a a f, kE1'd, Ex. h Q 0 Then h E l.a by xEA, and suppose t.x1kCf Let f kO• c omp dEGws and hEx by l'.xlkCf. Now lUt.xlkc hEx and regularity of a x imply kEx. We have seen that x is O-regular. To save space, we omit the proofs of (iii)and (iv). QED (proposition 1.6.2.) By Prop.l.6.2 we have the following connections with earlier papers. The notion of "i-finiteness" as defined in [AGNIJ,[ANIJ,CAGN2J coincides with regularity in Cs nLf . In Gws nLf "i-finiteness" a a a a Crs nLf lIi_ of [ANIJ ,[AGN2J coincides with regularity. In a a finiteness" of [AGN1J coincides with O-regularity. 2. Relativization Definition 2.1. Let {j,ECrs. Let ZC1f.t and let 1:.= lS8'z. a (i) g Q rlA(Z) Q c xnz : YS=A > • (ii) Rl Q Rl A g Rl(Z) (JL We shall omit the superscipts and A if there is no danger of confusion. prop.2.2 below says that regularity can be destroyed by rlz, 154 2.2. unless both 6[VJ Z=O and ZEA hold. Both of these conditions are needed by Proposition 2.2(ii) and (iii). Proposition 2.2. (i) Let and let Z E ZdeJ!,. Then 'l(QzeJ!, is regular. r e g ( ii) For every a2w there are aECs and ZEA such that a r e g. I Cs a r e g (iii) For every a2w and M22 there are an OLE Cs and M a Z E Zd Sb 10[. such that 'b(QzU is not regular. r e g Proof. Proof of (i): Let fJLECrs and let Z E zdOL Let a Let yER be arbitrary. Then yEA by Z E A. Let iEa. Then Z by ZE and therefore = = l l l = Thus 6(a)y. We show that y is regular in 1G. Let pEy and q E = Z be such that Then q E 1m r e g by ZC1{)(, and (1U6 (fJL)y) by y = 6'10 y. By yEA and -l!tECrs lJl· a we have that y is regular in a, and therefore qEy. We have seen that y is regular in &G, too. This proves that R is regular, since y was chosen arbitrarily. d Proof of (ii): Let a2w Let P = a x l , and Let fJ[, be the Cs with base 3 and generated by {X,Z}. Then a {X, Z} c smflL since and the same holds for X since XCZ. X and Z are regular by 6X=6Z=a. Therefore a is regular I(.ECS since by Theorem 1. 3. Let Then a = Z = a 2. Clearly XER and IJ. x=o (despite the fact that 6e1l. X a) . Since X"O and XH'R. by a2w we have that 1(, is not regular, by [HMTIJ4.3. (iii) of Proposition 2.2 is a consequence of Prop.4.11. QED (Proposition 2.2.) 2.3. 155 About Proposition 2.3 below see CHMTIJ2.10. Proposition 2.3. Crs RQCs 2 * 2. Proof. Let V {( 0,1 ), ( 1,2 ), ( 2,3 )} and 1: 6ff 24. Let mv Then OlECrs We show thatUEfRQCs Vie have 2. 2. A = {V,O,{(O,l)}, {( 1,2 ),( 2,3 }}. Let 'TtEcS2 and suppose ACN. Then x Q {(O,l) }EN and y {( 1,2 ),( 2,3 ) }EN. Then z d '" y'c1 (d0 1·cOX)EN. But z = {< 1,2 )} since 'ltEcsZ' Hence E {( 1,2 ) }ERlVTt for arbitrary rt c s 2 if Then zEfA shows {)[Iif RQ Cs 2. QED (Proposition 2.3.) 3. Change of base About Propositions3.4-3.5 below see CHMTIJ3.11(1)-(3). Prop.3.5 (iii) says that CHMTIJ3.6 does not generalize to GSa-s, but Prop.3.4 says that under some additional hypotheses CHMTIJ3.6 does gen- eralize to GSa' Prop.3.5(i) implies that the condition that one of the bases be finite cannot be removed from CHMTIJ3.6, and 3.5(iv) implies that Lf cannot be replaced with Dc in CHMTIJ3.6. Definition 3.1. 1. Let f be a function. Then we define ! ({foq : qEx and :X is a set of functions) . Note that f(aU) a(Rgf) for any sets U and a. 2. Let Let FE Is(m,h) with DoF=A. ( i) F is a base-isomorphism if (F '" Al! and base (0/,) C Dof) for some one-one f. (ii) F is a strong ext-isomorphism if F U base(-eJL) . a (iii) F is an ext-isomornhism if F rlA(V) for some v C 1 . 156 3.2. (iv) F is a (strong) ext-base-isomorphism if P = goh for some (strong) ext-isomorphism g and base-isomorphism h. (v) sub is the dual of ext, that is, F is a (strong) sub (-base)- 1 -isomorphism if p- is a (strong) ext(-base)-isomorphism. -1 (vi) F is a lower base-isomorphism if F = k "hot for some base- -isomorphism h and some strong ext-isomorphisms k,t. The above definition agrees with CHMTIJ3.5 and CHMTIJ3.15. Lemma 3.2. (i) Let f be a one-one function and let mECrs. Assume tEls eJ(, a and t*QECrs. Let KE{Gws, Gs , Ws , crsr e g, Gwsno r m, Gwswd, a a a a a Ct Ct Then A1t is a strong ext-base-isomorphism,and ()LEK implies t"'{)LEK. (ii) Let a>O. There is and h E Is (Q,U) such that a a h is not a base-isomorphism. ?roof. (i) follows easily from the definitions. Proof of (ii): Let Ct>O. Let x g IwUCtI. u g x+ x d= { qE CtU E Is ({R.,(t) such that h(x)= -x. OL and x satisfy the conditions of CHMTIJ3.18(i)c) because I AI :0; < I U I, and = = Then by CHMTIJ3.18(i)c) there is eWe U such that Clearly z = {qECtW gOEx} and B = Sg{z}. Since x+x c W we have I . Let f E Ww be such that f*x = and fof C Id. Ct Then t is a base- autorrorphism of (Ser ." by CHMTIJ3.1. By t (z) we have t E Let h = rlA(CtW)-lotorl(CtW). Then h E E Is (ffi,Q) and h(x)= -x and hoh C Id. Assume that h is a base- -isomorphism. Then there is k: U >-- U such that -x = hex) = {koq : qEx}. 3.3. 157 (3i QED(Lemma 3.2.) By Lemma 3.2(ii) it is meaningful to ask which isomorphisms of base-isomorphic algebras are actually base-isomorphisms themselves. From the proof of CHMTIJ3.6 it follows that every isomorphism between CsregnLf -s is a base-isomorphism, if one of them has base of power a a Gs -so a Definition 3.3. (i) Let eJLECA said to be of residually nonzerO charac- a for some such that each a J5. has a nonzero characteristic. 1- (ii) Let Ol.ECrs. \'Ie say that is base-minimal if is not a a strongly ext-isomorphic to any Crs except itself. a Note that if every subbase of J:5 has power residually nonzero characteristic. Proposition 3.4. (generalization of CHMTI J 3.6 to GSa -s) Let en., J3 E EGsregnLf be of residually nonzero characteristic. a a ('L) Assume that either {}L is finitely generated or ZdOl. is atomic. Then Of. is strongly ext-isomorphic to some base-minimal Gs a (2) Assume a and .fj are base-minimal. Then every isomorphism between a and is a base-isomorphism. (3) Assume that either eJL is finitely generated or ZdelL is atomic. Then every isomorphism between Ol. and J':5 is a lower base- -isomorphism. 3.4.1. 158 To prove 3.4 we shall use the following lemmas. Lermna 3.4.1. Let mEGsregnLf be finitely generated and of resid- a a ually nonzero characteristic. Then ZdOL is a t.orni c . Further, is finite if OL has a characteristic. To prove 3.4.1, first we establish two other lermnas. Lermna 3.4.1.1. Let mEGsregnLf be of characteristic n>O. Then a a r e g {'}(,C &EGs for SOITle monad i.c-iqnner a t.eo cfs. Moreover, if a is fin- a itely generated then so In fact, there is a function G:A IG I <::n+nl 1'1 x I and xESg (.16)G for every xEA. x x Proof. Let OtEGsregnLf be of characteristic n>O. Let J: g (Sf! lOt.. a a Let U I >-- Subb(m) for some I. Then (ViEI) IUil=n. Let k E n >-- U.. Let xEA be fixed. l Let m g I'1x. If m=O then let G(x) g {x} if otherwise let G(x) g O. Suppose m>O. Then a>O, hence OEa. Let + d x = i ml q qEx}. Let rEITln. Set N g U{aU. : kl.orEx+}. For every r l v U{N : rEmn}. Then IG(x) I<::n+nl 1'1 x I and We show r m d 0 that x E sg(.()G(X). For every rE n we set Zr = n{siYi iEm} . Clearly, Z = {qE lJL We show that x r -1 m = [{Zr·Nr : rEmn}. Set g k o (ITllq)E n. r i Thus k.orEx+, so Clearly qEZ also. Thus qEZ ·N l r r r· Conversely, suppose rEITln and qEZr·N Since qEN there is an r. r I iEI with EaU and k.orEx+. By qEZ we get mlq = k.or for q i l r J some JET. Since miO it follows that i=j and so mlqEx+. By regularity of x we get qEx. We have seen XESg(r)G(X). Assume A = SgX. Let G'-qr(J:.')U{GX: xEX}. Clearly, G and J$ satisfy the requirements of Lemma 3.4.1.1. QED(Lemma 3.4.1.1.) 3.4.1.2. 