Appendix A Prerequisites to Set Theory and General Topology
1. General remarks As mentioned in the Introduction, this book is intended to the reader acquainted with the basics of "naive" set theory. The modern (axiomatic and "model-theoretic") set theory is not evoked in fact.l We advice the reader to peruse the Introduction to the book "Topology" by K. Kuratowski. It is also helpful to consult other sources; for instance, the celebrated books by F. Hausdorff, P. S. Aleksandroff, or 1. P. Natanson as regards ordinal numbers and transfinite induction. A few remarks about notation and terminology are now in order. We use the following symbols of set-theoretic operations and relations:
u,n, \, E,~, C,:J .
We abstain from using the notations like ~, on presuming that the symbol C does not exclude the equality of sets. The empty set is denoted by 0; the cardinality of E, by cardE. The class of all x satisfying some property is denoted by
{xl·· .}.
To denote a function2 we usually take a single letter: f,g,r.p and so on. In this event f(x} is the value of f at a point x; the parentheses are sometimes omitted. We practice only the definition of mappings by the formulas of the type x --+ f(x}. At last, to denote a function on some set A, we use the record like {X"'}"'EA. It this case we speak of a family € == {Xa}aEA and call A the "index set." A particular case of a family is a simple sequence {Xn}~=l (the role of A is played by the set of natural numbers). The meaning of the terms "system," "totality," "collection," "tuple," etc. is always clear from the context (we usually imply the set of members of some family).
lPor instance, cf. T. Jech [1) and E. I. Gordon and S. P. Morozov [1). 2In this book the words "function," ''mapping,'' and "operator" are viewed as synonyms.
563 564 BOOLEAN ALGEBRAS IN ANALYSIS
The inverse of a function f is denoted by f~l. Irrespective of whether or not the inverse of a function f exists, the symbol f~l(e) always stands for the inverse image of a set e as well as f(e) symbolizes the image of e. The restriction of a function f to some e is denoted by fie. Assume given a family of sets {Eah'EA. The product of this family is the set of all families c == {ea}aEA satisfying eo E Ea for every 0: E A. This product is denoted by ITaEA Ea. In the case when A = {I, 2}, we obtain the product of two sets which is denoted by El X E2; this is the collection of all ordered pairs (el' e2), where el EEl and e2 E E2. If El = E2 = E then we write E2 instead of El x E2 and speak about the "product of two copies of E"; the analogous terminology applies to the general case in which all Be, (0: E A) coincide. To each value of the index 0:0 E A we assign the "projection" ?rao that sends an arbitrary family c == {ea} treated as a point in the product Ea to the element
The values of these mappings are called the "projections" or "coordinates" of c.
2. Partially ordered sets A partial ordering or partial order3 on a nonempty set f£ is a subset P C f£2 satisfying the following axioms: I. (x,x) E P for all x E f£. II. If (x,y) E P and (y,x) E P then x = y. III. If (x, y) E P and (y, z) E P then (x, z) E P. Thus, a partial order is a RELATION between the elements of f£. The axioms I and III express the REFLEXIVITY and TRANSITIVITY of this relation; the axiom II tells US that it is ANTISYMMETRIC. As a rule, we write x ~ y or y 2: x rather than (x, y) E P. Other similar symbols may replace ~; for instance, -<. The axioms I~III may be rewritten as I. x ~ x for all x E f£. II. If x ~ y and y ~ x then x = y. III. If x ~ y and y ~ z then x ~ z. The formula x < y (or y > x) means by definition that x ~ y and x -I- y. The expressions of the form a ~ b, a < b, etc. are called inequalities. A partially ordered set or poset is some set f£ furnished with some partial order P on f£; i.e., the pair {f£, P}. Most often we denote a partially ordered set by the same letter f£ as the underlying set; accordingly, we call the members of f£ "elements of a partially ordered set." This abuse of the language is very common in mathematics but is excusable only in the case when f£ is equipped with a single order. If every two elements in a partially ordered set f£ are compatible, Le., either (x,y) E P or (y,x) E P for all x,y E f£; then we say that f£ is linearly ordered. As an example, we may take every set of real numbers furnished with the conventional order. Likewise on the real axis, the set of all x satisfying the inequality a ~ x ~ b is called an interval and we denote it by [a, b].
