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).