159 Remark: If in Lemma 3.4.1.1 we replace "of characteristic n>O" with "of residually nonzero characteristic" then the conclusion becomes false. Namely let Then there is dEGsregnLf with all a a subbases finite such that no crsa 16.:'. ot is monadic generated. (Being monadic-generated is not preserved under p.) Lemma 3.4.1.2. Let mECAa be finitely-and monadic-generated. Suppose m is of nonzero characteristic. Then I Zd a I Proof. Let XCA be such that A = Sg X, IXI By [HMTJ2.2.24 we have that Zd elL = Sg (J:5f. ()I.,) C where C = {ax(Y'Z) : yUZ C X, x«a+1)nw}. Let the characteristic of OL be n>O. Then C(x)d(xxx)=O for every Let x>n. Then ax(Y'Z)=O for every since a (Y,Z)Cc( )d(xxx). x - x Therefore ICI Now we turn to the proof of Lemma 3.4.1. Let n Let fJtECA a Then Ot is of residually nonzero characteristic iff anofX'I'o. n Let mEGsregnLf be finitely generated and be of residually non- a a Then cff E zero characteristic. Let n Then Jj c for some f initely and monadic-generated , by 3.4.1.1. n - The characteristic of is nonzero and therefore by 3.4.1.2. Therefore IZd by e . Hence Zd.l:5'n is atomic. 160 3.4. by [HMTJ2.4.3. By : i< 1 Therefore (Zd c6'n is atomic) implies that 'P£ is atomic. Then Zd en. is atomic by Ol. -:;; J:. • QED(Lemma 3.4.1.) Now we turn to the proof of Prop.3.4. Let be of residually nonzero characteristic. Proof ofel): Suppose either a is finitely generated or Zd{lt is atomic. Then Zda is atomic in both cases, by 3.4.1. Let T g g At Zd A. Let Y:T be such that (VaET)[ClY(a) a and IY(a) I /Y(a) I=n. Then Cly(a) C a since a is regular and 6a=O. Let d U = URgY. Let Vl=Cl \ve rlwElsOL. Let Then U. show that 'R g 1tQwm, · d rlWEHQ (Ot,at) by ,'I E Zd Sb let. Let be arbft.rary, Let z = c(t:.b)b. Then zEZd A, by aSz for some a E At Zd A since Zd A is atomic. Then a=c(t:.b) (a·b), hence bnClY(a)*O by Y(a)ESubb({)l.) We have seen that rl is a strong W ext-isomorphism. Clearly, 'REGs We show that is base-minimal. Cl Suppose that VCU is such that r L Let & g (Cl V)1G. Let aET and a' g rl(ClV)rl(W)a. Then a' = Clvnwna = ClVnClY(a) d by Y(a)E Subb (OL) • Let n = [Y(a)1 IVnY(a) I=n which means by n Proof of(2): Suppose OlEGSregnLf is base-minimal. He show that Cl Cl At Zd A = {Cl y : YESubb(U)}. Let YESubb(Ol.) and U g Then hence ClUnb=O for some b E Then 0 * *bCCly. Then c(t:.b)b=Cly since Then At Zd A = {CJ. y : YESubb(U)} by Ol.EGSregnLf . Cl C" Now let be of residually nonzero characteristic. Cl Cl 3.5. 161 Assume that OL and £, are base-minimal. Let h E Is (UL,lj). Let yE Subb (tt) and let a uY. Then a E At Zd A, hence h(a) E At Zd B and therefore h(a)=u\'J for SOMe \"lESubb(.{s), by the above. Then We have I y I (3) of Prop.3.4 is a consequence of (1) and (2). QED (Proposition 3.4.) Proposition 3.4 above and Proposition 3.10 at the end of this section are in a kind of dual relationship to each other. Prop.3.5(i) below exhibits an asymmetry in this duality. Proposition 3.5. (discussion of the conditions in Proposition 3.4(3» (i) The condition" a be of residually nonzero characteristic" is necessary in Prop.3.4(2),(3). Namely: Let a'2:w and x'2:w. Then there are two finitely generated sat- isfying a.-c. below. b. VI, is not lower base-isomorphic to Js. c. There is no Gws sub-base-isomorphic to both eJt and.15. a (ii) The condition" a is finitely generated or ZdeJt is atomic" is necessary in Prop.3.4(3). Namely: Let u>l and ){>o. There are isomorphic which are not lower base-isomorphic. Moreover, no Crs is sUb-base-isomorphic a to both a and J:s (further it is hereditarily nondiscrete if x>l). (iii) "lower base-isomorphisIT\" cannot be replaced with "base-iso- 162 3.5.1. -morphism" in prop.3.4.(3). Namely: Let be arbitrary. Then there are but they are not If then every between GsregnLf -s is a base-isomorphism. x a a (iv) "Lf" cannot be replaced by "Dc" in Prop.3.4(2),(3). Namely: For every a2w there are isomorphic CsregnDc -s which are 3 a a not base-isomorphic. Proof. Proof of 3.5(i): First we prove a lemma. Lemma 3.5.1. Let Q be a CsregnLf with base U. Let F be an a a ultrafilter on I. Let udA d= ud >1_ F F ud ({qEa(Iu / F): (3kEPq){jEI : (k(i). : i Proof. Let everything be as in the hypotheses. Let c be an (F, (U : iEI ) ,a)-choice function such that c(i,Eu)=u for all i In fact, for any aEA we have so it is clear that ga ud(a) . Now suppose kEPq and {jEI: :( (ki ) i QED(Lemma 3.5.1.) We continue the proof of 3.5(i). Let a2w and x2w. Z denotes 3.5.2.1. 163 the set of integers. K g Sb Zq and U g ZxK. We define +: UXZ w U as follows: Let uEU and zEZ. Then u+z g (u(O)+z, u I L) ). x {qECtU ql = qo+l}, y g {qECtU qoCO) E q o(1)O} Qg yU { qECt U qoCO) 2 > max{z 2 : ZEQo(l)O} and qO(O) is odd}. glG'f{CtU){X,Q}, Clearly, x,y and Q are locally finite dimensional regular elements. Therefore no Gws sub-base-isomorphic to both {}t and Since obviously a e . c. b. in 3.5.Ci), this will prove Proposition 3.5.(i). Throughout the proof of 3.5.(i) we shall use the following notations. F is a fixed nonprincipal ultrafilter on w. U+ g wu /F • Let ud be as in the formulation of LeJTlffia 3.5.1. Let {)l+ = ud"'O/, and J:s+ Then m+, Js+ E CsregnLf base cm+) base (.16+) = u+ Ct a' and ud E Is(ffi,oC) , ud E Is C.Is, ll) . A+ and B+ are the universes of O!.+ and J5+. For every b E AUB we shall use the notation + + b+ ud(b) . Thus A+ = sg{x+,y+} and B+ Sg{ x ,Q r , Let gEwU • - d + Then 9 = g/F. We define + : u+xz U as follows: Let gEWU and zEZ. Then g+z d= (gCn)+z : nEw) IF. Claim 3.5.2. There is no Gws sub-base-isomorphic to both OL and a &. To prove this, we first establish two other claims. Claim 3.5.2.1. Let h E Is (J:s,Q) • Then (VqEh(x))qO(l) = ql(l). 1 Proof. We have x's x S; d and f',. (./6) x = 2. Therefore Ott= 2 1 2 1 h(x) and f',.(Q)h(X) = 2. Let bEA be arbitrary F hex) 's2 s d 1 2 such that f',. (l6)b 2. Suppose qEb and qol * q ll. We show that Ol. F d Let (H,y) M q l qo1. For every zEZ i 12• l * let z + =d « z ,« HU{ I l , y »: iEw ) IF.- Let k: u+ u" be a permutation of u+ such that k = {( s(z,H),z+),( ) 164 3.5.2.2. ZEZ} U R1Id, for some R. Such a k exists. + + Now ky =y , since for every we have p(O/E(z,M» E y+ iff p(O,Z+)Ey+ (and hence pEy+ iff since 6y+=1 and y + is regular, by 3.5.1). To see this equivalence, from the definition + of ud it is easy to check that p (0 / E (z, ) Ey iff {jEw : zEH}EF, and p(O/z+)Ey+ iff {jEw: zEHU{j})EF, so the equivalence follows. Also kx+=x+ since if pEx+ then (p(O)=E(z,M) iff p(l)=E(z+l,M» and (p(O)=z+ iff p(1)=(Z+1)+). Therefore A+lk C Id by A+ = sg{x+,y+}. Let p soq. Then pEb+ by qEb and by and k(PO)=PO=E(qO) qoHM k(P1) i" P l =s (q1) by q11=M. Also, kopEb+. Thus since ot is regular and 6b+=2, 2 + 1 + we have E b ·s2b -d and hence t= b+. s1 b+ PkPl 12, ot 2 i d 12· 1b Therefore (J(, b·s ud E IS(Ot,ot) . t= 2 i d 12 by QED(Claim 3.5.2.1.) Claim 3.5.2.2. Let V c aU be a Gwsa-unit and assume that rlV E E Is(ih,bi'Qv!S). Then (VI"lESubb(V»(3MEK) Zx{M} C \1. Proof. Let V C aU be a Gws -unit and let Assume that a rl E Is (.(1,'R) • Let al/p) E Subu(V). Let (m,M) EW. It is enough V to show that this implies {(m-l,M ),( m+l,M ) } c w, Let q d Jg = p(O/(m,M) )(l/(m,M». Then qEV. He have ce e Therefore Ox=c1x=1. 1/, l' '6/, c so (xnV) = (xnV) = V. Then q E o(rlvx)=c1(rlVx)=1, o E E by qEV and this means that qu xnV for some u. Then u = (rn-l,M by q oEx and uEI., Similarly, q E U VJ E C[1 (xnV) lmp'1'les (1m+,M )EW. QED(Claim 3.5.2.2.) Now we prove (c). Assume that theTe is a GWS sub-base-is6morphic to both and a m Then there is an 'R,EGws with unit V and with base Y c U such a that rl E Is (.(5,'R.) and there is f : Y >- U such that v d ?,-1 rl (fV) E Is (.fJl, . Let h = rl(rV) oforlv. Then h E Is (./;5 ,Ol,) • 3.5.2. 