3The word "partial" is often omitted. APPENDIX A: PrereIJ.uisites to Set Theory andGeneral Topology 565
Let !£ be equipped with some partial order. Each subset !£o c !£ may also be furnished with some partial order; to this end, we put
It is easy to see that the axioms I-III are satisfied. The so-defined partial order Po is said to be induced by P or induced from outside. Practically speaking, this means that all inequalities have the same sense in !£o as they have in !£. It might happen that !£o is linearly ordered in the induced order; in this event we call &:0 a chain. An element Xo of a partially ordered set !£ is maximal provided that the inequality x ~ Xo implies x = Xo. A significant role in mathematics belongs to the following Kuratowski-Zorn Lemma.4 Assume that a partially ordered set !£ possesses the following property: to each chain !£o c !£ there is some y E !£ such that x ~ y for all x E !£o. Then, to whatever element x E !£, there is a maximal element xo satisfying the inequality Xo ~ x. The Kuratowski-Zorn Lemma often replaces the principle of mathematical induc• tion in proofs, enabling us to avoid ordinal numbers. However, it is sometimes more natural to use the Zermelo Theorem or transfinite numbers which leads faster to the aim (for instance, cf. the proof of the theorem about the structure of a homogeneous BA in Chapter 9 of this book).
3. Topologies An important example of a partially ordered set is provided by the system of topologies 5 on some set R. By a topology we routinely mean a class T of subsets of R that is closed under all unions and finite intersections. The members of T are called open and their comple• ments, closed sets in R. The system T' of all closed sets also uniquely determined the topology of Rj the role of the latter may be performed by each class of sets closed un• der all intersections and finite unions. The pair {R, T} is a topological space denoted by the same letter R. We say that a topology Tl is stronger than a topology T2 whenever Tl J T2 (or, which is the same, T{ J Tn. In this event we say also that Tl majorizes, or dominates 7"2, or T2 is weaker than Tl, etc. For each nonempty set T of topologies on R, there always exists a unique WEAKEST topology among those dominating every member of T; in much the same way, there is a unique STRONGEST topology among those dominated by every member of T. We say that x is an interior point of a set V or that V is a neighborhood about x whenever there is an open set G satisfying x E G C V. Let W(x) stand for the collection of all neighborhoods of x. If Wo is such a set of neighborhoods that to each V E W(x) there is some Vo E Wo satisfying Vo C V then we say that Wo is a base of neighborhoods about x. Assume that to each point x E R there corresponds some base of neighborhoods Wo(x)j then the inclusion GET means that to each point x E G there is some V E Wo{x) satisfying V c G.