165 By Claim 3.5.2.2 we have that C3HEK)zx{r1} C IE Subb(V). We show that (3 LEK) f" (Zx{H}) C Zx{ L}. It is enough to show that (VzEZ)f(z,M)(1) = f(z+1,H) (1). Let uw(p) E Subu(V) be such that Let zEZ. Let CJ p(O/( z,M > ) (1/( z+1,M > ). Then q E Vnx and therefore fog E h(x). Then f(z,t1) (1)=f(z+1,M) (1) by Claim 3.5.2.1. Let LEK be such that C Zx{L}. Let y' Zx{M}. Consider the generator eleP1ent QEB. Let T g {uEy' (.3qEV)qO E VnQ) u and N { uEY' (3qEV)qO E u Then N = since .e is regular and 6Q = 1. ITI=INI=w by the definition of Q and by Y' = . Since h(Q) EA is regular and 6(d)h(Q) 1, by the definition of h we have (VqEuU) and J. Let t w >- and n : w >- f*N be two one-one such that (Vz E EZ)I{iEw Such t and n exist. In fact, well- order and in type w. Define t and n inductively. i i For i even, let t, = "least element of t.. j q(O/n)EFh(Q)+J. Let T {t+z : zEZ} and N {n+z : zEZ} . Then TnN = a by (V'zEZ)I{iEw : t.+z=n,}I u We show that A+1d C Id. Since t and n are one-one and RgtURgn C ZX{L} for some LEK we have (YqEa U+)(YzEZ){q(O/t+z),q(O/n+z)}- - n y + a since l{uEZX{L} : (3qE U U)quaEy} I By the definition of x we have 166 3.5.3. v=u+1). Let u,vEU+ By the definitions of T,N and d we have + v=u+1 iff d (v) =d (u ) +1. Therefore = x . By A+ then we have A+1d C rd. Therefore d h(Q)+= =h(Q)+. Now (VqEaU+)Cq(O/t)Eh(Q)+ and and d(t)=n contradict d h(Q)+ = h(Q)+. This contradiction establishes 3.5.2. QED(Claim 3.5.2.) Claim 3.5.3. Proof. h is one-one and (VzEZ)h(z+1)=h(z)+1}. Let MC Z. Define A G g { hEH M B G g { hEH M Proof. For every nEw is such that URgL=Z. Let MCZ be fixed. Let kEw(WXM) be d ' arbitrary. Let t = PJOok. Let T d < {t(n)+z : zEMnLn}U{ (t(n) + 2 +n)} nEw Then T E wSb Z. Let h(k,M) « (t(n)+z, w ,< Tn,k(n)l ») : nEw) /F zEZ ) . Then h(k,M)EH since EZU+ is a one-one mapping such that (VZEZ) h(k,M) (z)+l = h(k,M) (z+l). We show that Recall that y + and Q+ are regular + + a + elements and /',y = /',Q = 1. Let qE U . Let zEM. Then 2. 2 Therefore h(k,M)EGB M. Let k,dEw(wxM) be such that {iEw : ki*di}EF, i.e. k/F*d/F. We show that h(k,M)*h(d,M). Let D g {iEw : k,td,}. Then l l 3 • 5 • 3 • 2 • 167 k d (\fnED)( k(n)O, (Tn,k(n) 1) ) i- (d(n)O, (Tn,d(n) 1) shows h(k,M)O i- w i- h(d,M)O. By Prop.4.3.7 of [CKJ we have IWx/FI=x since every nonprincipal ultrafilter on W is W -regular. Therefore B W = IGMI = x • QED(Claim 3.5.3.1.) Now we define an equivalence on Sb Z. Let Z. Then we define L Miff (3zEZ)L={r+z: rEM}. Claim 3.5.3.2. Let L,M E Sb Z be such that L. M. Then Proof. Assume RghnRgk * O. Then Rgh = Rgk by h,kEH. Then there is zEZ such that h(O)=k(z)=k(O)+z. Let this zEZ be fixed. Then by the definition of we obtain for every rEZ and qEuu+ rEL q(O/h(r)Ey+ q(O/k(r)+z)Ey+ r+zEM. Then M = {r+z : rEL} showing M L. The proof for GB is entirely analogous. QED(Claim 3.5.3.2.) Let W C Sb Z be a set of representatives for the partition A+ Sb Z For every LEW let G C GA be such that L-L A A+ (VhEGL) (3kEGL )Rgh=Rgk and h=kJ. vie define B+ B G C G similarly. Then still IGB+I = IGA+I x W (since for a L-L LL A given hEG there are only countably many with Rgh=Rgk L see the proof of 3.5.3,2). Claim 3.5.3.3. Then LCZ and hence Then M {z+r: zEL} for some rEZ. Let k £! (u+ (z-r) : zEZ ) . Then k E GA Then there is h E M, E A+ such that GM Rgh=Rgk. This proves existence. Uniqueness follows from the definitions of Wand The proof for is entirely analogous. QED(3.5.3.3.) 168 3.5.3.4. For every v=W let be a one-one and onto function. Let p U{gL : LENl. Then B+ >- U{G : LEW} is a one-one and onto funtion by Claim 3.5.3.2. L A+ Let R {(h(z),p(h)z > hEU{ G : LEW} and ZEZ} . L Claim 3.5.3.4. R : U+ >-- U+ is a permutation of U+ and R E E Is «(JL+,l5+) • Proof. Let uEU+ be arbitrary. By Claim 3.5.3.3 there is a unique hEU{GA+ LEW} such that uERgh. Let u=h(z). Let vEU+. Then L (u,v > ER iff v = p(h)z. Therefore R is a function with domain 1 U+. A similar argument, using GB instead of GA, shows that R- is a function with domain U+. These statements prove that R: U+ >-- >-- U+ is a permutation of U.+ + + + + We show that R x = x and RY = a . By Lemma 3.5.1 we have x+ = {qEau+ : Q1=qO+l}. Therefore R x+ = x+ since u=v+1 iff Ru=Rv+1 holds by the definition of R. Let u=qO' Then and Ru=k(z)J. By the definition of GA we have qEy+ zEL L =* q(O/k(Z))EQ+ =* q(O/Ru)EQ+ RoqEQ+ since y+ and 0+ are regular elements. Therefore R y+ = Q+. Thus RE by A+ = Sg{x+,y+}, B+ QED(Claim 3.5.3.4.) We have that R E is a base-isomorphism between ud*a= =« and ud'\.(6 = &+. Then {]L+ is ext-base-isomorphic to both ell, and cIS. Thus a Actually, QED(Claim 3.5.3.) By these claims, Proposition 3.5.(i) is proved. Proof of 3.5(ii): We shall need lemmas 3.5.4, 3.5.5 below. Lemma 3.5.4. Let There are m,£EBA with such that 3.5.5. 169 -isomorphic to any .r: with 11J:: II lULl . Proof. Notation: For any set H we define to be the unique Boolean set algebra with universe Sb H. We base the proof on the well known result (*) below, see Hausdorff[HJ. x. U:) Let x=lxl and S=2 Then J:fr S BA ;; .a:=. 'P(){) for some elL such that Let be a cardinal. By (*) there exists an m:=. p(y) with aEy+} a BA. Let 1: :=. -p(y+) be BA-freely generated by {xa d with ICI=y+ and IZI=y+. For each aEy+ let v = X " a a Clearly {Ya : aEy+} BA-freely generates some f'(y+), since Izl>y. Let V:='y+ with IVlsy. Then there is an aEy+ such that VCa and hence vnYa=O' Thus Observing 541' + BA ;; 16 completes the proof of 3.5.4. en.;; (y ) QED(Lemma 3.5.4.) Lemma 3.5.5. Let nEw and a,.[ be two CA -s both of cha- a racteristic n , Let .f5:=' 'Pm and f E Ism (£5,'P s:», Then there exists with f C F. Hence (5<}COO Dof ;; (;0lJ::) Rgf • Proof. Assume the hypotheses. Let GC B be arbitrary. Let L -w At G. Then ILI xEQ) E Is(q.,PaEL fUa{V and «x'f(a) : aEL) xER) E Is (4(, P aEL . Let aEL. Then Mn are both of characteristic n, hence by [HMTJ2.5.25 a there exists ha E By [HMTJO.3.6(iii), « ha(Pa) : aEL pEP aEL(Rla0-» E IS(PaEL 'lUaq,PaEL 'R'tfa'R) and hence d F = ([{ha(x'a) aEL}: xEQ) E Is (f{,1(.) • Let aEL. Then F(a)= =ha(a)=f(a) proves L1f CF and hence G1f C F. We have proved statement (*) below. 170 3.5. By [HMTJO.2.14, statement (*) implies the existence of FE E with f C F. QED(Lemma 3.5.5.) Now we turn to the proof of 3.5(ii). Let a>l and x>O. Let y=lyl;:::x+w. By Lemma 3.5.4 there are Boolean set algebras or.;;:,{j' with 11al=y and such that (**) below holds. (1n") (VU::1&)CIUlsy"" rlu9'Is&:J. Define f (U{a({y}xx) : yEx} xEA) and k 2 ( U{ a ({ y} xx ) : yEx} g XEB ) Let .c: f (im) and rt& k( . Then f E E Ism(eR., 1"[.) , k E and E Gs There is n such Ism(J1', L,E} x a g that both .t and C} are of characteristic n. Hence n g f"'A ;;: k"'B g 'P, by Lemma 3.5.5, since 'Pi:. ?- f"'OL - - k'''.,t:s:: 7J l' . Assume that some Crs is sub-base-isomorphic to both a q,. 1t -y.. ()[. ()[. and Then Ibase(q) Islbase(n) 1=lbase(f1 ) 1=11 1·x=y·x=y. Hence there is a with Ibase(V) Isy and rlvEIsP. Then rlv.k E E Is.t!s. Let U (pjO)"'base(V). If rlVk(x)*O then xnU*O since then (3yEx)Vn a({y}xx)*O. Therefore rl EIsJ1'. By IUlslbase(V) Isy, U this contradicts property (k*) of formulated above. We have derived a contradiction from the assumption that is -isomorphic to both and Hence no Crs is sub-base- 1L 1'>. a -isomorphic to both 7t and 1>. Let f be a one-one function and The base-relation a induced by f on AxB is defined to be fAB g {( x,y )EAxB : --=1 : rex) :: y and (f )y C x l , Now: ( ,', ,', ,',) and Jf are lower base-isomorphic iff there is a one- -one function f with fAB E Is(<<,$). By (1::':"') and by the above, 1(, and 'P are not lower base-isomorphic. Proof of 3.5(iii): 3.5. 