4K. Kuratowski [1] and M. Zorn [1]. This proposition is equivalent to the axiom of choice and also to the so-called "Hausdorff maximality principle" (J. Kelley [2]). 5N. Bourbaki [1]; K. Kuratowski [2]; and J. Kelley [2]. 566 BOOLEAN ALGEBRAS IN ANALYSIS
We may thus uniquely recover the topology 7 from available bases of neighbor• hoods about all points. This idea is often used for the INITIAL INTRODUCTION OF A TOPOLOGY. Assume given a family {!Uo(x)} whose every member !Uo(x) is some class of subsets containing the point x. Assume further that the following hold: I. If V!, V2 E !Uo(x) then there is a set V E !Uo(x) lying in Vi n V2. II. Given V E !Uo(x), we may find V' E !Uo(x) so that each class !Uo(x/), x' E V', contains at least one set included in V. Then there is a unique topology 7 with respect to which each system !Uo(x) is a base of neighborhood about x. If to each point x there is a COUNTABLE base of neighborhoods then we say that the space under consideration satisfies the first axiom of countability. The second axiom of countability requires that there exists a countable system <8 of open sets such that each open set is a union of some family of elements of <8. A topological space is called sepamted or Hausdorff provided that to each pair of distinct points Xl and X2 there exist disjoint neighborhoods of Xl and X2 (the "Hausdorff axiom" ). If Ro is a subset of a topological space {R, 7} then the system 70 of all sets of the form G n Ro, G E 7, is some topology on R.() that is called natuml or induced (from outside). The set Ro, furnished with the topology 70, is called a subspace of the original topological space. A class of sets u C 7 is called a cover or covering of E provided that
EC Up. PEer
A topological space is called compact if E is separated and each open cover of E includes a finite part that is a cover of E too. Such a space is shortly called a com• pactum.6 For a separated space to be compact it is necessary and sufficient that the intersection of every centered system of its closed subsets be nonempty (a system of sets is centered whenever it has the finite intersection property; Le., the intersection of each finite subsystem is nonempty). Each closed subset of a compact space is com• pact too in the induced topology. Note finally that every compact space is not only separated but possesses the important normality property: every two closed disjoint sets are included in open disjoint sets. Assume given the two topological spaces {R1, 7t} and {R2, T2}. A mapping f from Rl to R2 is called continuous with respect to the topologies 71 and T2 provided that f- 1 (G) E 71 for all G E 72. If the topologies are implied then we simply speak about a continuous mapping. Consider some family {{ Ra , 7 a} } aEA of topological spaces, and put
R= n Ra. aEA
We may consider various topologies on R: the most important are those that guar• antee the continuity of all mappings 7ra (a E A). Mostly we use the weakest of these topologies; it is called TychonofJ. The set R, equipped with the Tychonoff topology, constitutes the product of the topological spaces Ra. The topology of this product
60r a bicompactum, the latter is the historically first term coined by P. S. Aleksandroff, the founder of the theory of compact spaces. APPENDIX A: Prerequisites to Set Theory andGeneml Topology 567 is the weakest among those in which all sets of the form 7l";;l(G), GET"" are open. Assume that {R",} is finitej A = {I, 2, ... ,m}. Bya continuous function in m vari• ables we always mean an arbitrary function on the product of the topological space Rl, R2, ... ,R"", and continuous with respect to the Tychonoff topology. (The values of the function may belong to an arbitrary topological space.) In particular, those continuous functions are most important that send a topological space or its "powers" R2, R 3 , ••• to Rj in this event we usually speak about continuous operotions on R. As an example we may take the multiplication operation or, in the "additive" ter• minology, the addition operation on some group r furnished with a topology. The continuity of this operation (as a mapping from r2 to r) together with the conti• nuity of the taking of the inverse means by definition that r is a topological group (L. S. Pontryagin [1]; J. Kelley [2]). A more general concept is that of "uniform space" (J. Kelley [2]). In this book we mostly pay attention to partially ordered sets of a special form (Boolean algebras) which are furnished with various topologies. These topologies must be properly compatible with orderj our constant preference is the joint consideration of order-theoretic and topological properties. Appendix B Basics of Boolean Valued Analysis
1. General remarks Boolean valued analysisl is a branch of functional analysis which uses a special model-theoretic technique that is embodied in the Boolean valued models of set theory. The term was coined by G. Takeuti. The invention of Boolean valued models was not cormected with the theory of Boolean algebras but rather has revealed a gemstone among the diverse applications of the latter. It was the celebrated Cohen forcing method for solving the continuum problem whose comprehension gave rise to the Boolean valued models of set theory. Their appearance is commonly associated with the names of D. Scott, R.. Solovay, and P. Vopenka. Boolean valued analysis consists primarily in comparative analysis of a mathemat• ical object or idea simultaneously in some standard and some Boolean valued models which is accomplished by a special technique of ascending and descending.