171 and r e g 1JLu,( there are 5 cases: V'dE{{4},{3,l},{2},{2,1},{1}}. By fJI.';;J:s we have If then and we are done by CHMTIJ3.6. If a then OL ,!:s are base-minimal and we are done by 3.4 (2) . Let Va={2}. If IAt Zd A/=2 then VI. and .!6 are base-m.inimal, and we are done. Suppose IAt Zd AI=l. E IseJt. and similarly for Hence ('t/yE Subb (Ol) ) ('t/v, E E '6U.(a ).15. 'W.(aY)V/,E Cs r e g by OLE GsregnLf and H a a a similarly for Hence we are done by CHI1TIJ3.6. The cases vOlE{ { 2 , 1} ,{ 1} } and x<4 are similar to the above ones. Proof of 3.5(iv): Let Let HC a be such that H2(P)} Let p Hx1 and £ g Define X g {fEa3 H1fE and y g 1.fEa3 : H1fEH2}. We let eJt g {X} and J6 g {y}. Then nDc by and by CHMTJ2.1.7. a a Now we show that a and are regular. To this end, we use Theorem 1.3. Clearly, EGwsno r m. Let xE{X,Y}. Now x was a defined so that x is regular. Now we check that x is sm.all. Let KC6x be infinite and let r C a. Let Then since -w a Then for any gEa3. Hence cic(f)x=O. We have seen that x is small. Then Theorem. 1.3 implies that fJL and are regular. Next we show fJL';;&. Let xE{X,Y}, gEx and Q g a3(Q). we shall use Prop.4.7 to show rl E Clearly and x Q is Q-wsmall by Remark 4.10 since we have seen that x is small. Let r-c a Then fCr/sJEx iff iff -w ' qcr/sJEx. Thus condition (ii) of 4.7 is satisfied. Cond.(i) is satisfied since I {x} 1=1. is simple by ECS (see ll'!.u-(.O .r a CHMTIJ5.3) and Then by Prop.4.7(II). 172 3.6. d Let q = o x l . Then rl E Is elL and by the above. Q Therefore It remains to show that OL and are not base-isomorphic. Suppose f : 3 >-- 3 is such that t E Let Z tX. Then ZEB and, clearly, Zi'Y. For every n<3 let Wn a 3 « n : i we have seen that rl(Wn) E Isis for nE2. Hence f2=2 since W(f2)nZ=O (by W2nx=O). Then fOE2 and f*2c2. Thus rl(WfO)E EIsJ:s and W(fO)nZ=W(fO)ny. A contradiction, since Zi'Y. QED Proposition(3.5.) The algebras 11., l' in the proof of 3.5 (ii) were generated by uncountably many elements. Problem 3.6. Let Do there exist two countably generated isomorphic both with all subbases finite such that they are not lower base-isomorphic? [HMTIJ3.19 refers to Propositions 3.7, 3.8 below. Prop.3.7 below implies that the condition cannot be replaced with any weaker condition in [HMTIJ3.18. With respect to the condition lal$x in 3.7, note that it follows from IAI$x for non-discrete {[. The condition IAI$x was noted (in [HHTIJ3.19) to be necessary in [HMTIJ3.18. Proposition 3.7. Let U be a cardinal. Let the cardinals x and be such that and Assume Then there is an mE CsregnDC such that (i) - (ii) below hold. U a a (i) IAI $x and ;\=u{ Il'>xl+ xEA} . ( ii) For any I'lCU for any Crs if 1\'11 =x then a is not ext-isomorphic to any Cs . Is and hence x a 3.7. 173 Proof. Let u,» and A be as in the hypotheses. Then there is Ivl=v = I I K exists by x Then KC U'. Let n: HU' - U' be such that In*CHK) I>x. Such a function n exists because IHKI = xV>x and IU'I>x. We define HU'}. x {gEau go = nCH1g) and H1qE T {{ gEau qo=u} uEK} . For every cardinal such that let be such that vnL cH and Let u-'- y Q {qEaU : (v'iEL )g.=k}. u u 1 Y {y : and u G Q {x}UTUy. Let Of, Clearly mEDCa since and I v-CHU1) I First we show that is regular. We shall use Thm 1.3. Clearly, every element of G is regular. Clearly, TUy C Sm{t. We show that Thus x E sma. Then by Thm 1.3 we have that is regular. Next we prove A = u{ I"'al+ aEA}, and For every cardinal we have Therefore by YCG and by we have U{ Itlal+ : Next we show that (YaEA) l"'al Then a E Sg GO for some For every finite subset Go of G there is a cardinal such that Le. G cDm 0- CFor the notation Dm see section 0.) Then a enm by Dm E u u u E Suo\' , Le. Then I"'a I It remains to show that IGI=x since ITI = IKI x and IYI I a I x by A ::; s lal+. Therefore IAI xUwulal x , by [ m1T ] 0 . 1 . 19 , 0.1.20, It remains to show that for a ny if /,1/ =x then rlACal"l)'f 'f HomCtJI., 'Rf( aW).() . Let l"cU be such that 11'11 = x . Let V aw. 174 3.8. Let 'R 1rly U. First we show that if K 'I I'J then r ly'9'Hom(m,'R.) . Suppose u E Let Z d= { qEa U : qo=u} . Then zETCA by uEK, [YJ and c However, rly(z)=O by Oz=1. u'9'I'J and hence Co rly z t t rly(coz). Thus rly'9'Hom(m,'R.) if K'IW. Assume KCI'J. I'le show [YJ a H that Co (rly(x» t rly(cox). Clearly cOx= {qE D: H1qE D'}. Therefore HK ={H1q : qEawncox} by KcWnU'. Thus there is qE EynCoX such that n(H1q)'9'I'l, by Inft(HK) I>H-IWI. Therefore q q [YJ a [YJ '9' Co (rlyx) since (VfE Therefore (rlyx) showing that QED(Proposition 3.7.) Proposition 3.8 below show.sthat the hypothesis cannot be omitted from [HMTIJ3.18 even if we replace ext-isomoprhism with ordinary isomorphism or homomorphism. Proposition 3.8. For each and cardinal H>a there is an of power I a I such that => IWI>a]. proof. Let and let U be a cardinal such that U>a. Let be such that IHI-a and Let X {qEau : Then IAI=lal and {}LECs nDc a a since L'>X-llJH. Next we show that for every JjEGs the hypothesis Hom a «()f.,.6) to implies that every subbase of has power >a. Suppose h : a- d E Gs . Let y = heX). Then since .16 a s -d(Q) and h E Hom({){,Is) • Suppose there is a subbase W of .h Oi such that (Then ,"ItO since no subbase is empty if ,-,*0. ) Then there exists a qE'-'W:='f'5 such that Now, q'9'c since (VwEW) =w oY tt Therefore But we have c o(a) X=l (by IUI>,-,), contradicting 3.9. 175 EHom(a,15). h We have seen that HomWl,Js) '*0 and 16 EGs a imply that every subbase of J5 is of power >a. r e g It remains to show that is regular. ECs will be proved a a a in section Reducts, in Prop.8.24, because it uses methods of that section. QED (Proposition 3.8.) We conjecture that [HMTIJ3.18 remains true if the condition "IA I is replaced by the weaker condition "a can be generated by elements and (VxEA) x2: 16x In. In particular: Conjecture 3.9. Let {XEK, S:='base Assume Then we conjecture that Ol is strongly ext- isomorphic to some .,t6E with S:='base(.(s). xK We note that the above conjecture is interesting only in the case when x< I a I. Proposition 3.10(i) below was quoted in [HMTIJ7.30(g). Proposition 3.10 is an algebraic version of the various model theoretic theorems to the effect that elementarily equivalent structures have elementary extensions. Note that (i) of Prop.3.10 below is stronger than the quoted model theoretic result since the elementary extensions in 3.10(i) are identical not only isomorphic). In this connection see also Problems 3.11, 3.13. Proposition 3.10. Let be such that eJL ';;./3. Then a statements (i)-(vi) below hold. (i) Assume Cs and Let a2:w. Then 00 a there is a .t E",cs ext-isomorphic to both Ol and J':5. a 176 3.10.1 (ii) Assume and Then some is ext-base-isomorphic to both vt and .f:s. (iii) Assume and a*i. Then there is some [Ecrsa strongly ext-isomorphic to both {Jl and ,fs. Let no r m, wd, r e g}. KE{ Gwsa' Gwsa GWS aCrs a If PVl:5EK then J:EK. c omp (iv) Assume ·OLlsEGws have disjoint units and base ({)l.) =base (rh) • I a Then there is .,tEG\"Scomp, with the same base, ext-isomorphic a to both Ol and cI::5 • (v) Let There are isomorphic such that no WS a is ext-base-isomorphic to both (JL and .16. -1 (vi) Let hEIs «(}t,;!j) and let 0[,& be as in (i). Then h=k.t for some ext-isomorphisms kEIs(,t,Jj), tEIs(,t,OD and J:. ECs a Proof. Proof of (ii): First we prove a technical lemma. Lemma 3.10.1. (Existence of partial base-isomorphisms) Let and let fEIs (or,is) . Let and Let R Q U{q.*6X. l l Then there is a one-one function t R >- base (.f6) such that Proof. Assume the hypotheses. We shall prove the lemma by induction on n. 1. Assume n=O. Let d Q TI{d.. : q(i)=q(j) and i,jE6x)' lJ ·TI{-d .. : q(i)*q(j) and i,jE6X}. Now d is a term in the language lJ of CAa-s, since 16xl Q {( q(i) ,k(i) : i E 6 x } . Then t : Rg(6x1q) >- is a one-one function. Let be such that Then by t;x1toq:'k. Thus pEf(x) by [HHTIJ1.13 since 6x=6f(x), kEf(x), and f (x)EB is regular by 3.10.1, 177 EA, 2. Let nEw and assume that 3.10.1 holds for n. Let x o, ... ,xn+ 1 Let : Let , be a finite 1 permutation of a such that Such a , exists since are finite by dELfa. Let and a2:w it follows that s , is a derived operation both in -a -ol,ECs one we and see CHMTJ1.11.9. By CHMT J 1. 