2. Boolean valued models Now we briefly present necessary information on the theory of Boolean valued models. All details may be found in a book by A. G. Kusraevand S. S. Kutateladze2 and the literature cited therein. The universe of discourse of Boolean valued analysis is a Boolean valued model of ZlFC. To sketch its structure, we start with a complete BA B. Given an ordinal 0:, put
viB) := {x I x is a function 11(3,8)(,8 < 0: II dom(x) c V~B) II im(x) C Bn. Thus, in more detail we have V o(B) '-0.- , V~!>l := {x I x is a function with domain in viB) and range in B}j ViB) := U V~B) (0: is a limit ordinal). i3<01 lThis appendix is compiled by S. S. Kutateladze. 2A. G. Kusraev and S. S. Kutateladze [2J.
569 570 BOOLEAN ALGEBRAS IN ANALYSIS
The class V(B):= U V~B) aEOn is a Boolean valued universe. An element of the class V(B) is a B -valued set. It is necessary to observe that V(B) consists only of functions. In particular, 0 is the function with domain 0 and range 0. Hence, the "lower" levels ofV(B) are organized as follows:
VbB) = 0, V~B) = {0}, V~B) = {0,({0},b) I b E B}.
It is worth stressing that a: ~ {3 ---> V~B) C V~B) for all ordinals a: and {3. Moreover, the following induction principle is valid in V(B):
(Vx E V(B» «Vy E dom(x» 'P(y) ---> 'P(x» ---> (Vx E V(B» 'P(x), where 'P is a formula of ~lF([j. Take an arbitrary formula 'P = 'P(Ul, ... ,Un) of ~lF([j. If we replace the ele• ments Ul, ... , Un by elements Xl, ... , Xn E V(B) then we obtain some statement about the objects Xl, ... , X n . It is to this statement that we intend to assign some truth• value. Such a value ['¢) must be an element ofthe algebra B. Moreover, it is naturally desired that the theorems of ~lF([j be true, Le., attain the greatest truth-value, unity. We must obviously define the truth-value of a well-formed formula by double in• duction, on considering the way in which this formula is built up from the atomic formulas X E y and X = y, while assigning truth-values to the latter when X and y range over V(B) on using the recursive definition of this universe. It is clear that if 'P and '¢ are evaluated formulas of ~lF([j and ['PI E Band ['¢I E B are their truth-values then we should put
['P /\ '¢] := ['PD /\ ['¢D,
['P V '¢] := ['PD V [,¢D,
['P ---> '¢] := ['PD ---> [,¢D, [""'PD := C['P)' [(Vx) 'P(x)l:= /\ ['P(x)D, XEV(B) [(3x) 'P(x)]:= V ['P(x)D, xEV(B) where the right-hand sides involve the Boolean operations corresponding to the logical connectives and quantifiers on the left-hand sides: /\ is the meet, V is the join, C is the complementation, while the implication ---> is introduced as follows: a ---> b := CaVb for a, b E B. Only such definitions provide the value "unity" for the classical tautologies. We turn to evaluating the atomic formulas X E y and X = Y for x, y E V(B). The intuitive idea consists in the fact that a B-valued set y is a "(lattice) fuzzy set," Le., a "set that contains an element z in dom(y) with probability y(z)." Keeping this in mind and intending to preserve the logical tautology of X E y ..... (3z E y) (x = z) as well as the axiom of extensionality, we arrive at the following definition by recursion:
[x E yD:= V y(z) /\ [z = xD, zEdom(y) APPENDIX B: Basics of Boolean Valued Analysis 571
[x = yD:= /\ x(z) --+ [z E y) 1\ /\ y(z) --+ [z E xl zEdom(x) zEdom(y)
Now we can attach some meaning to formal expressions of the form
(3) [x = y] 1\ [y = z) ~ [x = z],
(4) [x = y] 1\ [z E x] ~ [z E y],
(5) [x = yJ 1\ [x E z] ~ [y E zl It is worth observing that for each formula