11. 10, and by a a Ex By CHMT J 1. 11. 12 (x) we have have p n+ 1 iff = D. Let Recall the notation f[H/kJ from section O. We define -1 x. ·S x and Then since {1L is regular. Let h , ) . hiEYi R1- 1 1 Then, by our induction hypothesis, we have a one-one function t 1: R1 >- with the property that pEf(Yi)J. Let R = : and let t: R >- >- be anyone-one function such that Let and let be such that We show that p E E f(x ). i Suppose first Let p' Now Dop'=a since by J3ECs , by Therefore a and Then p'Ef(y since i) Thus p'Ef(x since f(y = f(x ·s,f(x by We i) i) i) n+ 1) show Let H g It is enough to show -1 Now, by and hi = qiCD/qn+1°' J. Therefore Hl t10h .'Ctoh. by 1- 1 Then Hlp' = Hlt 0h = Hlt oh = HltoQi = Hlp (as desired) by the above and by 1 i i t oQ H = p' 6XilP::. i · -1 -1 Suppose next i=n+1. We have Dl(t0 1 Hlp' = Hlt oh = HltohO = Hltoqn+lOT-l HlpoT- by Rg(Hlh 1 O O) -1 CRnR Then pOT Es by p'Es 1. Tf(Xn+ 1) Tf(xn+ 1)EB and Then pEf(x by [HMTJ1.ll.10. n+ 1) QED(Lemma 3.10.1.) Now we turn to the proof of 3.10(ii). Let Cl2:W. Let eJl,h E be of bases U,W respectively. Let Let I be any set such that III 2: I BUWUwI . Let F be a regular ultra- filter on I. Let U+ = I u / F. Let A + = udFEIs (U,U) be defined as in Lemma 3.5.1. Let e Then r e g ud- 1c is a strong ext-base-isomorphism and .(}(,+ECs by Lemma e Cl 3.5.1. Let K : qExEB}. Then IKI IBUWUw I III by .[sELf Let ECF be such that lEI I I I and (ViET) ({ ZEE : iEZ} Cl is finite). Such an E exists since F is III-regular. Let m : K >- E be one-one. Define for all iEI. Fix iEI. Then G is finite. Then i R.C i') since £:s ELf By Lemma 3.10.1 there is t. R. >- U 1.-W c 1. 1. such that (VpEClu) (V( x,q ) EG.) ct. oqCp => pEf(x)J. Let bEv,(IU) be 1. l- d -. - such that To prove boqEud(fx) it is enough to prove { iEI (bq(j) i : j ZEEcF and )EG .. Thus (ViEZ)q": .. Let iEZ 1. - l and Then qjER and hence bq(j). = t. (q(j)). Thus i 1. l t. 0 (Ax l q ) C (bq(j). : j (*) (VxEB) n(x)Cud f(x). Since udof is a homomorphism, (*) implies that b is one-one as the following computation shows. Let v,wEW be different. Then 3.10.2. 179 there lS. pEaWsuc h that pO=v and Thus By (,',) (a+) then b.pEud f(-do 1) = -dOl showing that b(v)*b(w). Then by [HMTIJ3.1 we have and is a base-isomorphism. a Let V = b(aW). Let be fixed. Then bX vnud(f(x)). By (*) we have fe-x) -ud f(x). By we have -ud f(x). Thus bx Vnud f(x). This proves B1b = E A+ -1 n+ d g r Iv.u. d f IS\cv,b':..:,). Thus rl = b. (ud s f ) Els(l)\, ,b"o<.:». Let v d -1 "'" -1 01- = b . Then u = riV. (B1b) Els (at ,15) is a strong ext-base-iso- morphism. We have proved (ii) since is strongly ext-base-iso- morphic to both Ol. and 1.5. We have also proved Statement 3.10.2 below, which shows the "(ii)-part" of 3.10.(vi). r e gnLf Statement 3.10.2. Let eJ1,/sEC s and f Els(00,OL). Then there a a r e g is a Cs and two strong ext-base-isomorphisms eEls (eJ1+,a) a m+ Proof of (iii) and (iv) : Let m,fsECrs have units V and ,'I a and have bases U and y respectively. Assume {j[';;!S and VnW=O. If a=O then 0l.';;15 implies fJt=/5 and we are done. Let fE Is(Ol ,f5). Let ;j' with unit 'Jl.M. Let X g {xUf(x) be the full Crsa Then XESu:2) by f EHom(Ol.,l5) and VnW=O. Let £ be the Crs with unit VUW and with universe X. a Uny=O) Wl,15EGvlS comp and Suppose that either (a22 and or a J:: U=Y). Then = = O. Therefore rlVEIs (J: ,ot) and 1:. rlwEls(.t,!s) since by f EIs(Ol,1:5) we have x=o iff xUf(x)=O iff f(x)=O. I.e., ;: is ext-isomorphic to both til and ./5. If a,,tSE c omp E Gws and U=y then .tEGwsc omp with base U. So far, (iv) a a has been proved. a22 and Uny=O. Then and shOWing that £ is strongly ext-isomorphic to both and .l5. Let KE {GSa' GWS GwSnorm ' Gwswd} If on "EK then a, a a. 180 3.10. clearly cfEK. Let m,.f!,ECrsr e g. Ive show that is regular. Let a yEC and pEy, be such that There is xEA such s: J: that y=xUf(x). By rlvEls(.(,Ol.) and 6y=6x=6f(x). Suppose pEx. Then qEV by p(O)=q(O) and base(V)nbase(W)=O. Therefore qEx since x is regular and Similarly pEf(x) implies qEf(x). Proof of (v): Let a2:w, p=a1Id,d q W aa(q). Let .15 0ftII,( (;'6' W). Then ut;;; Is by cHMI'IJ 3.22. Suppose £. EWsa is ext-base-isomorphic to both Ol. and J's. I'le shall derive a contradiction. Let {=au(r). Let the corresponding sub- base-isomorphisms be induced by f: a >- U and g : a >- U, i.e. let rlJ: (tV)Els(r,t*a) and Note that base(a)CDof and base (k,) :':Dog. Then "u (r)nrV;fO and "u (r)ngW;fO. Let p'EaU (r)n'tV and q'EaU(r)ngw. Then p'=fop" for some p"EV=aa(P) and therefore IRgp'l IRgp'l Proof of (i): Let be of bases U,Y respectively. Assume .eJl ;;; J's r Uny=O and a2:w. Let x be a cardinal such that x=x 1a l and x>IAUBUuuYI. Then by CHMTIJ7.25(ii), eJl and are sub-isomorphic to ot, .(s+ECs respectively such that Ibase(dt) 1= a =1 base (.(s+) I=x. Then rl (aU)Els wt ,00 and Let H uuyux. Then IHI=x and therefore there are ';)', qJ!Ecs ext- a -isomorphic to -or. , f:{ respectively and such that base( ;;)1) = base( f}') = H. Let V u{ aH(p) : pEaU} and W u{ aH(p) pEay}. Since VEZd SbaH we have EHo (3)', ,3 ) for some Crsa ;:; with unit V. Then rlDCaU)EIS(';J,f)L) by CHr1TJO.2.10(ii) since by D' aUCV we have rlD(au) =rl (aU)Els( :J', OL). That is, .vL is sub-isomorphic to ;j EG\·'Scomp. Similarly, ,J5 is sub-isomorphic to a some with unit W. By Uny=o and a2:w we have 3.11. 181 vow-c. By we have and There- fore there is 'l1EGws comp with base H ext-isomorphic to both .::J and a {}; this was proved as 3.1Q(iv). comp By (1) in the proof of [HMTIJ7.17, every Gws is sub-isomorphic a to some CS Therefore, by there exists ECs ext-iso- a' s: a c omp. morphic to n E Gws Then J: is ext-isomorphic to both ell and 00 a since n is ext-isomorphic to both Q and £ and the composition of ext-isomorphisms is again an ext-isomorphism. QED (proposition 3.10.) r e g Problem 3.11. Let Let Vl E Cs and let 01.-;;',,$. Does , I:J 00 a r e g there exist ECs ext-base-isomorphic to both and f6 ? a a We shall need the following definition in formulating Problem 3.13 as well as in subsequent parts of this paper. Definition 3.12. Let with base U. Let F be an ultrafilter a on 1. Let c : Then Problem 3.13. Let t':sECS Assume VI. -;;; ,/$. Are there ultra- Vt, a. filters F,D, an (F, are base-isomorphic and udcF E Is VI., ud dD E IsJ:s? We know that the answer is yes if Ut,J:sECSregnLf. A positive a a answer to this problem would be an algebraic counterpart of the Keisler- Shelah Isomorphic Ultrapowers Theorem. See also 3.10 and 3.11. Let hEIs(vt,i5). Are there F,D,c,d as above such that h B -1 A . (UddD) ofoUdcF for some base-isomorphism f between udcF"Vl. and i uddD ' 16? By (1) in the proof of [HMTIJ7.17 and by the proof of [HMTIJ7.13 182 3.14. c omp we have that every Gws is sub-isomorphic to some Cs and every a a reg Hs is sub-isomorphic to some Cs . Below we give a rather direct a a c omp and simple construction of sub-isomorphisms from Gvrs -s into a Cs - s and from Ws -s into Csreg_s. This construction is useful e.g. a a a c omp, when to a given concrete mE Gws x c omp Theorem 3.14. Let a,x be ordinals. Let W be a Gws -unit with a base x and let pEW. Let F be an ultrafilter on I Sb a. Define w {rEI: pcr/qJEx}EF} x c W)• rEI}. Then (i)-(iii) below hold. (i) Let x and {pCfl,x/sJ xEA, sEV} c ax(p). Let f rlV.h. Then fE EIsm( eJt, I5(rV), f is a sub-isomorphism if p=.V, e'VL is regular and f = ( {qEV pLfI,X/qJEX} xEA) . d a Proof. Assume the hypotheses. Let g = ({qE x : (rEI: pcr/qJEx}EF} Claim 3.14.1. Let x C Q and i, jE a . (i) x::'gx. CQJ a ; l' ) Q ( gEHom « Sb ,n , , D i j') (Sb x ,n a x r D(x)i j» . 