V(B) F= X = Y 1\ 3. Principles of Boolean valued analysis In a Boolean valued universe V(B), the relation [x = yD = 1I. in no way implies that the functions x and y (considered as elements of V) coincide. For example, the function equal to zero on each layer V~B), where a 2': 1, plays the role of the empty set in V(B). This circumstance may complicate some constructions in the sequel. In this connection, we pass from V(B) to the sepamted Boolean valued universe V(B) often preserving for it the same symbol V(B); i.e., we put V(B) := V(B). Moreover, to define V(B), we consider the relation {(x, y) I [x = y] = 1I.} in the class V(B) which is obviously an equivalence. Choosing an element (a representative of the least rank) in each class of equivalent functions, we arrive at the separated universe V(B). Note that [x = y] = 1I. --+ [ (1) Transfer Principle. All theorems ofZlFC are true in y(B); in symbols, y(B) F a theorem of ZlFC. The transfer principle is established by laboriously checking that all axioms of ZlFC have truth-value 11. and the rules of inference preserve the truth-values of formulas. Sometimes, the transfer principle is worded as follows: "y(B) is a Boolean valued model of ZlFC." (2) Maximum Principle. For each formula 'P of ZlFC there exists Xo E y(B) for which [(3x) 'P(x)] = ['P(xo». In particular, if it is true in y(B) that there is an x for which 'P(x) then there is an element Xo in y(B) (in the sense of Y) for which ['P(xo)D = 11.. In symbols, y(B) F (3x) 'P(x) ~ (3xo) y(B) F 'P(xo). Thus, the maximum principle reads: (3xo E y(B» ['P(xo)] = V ['P(x» XEV(B) for each formula 'P of ZlFC. The last equality accounts for the origin of the term "maximum principle." The proof of the maximum principle is a simple application of the following (3) Mixing Principle. Let (bE)EES be a partition of unity in B, Le. a family of elements of a Boolean valued algebra B such that VbE = 11., (Ve,." E 'S)(e =1=." ~ bE f\b., = 0). eES For each family of elements (xE)EES of the universe y(B) and each partition of unity (be)eEs there exists a (unique) mixing of (xe) by (be); i.e. an element x of the separated universe y(B) such that be ::; [x = xe) for all e E 'S. The mixing of x of a family (xe) by (be) is denoted as follows: x = mixeEs(bexe) = mix{bexe leE'S}. 4. Ascending and descending The comparative analysis mentioned above presumes that there is some close in• terconnection between the universes Y and y(B). In other words, we need a rigorous mathematical apparatus that would allow us to find out the interplay between the interpretations of one and the same fact in the two models Y and y(B). The base for such apparatus is constituted by the operations of canonical embedding, descent, and ascent to be presented below. We start with the canonical embedding of the von Neumann universe. Given x E y, we denote by the symbol xII. the standard name of x in y(B)j i.e., the element defined by the following recursion schema: 011. := 0, dom(xll.):= {yll. lyE x}, im(xll.):= {11.}. APPENDIX B: Basics of Boolean Valued Analysis 573 Observe some properties of the mapping x >-+ x" we need in the sequel. (1) For an arbitrary x E V and a formula 'P of ZlF!C we have [(3y Ex") cp(y)] = V{[cp(z")] : z Ex}, [(\fy Ex") cp(y)] = /\ {[cp(z")] : z EX}. (2) If x and y are elements of V then, by transfinite induction, we establish x = y ..... V(B) F x" = y" . In other words, the standard name can be considered as an embedding of V into V(B). Moreover, it is beyond a doubt that the standard name sends V onto V(2), which fact is demonstrated by the next proposition: (3) The following holds: (4) A formula is called bounded or restricted