3.14. 183 (iii) If H Proof. (i). Let x:::Q and qEx. Let r DO(p--q). Then Irl Then qEgxngy iff qEg(xny) since F is a filter. qEgx iff {rEI: pcr/qJEx}EF iff {rEI: iff since {rEI: {i,j}:::r}EF. (iii). Suppose H QED(Claim 3.14.1.) d Proof of (i): Let x Proof of (ii): Let H by 3.14.1(iii), hence gx = : {rEI: pcr/qJEx}EF}= =hx. Now 3.14.1(iii) completes the proof of (ii). Proof of (iii): Let C a x and f be as in the hypot- heses of 3.14(iii). First we show that fx = Vnhx = {qEV : pCL',x/gJEx}. d Let qEV. Let s = pCL',x/qJ. Then sEQ by the hypotheses and there- g fore r ) is finite. Let Z g {L',EI : rCL',} . Then ZEF and (V L',EZ) pCL', / s J=s. By t,x1s::;,q and regularity of fJL we have pCL', / s JEx 184 3.15. iff for every Now, qEhx iff : iff and iff We have seen fx={qEV Next we show fEHOr:l(Vl,(;'&V). By 3.14.1(ii) and by it is enough to show that f is a homomorphism w.r.t. the cylindrifications. Let iEa, xEA and qEV. Then qEf(cix) iff iff iff (]bE x) p[ { UUAx / iff i (3bE x ) p Lzxx / qb JEx iff iff qEcifx, by c;x)v=v. Let qEV and Let f : qi"si}' Then Ifl SEQ. Thus q' g q[f/sJEV by ( x )v=o. Now JEx by C :: J. r .«. q'Efx. We have seen that fEIsm( fJl, CSfsV). If pEV then Q::V and thus which implies that f is a sub-iso- morphism. It remains to show that f*m is regular. Let xEA, sEfx and qEV be such that C q. By we have = and therefore qEfx by sEfx = {gEV : QED (Theorem 3.14) Corollary 3.15b) below is a generalization of [HMTIJ3.22 and 7.27. For the necessity of the conditions in a) and b) see [HMTIJ7.30a)b). Corollary 3.15. a) Every is sub-isomorphic to some with the same base: and every with nonempty base is ext-isomorphic to some Ws nLf Thus H(Ws nLf ) = I CsregnLf . a a a a a a b) Let OLEWs have unit au(p). Let HC a be such that ('r/xEA) a and let qEaU be such that H1P::q. Then fJL is iso- morphic to some J':s EWs with unit au(q). Moreover a : xEA ) EIs ({Jt , J!s) . Remark 3.16. (i) Thm. 3.14(ii) becomes false if the condition x (ii) By Prop.5.6(i) we have that for all there is VE E Zd Sb such that h does not preserve regularity. Hence Thm.3.14 F (ii)-(iii) do not extend to (Gwscomp) reg from Ws . a a Remark 3.17. (1 ) Let Let Lea. Then there are {)LE Ws nDc and a a Ws such that (i)-(iv ) below hold. a (i) Cs nHeJt = Cs for all )J a 0 a (ii) ('I 11. ) (Vcf=1'lL ) I base rn (iii) If then c!5 EDc . a (iv) The proof is an easy modification of [HMTIJ7.30a). The basic change is to replace "max{wq : O<)J (2) If we replace "wsa" by "K" in [HMTIJ7.30a)(4) then we obtain statement (*) below. There is Ol EK with IAI such that for all ,l5 EKnHeJt Let Then (*) is true for KE{Ws by the proof of a, [HMTIJ7. 30a) . Let K = Ws nDc . Then (*) is true iff + . In a a particular: let lal=w, and mEWS nDc with Then there a a are and gE 1..(5, with Rgg=base ). The r e g• proof goes by iterating the proof of [HMTIJ3.18 and using Ws CICs a- a We omit it. The case of is immediate by 3.17(1)(ii)-(iii) above, choosing 4. About Propositions 4.1-4.3 below see [HMTIJ4.2 statements (1)-(4). 186 4.1. Proposition 4.1. Let J:. be a full Gwsa. Then (i) and (ii) below are equivalent. (i) £ is norMal. (ii) Every subalgebra of J:. generated by a set of locally finite dimensional regular elements is regular. Proof. (i) (ii) follows from ThM 1.3. Next we prove (ii) (i) . Suppose .J:. is a full Gws and J:. is not normal. We shall exhibit a a regular xEC such that 6x=1 and cOx is not regular. Since £ is not normal we have a?:w, by [m1TI J 1. 6; furthermore, there are two subunits ay(p) and aw(q) of .c such that ynw*o Let bEynvl Let the unit of .c be V. d = {kEV : k(O)=a}. Then xEC since is full, and 6x=1 since PoEx and Clearly, is regular. a x Now we show that is not regular. 6(C by [HMTJ1.6.8. OX)=O, qb°EV by and since and 6[VJ(aW(q))=O by the definition of subunits (see Def.O.1). Thus At the same time o o 0 since PaEx. Now the two sequences qb and Pb are both in v, they coincide on but one of theM is in while the other is not. Thus is not regular. QED (Proposition 4.1.) Proposition 4.2 below implies that there is a Gws the minimal a subalgebra of which is not regular. It also implies that the condition "full" is necessary in Prop.4.1. Proposition 4.2. Let mEGWs Then (i)-(iii) below are equivalent. a. (i) 'IT\J..I, ( {)L ) is regular. (ii) (iii) c( ){)d(){x){) is regular (in vt) for every x Proof. vie may suppose a2w by [HMI'IJ1.17, 1. 6. Let -vt EGws with a unit V. Let Subu(V) = {ay. (pi) : iEI}. Let x First we prove (ii) # (iii). Suppose that (ii) holds. Let x Let kEax' gEV and k(O)=q(O). Suppose Then by kEa Then IY 12X by (ii) since y.ny *0 by k (0) =q(O) . x n l n Therefore qEa i.e. a is regular. Suppose that (ii) fails. Then x' x there are two subunits ay(r) and aweS) of V such that ynl'NO and IWI>IYI Suppose (iii), i.e. suppose that is regular for every a x x Let X {ax,di j : x QED(Proposition 4.2.) Proposition 4.3. Let a2w. Then (i)-(ii) below hold. (i) The greatest regular Lf subuniverse of a Gws need not exist a in general. Namely: There are an (Jt EGws a and elements x, y of Ol such that 6X=6y=1 and both {x} and {y} generate regular subalgebras in but I x , y} does not. (ii) The greatest regular Lf subalgebra of a Gws may exist even a if regular elements do not generate regular ones. Namely: There are an Ol EGws and xEA such that 6X=1, x is regular, a cox is not regular and 'l'tlM,(U) is the greatest regular sub- algebra of Ol 188 4.4. d g Proof. Let a<::w, ){, r = ( a ){ and let V g aa(p)Ua(a+a) (r), .r g i5&V. g Proof of (i): Let x g {qEV even} and {qEV : <:1 0 is y (<:1 Ea <:1 is odd) and (<:1 <:: a is even)}. Now Sg{x} and 0 0 0 <:1 0 Sg{y} are regular. This can be seen by using 1.3.5Ci) and CHMTJ2.2.24 since However, {x,y} is not contained in any regular sub- algebra, since aCa+a)(r) which is not regular. Ci) is proved. Proof of (ii): Let x {<:1EV : <:1 <:: a } and {xl , Now 0 a g x is regular and c x = aCa+a)Cr) is not regular. We show that 1'ttI\( {It) is the only °regular sUbalgebra of a 'ffiII, ( a) i s regular by Prop.4.2, since la+al=lal. Let yEA, vie show that y generates an irregular elenent. By CHMTJ2.2.24, A = = sgC08tVL){x"d" : i,jEa}, where Xl' g {qEV: <:1 Then yEA l lJ l,<::a}. implies that y L C IT IT -xv(i,j) j Case 1 (Vj Case 2 (3j Thus, in both cases, y generates a(a+a)Cr) which is a non-regular element. Thus Cii) is proved. QEDCProposition 4.3.) Remark 4.4. By Prop.4.1 we have that locally finite dimensional 4.4.1. 189 regular elements generate regular ones in every normal Gws . a If we do not suppose normality of Gwsa -s then cvlindrifications can destroy regularity of locally finite dimensional elements, but only cylindrifications.: See the counterexamples in 4.1-4.3 and see 1.3.5 which implies that locally finite dimensional regular elements always generate regular ones in the "cylindrifications-free reduct" (A,+,',-,O,l,d,,), 'E of VtEGws. (Note that in Crsa-sonly lJ r , J a a negation (apart from 0,1) preserves regularity of locally finite dimensional elements.) If we do not require locally finite dimensionality then again the Boolean operations can destroy regularity, already in Cs .: a Proposition 4.4.1. Let a2w and x>l. There is 01. ECs nDc of a a base x such that regular elements generate nonregular ones in in fact there are disjoint regular x,yEA such that xUy = x Y is not regular. Proof. Let a2w and x>l. Let HC a be such that d Y = Let be the Cs with base x generated by {x,y}. m a 6x=H, 6y=H and hence x and yare regular. However = xUy = R tit is not regular since 6R=0 and REl'{l,O}. QED (Proposition 4.4.1.) To construct regular algebras we shall frequently need Propositions 4.6, 4.7 and 4.9 below. They are closely related to Thm.l.3 and they address the question "which (not necessarily finite dimensional) regular elements generate regular ones". Definition 4.5. Let V be a Gwsa-unit and let Then x is defined to be Q-weakly small (Q-wsmall) in V iff for every infinite KC6[VJx we have 190 4.6. (\if C a) (VqEQ) (38 C -w -w x is said to be weakly small (wsmall) in V if x is V-wsmall. Note that weakly smallness is a weaker property than smallness, since x is small in V iff for every infinite we have (Vf ::'W a) (30 ::'W K) (VqEV) c(0) {q} 1 c(r)x. Theorem 1.3 says that small regular elements generate regular ones in normal Gwsa-s. The next Proposition 4.6 says that weakly small regular elements generate regular ones in normal GWsa-s, if they (the generator elements) are "very disjoint". no r m Let {JtEGws be generated by a set G of Proposition 4.6. a weakly small regular elements. Assume that (Vx,yEG) (x*y a is regular. Proposition 4.6 is a special case of the next Prop.4.7. Recall the notation Dm from def.l.3.1. H proposition 4.7. Let with unit V be generated by a Let Q_CV,u A[VJQ=Oan d suppose th a tItevery e emen 0 f G is Q- wsmall. Assume conditions (i)-(ii) below, for every yEG and f C a. -w (i) c(f)xnc(f)y=O. (ii) (VfEy) (3qEQ) (Vp) [f r r /pJEy iff q[ r /pJEyJ. Then statements (1)-(111) below hold. (I) elL is regular if every element of G is regular. ex (II) rlQ is an isomorphism, if is simple, and if (VyE (III) (VH::'a)DmHnIg({}{,) To prove this proposition, we need two lemmas. 4.7.1. 191 no r m Lerruna 4.7.1. Let a2w and let I'Jl EGws be generated by GCA. a OL Let Q::'y9 satisfy conditions (i) and (ii) of Prop.4.7. Then for every zEA, statements (I) and (II) below hold. ( II) There is 8 C a such that for every 8 ere a we have -w "-w (VfEy) (3qEQ) (Vp) [f[ r /pJEZ iff q[ r /pJEZJ. To prove Lerruna 4.7.1, we need in turn a definition and two lemmas. 4.7.1.1, 4.7.1.2 below will be used in subsequent parts of this paper too. Definition 4.7.1.1. Let p s >- a be one-one. Define rbP g ( « f ., 1£ f is a function and RgP::'Dof). Pl Crs -unit. Lerruna 4.7.1.2. Let P: S >- a be one-one. Let V be a a Then (i) and (ii) below hold and rdPV is a crsS-unit. (i) rdP E IS('b?,VPCSf:rV, G'6'rd PV) and rb" V >-- rdPV, if s*o. (ii) If V is a Gws -unit with subunits iEI} then a d Proof. Let P:S >- a be one-one. Let H = Rgp. Let V be a crsa- -unit. Notation: For every gEV we denote g' g Proof of (i): Let qEV be fixed. Define V(q) g {gEV : g' ::. q} and Y(q) g {gop gEV(q)}. Let f(q) g ({gop: gEx} xESbV(q) ). Then f(q)EHO('R.VPG&V(q),Gh(q» by [HMTIJ8.1., since (\,fgEV(q»(gop)+= d = g and Rgf(q) = SbY(q). Let b(q) = «u,q') : uEbase(Y(q») Then b(q) is a one-one function on base(Y(q» and therefore b(q) r------defines a base-isomorphism b(q), see [HMTIJ3.1. 192 4.7. L 3. Let W(q) = b(q) (Yr q ) and h(q) = b(q)of(q). Now (5%W(q) ) and h(q) = SbV(q)lrd P since h(q)x = b(q)(f(q)x) = = b(q){gop gEx} = {b(q)vgop : gEx} = {rbP(g) : gEx} = rdPx for every x C V(q). Let W g U{W(q) : qEV}, it g 1W P G:&-V, n g !%vl. Let g,qEV. Then clearly q'=g' iff V(q)nV(g)*O iff V(q)=V(g) iff W(q)=W(g) iff base(W(q»nbase(W(g»*O. Therefore t:. ( 1t ) W( q ) =0 i i i because if iES, gEW(q) , gaEW then gaEW(q) since RggnRgga*O by i a>1. Also, t:. ( 1;(, ) V(q ) =0 since iEH, then giEV(q) if gEV(q) ,gaEV a by g'=(gi), . Therefore 1( a we may apply [HMTIJ6.2 to and n. By [HMTIJ6.2, [HMTJO.3.6(iii) and by t5&W(q)) we obtain that rdPEHo ('it, n). rdP is one-one because is one-one on V. Proof of (ii): Let {ayiPi) : iEI} = Subu(V). Let qEV. Let J g {iEI : q[H/p.JEay!pi)}. Then V(q) = {q[H/gJ (3iEJ)g E Hy!Hlpi)} 1 1 1 op) and therefore Y(q)=U{Sy>pi iEJ} . Let W. g Sy>piop) for iEJ. 1 1 1 Let gEwinwk for some i,kEJ. Then and therefore ayiPi) = which implies Wi=Wk· This shows that y(q) is a Gws -unit with subbases {Sy!piop) iEJ}. This immed- a 1 iately yields Lemma 4.7.1.2(ii). QED(Lemma 4.7.1.2.) and HC a. Define the function t(f,q,H) V- - aU as follows. Let sEV. Then £CH/ S J if 1q::s t(f,q,H) (s) :[H/SJ if l£::s. { otherwise Lemma 4.7.1.3. Let V be a Gwsa-unit, a2w. Let f,qEV be such that the bases of the (unique) subunits of V containing f and q coincide. Let H C a and P : IHI >-- H. Then t(f,q,H)* E -w 4.7.1. 193 Proof. Let V,f,q,H and P be as in the hypotheses. Let Subu(V) Ct y (,p i ) 'EI} { l : l • Notation: For every gEV we denote if w=' Let W rdPV. For every wEbase(W) define b(w) \:! < u,f'> if w='< u,q> { w otherwise Then b : base(W) >-- base(W) and b(W) = W since fCH/pJEV iff qCH/pJEV and vi = u { IHI (y, x { g , }) : iEI, gECty(pi) } by Lel'1ITla 4.7.1.2 l l (ii) and by IHI Lemma 4.7.1.2(i) we have that rdP E Is(m p g&V, Qrl'l). Therefore it is enough to show t(f,q,H)* = rd P- 1 obordP. By rdP =' rbP* it is enough to show t(f,q,H)g = rbP-1(borbPg) for every gEV. Let gEV 1(bo« be such that ,q'}. Then rbP-1(borbPg) = rbP- g "g'> pl p-1 iEIHI» rb « QED(Lemma 4.7.1.3.) Now we turn to the proof of Lel'1ITla 4.7.1. Let with unit V, A =' Sg G and assume that ncvcv satisfy conditions ---'- (i),(ii) of 4.7.1. Let zEA. It is enough to prove (II) for z, since (I) follows from (II). Let 8 be a finite nonempty subset of a such that z E Sg( 0 m)G. Let 8 c r a be arbitrary. Let fEy and let qEQ be such that (Vp)Cfcr/pJEy iff qCr/pJEyJ. Then since Then the bases of the subunits of V containing f and q coincide since V is normal and r*o, Lemma 4.7.1.3. By definition of t(f,q,r) we have that t(f,q,r)*x='x iff (V'p)CfCr/pJEx iff qcr/pJExJ. Therefore G1t(f,q,r)-:' c Id since t(f,q,r)*y='y by the choice of q, and if then 194 4.7.2. subalgebra of generated by G. Then and zER by z E sg( WP {)l)G. Then i.e. (Vp)Cfcr/pJEz iff qcr/pJEzJ. QED(Lemma 4.7.1.) no r Lemma 4.7.2. Let {]l EGws l'1 with unit V be generated by GCA. a Let ",[VJQ=o. Assume conditions a. and b. below. a. (\1'zElg ({)l) znQ;"O. b. For every HC el and zE Ig ( fll ) we have that ('iqEznQ)('t!0 a)(38 Then statements (I)-(III) below hold. (I) Ell is regular if every element of G is regular. (J[ ( II) rlQEls.eJL, if TffiI.( Ot) is simple and (VyEG) (III) (\1' H'::a) Ig (m ) (G---Dm n DI'1 O} . H) H=( Proof. Assume the hypotheses. Proof of (II): Suppose 'l/fl(( {)l ) is simple and (VyEG) I"'y I Let Then x=dEllz for some dEMn({I() and zElg( eJL)G, by [HMTIJ5.1. 2ase 1 zEMn( -ell). Then xEMn( -ell) • By '" [VJQ=O we have rl E Q : Ho(m.u.( {JL) ) '. We have that 'ffilt( {)l) is simple. Therefore xnQ = rlQ(x) ;" 0 by xEMn(VL )--{O}. 2ase 2 If d=O then By condition a. and we are done. Suppose d;"O. Then dnQ;"O by dE If then qEd-z C C dEllz=x and we are done. Suppose gEz. Then gEznQ and by condition b. we have that (38 £ z. Let qCO/pJ z. Now q[8/pJE Ed since qEd and 8n"'d=O. Therefore q[8/pJEd-z':: dEllz = x. By qEQ and c(8)Q=Q we have q[O/pJEQ. Therefore qCO/pJExnQ showing that xnQ;"O. (II) is proved. Proof of (III): condition a. Let gExnQ. Let 0 C a. Then -w 4.7.2.1. 195 (3 o C -w 1 x, by condition b. Therefore enL\,x oF 0 by qEx, and thus L\,x HUrl. Since rl C a was arbitrary, this shows -w i.e. x 'f- Dm H· This proves (III) . To prove (I) we need a lemma: Lemma 4. 7 • 2 . i. Let fj[ECA be generated by GCA and let HC a. a Consider conditions (i)-(iii) below. (1) (ii) = Sg( OL) (iii) Sg(GnH-dim):: H-din, where H-dim g Mn( OL)U{xEA : lL\,xalHI Then (i) (ii) and C(VHESba) (i) holds] (VHESba) (iii) holds. Proof. Let {}1,ECA and let HC a a. Proof of (i) (ii): Assume DmHnI={O}. By Fact(*) in the proof of 1.3.3 we then have Dm 2:s g( Vl) H we have Dm = sg(GnDm H H)· Proof of (i) (iii): Suppose a and G satisfy (i), i.e. ({]l ) = {OL Let G' cG be arbitrary and let a' g g First we show that G' and OL' satisfy (i), too. Let Dm g Dm VL'). = {yEG' : G--Dm and by {ll' C H' H( H, a then Ig( :: By Dm 2: Dm and by H' H Dm = {a} then we have Dm 'nIg( {O}, H H H H as desired. Now we turn to the proof of (iii) . Let G' g GnH-dim. Let xESg G' be arbitrary and let K g L\,x. We show that IH al KI2W implies xEMn ( en i , Suppose IH al K I2w. Now (1) G'nDmK::Mn(eJL). For, note that H-dim C Dm and hence Sg(H-dim) Dm So xEDm H, 2: H. H is finite and hence is infinite. If yEG'nDmK then so is infinite. Hence yEMn(VL) by 196 4.7. Thus (1) holds. Since G' and ffi' = satisfy (i), we have that Dm {Il') = sg( {)l')(G'rlDmK( eJl.'» Mn( ot'), by (i) => K( (ii). By xEDm m') then xEMn( m'). We seen Sg(GrlH-dim) C K( C H-dim. QED(Lemma 4.7.2.1.) Now we can prove (I) of 4.7.2. By (III) and by Lemma 4.7.2.1 we have eJL EGws no r m we can apply a Lemma 1.3.6 which yields that m is regular if every element of G is regular. QED(Lemma 4.7.2.) no r m Now we turn to the proof of Prop.4.7. Let {]L EGws be gen- a erated by GCA. Let QEZd g&lm . First we show that conditions (i)- (ii) of 4.7 imply that 4.7.2.a holds, i.e. (VZEIg(Q)G--{OI)zrlQ*O. Let zEIg({]t)G, Z*O. Then zrlc(r)Y*O for some r a and yEG. The conditions of Lemma 4.7.1 are satisfied by Qrly and y. Therefore 4.7.1 (I) says that (3f a)zrlc(f)Q*O. By we have Q = c(f)Q, therefore zrlQ*O. Assume now condition (i) of 4.7 and assume that every element of G is Q-wsmall. We show that then condition b. of 4.7.2 is satisfied. Let HC a and zE Ig (Vl ) (G--Dm Let qE zrlQ and (l a H). be arbitrary. By zEIg ( {}L ) we have that zsc (r)::::: Y for some f a and Y G--DmH. By qEznQ then qEc(f)ynQ for some yEY. d Let K = By yqDmH we have tKt2w. Then (38 K)c(8){q) c(f)Y' since Y is Q-wsmall. By condition (i) we then have c(8){q} z, since z C [{c(f)y : yEY} and c(8){Q} c(8Uf)y by qEc(f)Y' We have seen that condition b. of 4.7.2 is satisfied. Hence 4.7.2 yields the conclusion of 4.7. QED (Proposition 4.7.) 4.8. 197 Definition 4.8. Let V be a Gws -unit and let xcV. a (i) Let qEV, rCa and KC a. Then x is defined to be -w (q,r,K)-small iff C30 c c -w -w (ii) x is defined to be irreversibly-small (i-small) in V iff ('VqEx)(vr c a) (V'K:='il.[VJxHIKI2:w '* x is (q,r,K)-smallJ. -w Note that i-smallness is a stronger property than wsmallness. Proposition 4.9 below shows that the condition of disjointness can be eliminated from Prop.4.6 if we change from wsmall to i-small. Proposition 4.9. Every normal Gws generated by i-small regular a elements is regular. Proof. We may assume a2:w by [HMTIJ1.17. Before giving the proof of 4.9 we need a definition and a claim. Definition 4.9.1. Let otEGws and xEA. Tnen x is said to be a (;:) -small iff Note that appears to be a stronger property than i- smallness. Claim 4.9.2. Let OlEGws and let xEA. Then x is i-small iff a x is U:)-small. Proof. It is enough to prove that i-smallness implies (*)-smallness. Suppose x is i-small. Let qE1m rCa IKI2:w. , -w We have to prove that x is (q,r,K)-small. If (Vil. a)q c(il.)x then we are done. Assume q E c(il.){p} for some il. a and pEx. 198 4.9. Then x is Cp,rUL,K)-small and therefore h E C30 C w C3kEcCKnO) (p}) CV" c::.(J cCQ)x. There is E CCKnO){q} such that k E cCL){h} since k E cCKnO){P} C C cCKnO)cCL){q} = cCL)cCKnO){q}. It is enough to show that Then c::.w q Let "c::.w k cCSlUl;)x since "UL C by Then by k E cCL){h}. We have seen -w that x is Cq,r,K)-small. QED(Claim 4.9.2.) nor m Now we turn to the proof of 4.9. Let {JLEGWs be generated by a GCA and assume that every element of G is i-small. Let HC a. Let LCa be such that Define Vt sri» Q s g {xEA C'r/qE1 )(vr C a) X is -w We shall prove the following statements about S.: Cl) S is closed under +. (2) C S. (3) CVzEIg S) (Vct=-l(Jt ) (30 c::.w c(0) l q I 'I z. Vt Proof of (1): Let x,yES. Let and rea be arbitrary. ct=-1 -uJ and Then (38 c::.w (q}) CV" c::. w h cCrl)x, by xES, Then C3L c::.w {h l ) CV" c::.w k c(,,)y, by yES. oUL c::.w and k E Let "c::.w Then k ce,,)Y by rl c::. w Also k cC,,)x since hE C(L){kJ and h x+yES. cC"UL)x by "UL c::. w This shOWS that k cC") Cx+y), thus Proof of (2): Let Then by and therefore by Therefore x is for every qE lot and r c::. a, since x is (*)-small by xEG. Then w xES since if x is (q,r,K)-small then x is (q,r,M)-small for every M:JK. Proof of (3): Let zEIg S. Then for some r c::. w a and yES, by (1). Let qE 1& Then (3 0 C]hE by yEs. Then h ccr)y since 4.10. 199 f C complete the proof of (3). -w Now we show that the conditions of Lemma 4.7.2 are satisfied with Only condition b. is not immediate. Let zEIg( ell) (G-DmH), and a. Then zEIg ({)L) S (HU,,) by (1) and (2) and then "c-w (]8 c(8){q} I z by (3). Thus b. holds, so 4.7.2 yields the conclusion of 4.9. QED (Proposition 4.9.) Remark 4. 10. Let V be a Crs -unit and let xcV. Consider a conditions (i)-(iii) below. (i) x is small in V. (ii) x is weakly small in V. (iii) x is irreversibly small in V. Then (i) '* (ii), (iii) ". (ii), but (i) 'i> (iii), (iii) t> (i). In fact, let let x = {qEaw : (3nEw) [qn'"O, and let y = {qEaw : (]nEw) (I/k>n)qk=n}. Then x is small but not i-small, and y is i-small but not small. 4.10.1. G is said to be a set of hereditarily disjoint elements of {)L iff (l/x,yEG) [x,"y a set of hereditarily disjoint small elements of Sb V vli th ('ix:2 EG) II', I To prove that UG is wsmall, let qEV, f a, K C l',(uG) with IKI2w. If g c(f)UG then we are done. Assume Since ILI2w. By wsmallness of x there are 8 Land pE c(8) {q} such that p c(f)x. Let y E By hereditary disjointness proving that uG is wsmall. We have proved a strenghtened version of 4.10.1. However, not every wsmall element can be obtained from small ones 200 4.11. by using 4.10.1. An example of this is the following wsmall element: x ={qE w2 : (E)3n w [gn=gn+l=l, (Vk>n+l)gk=OJ}. Indeed, l'f V't x then y and z are not hereditarily disjoint. Propositiom4.11, 4.13 below are applications of Propositions 4.6, 4.7, and 4.9. For more applications see sections 5 and 6. About Propositions 4.11-4.13 below see [HMTIJ5.6(15). In [HMTIJ5.6 (15) it is announced that the construction given in [HMTIJ5.6(4) can be modified to show that (Vo."2w) (Vx"22) (3 V!.E x 0. This modified construction is given in Prop.4.11 below. (Actually, there.) Prop.4.11 also says that regularity can be destroyed by relativization with a zero-dimensional element; see Prop.2.2(iii) . Proposition 4.11. Let o."2w and x"22 be arbitrary. There are an mE CSregnDc and a such that x 0. 0. R1 Ol E (GwscomPnH.eJL and W a a Proof. Let a"2w be such that IHI=w and 100-HI"2w. Let h : w >- H be one-one and onto. Let pEw+1(Hx) be such that (Vi and x"22. Let isw. Define and x U{R, isw} . l Let .[ g l5&o.x and m. g ){x}. Note that ('Vi Claim 4.11.1. Proof. 6x = H, and therefore -eJ!.EDc by We show that a Q is regular by using Prop.4.7. Let Q g {qEo.x : (3i EHx(pi)}. Then QEZd.[. Let Gg{x}. Then O[,EGws no r m is 0. 4.12. 201 generated by G. Condition (i) of 4.7 is satisfied, since IGI=1. We show that condition (ii) is satisfied by x and Q.: Let fEx and r C Ct. If fEQ then we are done. Assume Then fER and for -w w r every g we have fCr/gJEx iff r 1gE M. Let nEw be such that no r C h'-'n. = 0 and therefore for every g we have iff By PnEQ the above shows that condition (ii) of 4.7 is satisfied. \'1e s how that x is Q-wsmall.: Let K:=H, and let r =:w Ct, and qEQ. If q c(r)x then we are done. Suppose Let L rUh"n). L"O by Then (3 nEw) qEc (r) R n · g and IrUh*nl