if each bound variable in it is restricted by a bounded quantifier; i.e., a quantifier ranging over a specific set. The latter means that each bound variable x is restricted by a quantifier of the form (\fx E y) or (3x E y) for some y. Restricted Transfer Principle. For each bounded formula cp of7llF!C and every collection Xl, ... ,xn E V the following equivalence holds: Henceforth, working in the separated universe V(B), we agree to preserve the sym• bol x" for the distinguished element of the class corresponding to x. (5) Observe by way of example that the restricted transfer principle yields the following assertions: "4> is a correspondence from x to y" ...... V(B) F "4>" is a correspondence from x" to y""; "f is a function from x to y" ..... V(B) F "f" is a function from x" to y"" (moreover, f(a)" = f"(a") for all a E x). Thus, the standard name can be con• sidered as a covariant functor of the category of sets (or correspondences) in V to an appropriate subcategory of V(2) in the separated universe V(B) . (6) A set X is finite if X coincides with the image of a function on a finite ordinal. In symbols, this is expressed as fin(X); hence, fin(X) := (3n)(3J)(n E W 1\ f is a function 1\ dom(f) = n 1\ im(f) = X) (as usual w := {a, 1, 2, ... }). Obviously, the above formula is not bounded. N everthe• less there is a simple transformation rule for the class of finite sets under the canonical embedding. Denote by .9'fin(X) the class of all finite subsets of X: .9'fin(X) := {Y E .9'(X) I fin(Y)}. 574 BOOLEAN ALGEBRAS IN ANALYSIS For an arbitrary set X the following holds: Given an arbitrary element x of the (separated) Boolean valued universe y(B), we define the descent xl of x as xl := {y E y(B) I [y E x] = 1}. We list the simplest properties of descending: (1) The class xl is a set, Le., xl E Y for each x E y(B). If [x -I- 0] = 1 then xl is a nonempty set. (2) Let z E y(B) and [z -I- 0} = 1. Then for every formula cp of ~lFC we have [(\Ix E z) cp(x)l = 1\ ([cp(x)] I x E zl}, [(3x E z) cp(x)) = V([cp(x)) I x E zl}. Moreover, there exists Xo E zl such that [cp(xoH = [(3x E z) cp(x)]. (3) Let 4> be a correspondence from X to Y in y(B). Thus, 4>, X, and Y are ele• ments ofy(B) and, moreover, [4> C XxV] = 1. There is a unique correspondence 4>1 from Xl to Y 1 such that 4>l(Al) = 4>(A)1 for every nonempty subset A of the set X inside y(B). The correspondence 4>1 from X 1 to Y 1 involved in the above proposition is called the descent of the correspon• dence 4> from X to Y in y(B). (4) The descent of the composite of correspondences inside y(B) is the composite of their descents: (111 0 4>)1 = 1111 0 4>1. (5) If 4> is a correspondence inside y(B) then (4)-1)1 = (4)l)-1. (6) Let Idx be the identity mapping inside y(B) of a set X E y(B). Then (Idx}l = Idx •. (7) Suppose that X, Y, I E y(B) are such that [I : X -+ Y] = 1, Le., I is a mapping from X to Y inside y(B). Then 11 is a unique mapping from Xl to Y 1 for which [flex) = I(x)] = 1 (x E Xl)· By virtue of (1)-(7), we can consider the descent operation as a functor from the category of B-valued sets and mappings (correspondences) to the category of the usual sets and mappings (correspondences) (Le., in the sense of V). (8) Given x!, ... , Xn E y(B), denote by (Xl, . .. , xn)B the corresponding ordered n-tuple inside y(B). Assume that P is an n-ary relation on X inside y(B)j Le., X, P E y(B) and [P C X n "] = 1, where nEw. Then there exists an n-ary relation p' on X 1 such that Slightly abusing notation, we denote the relation pI by the same symbol P 1 and call it the descent of P. APPENDIX B: Basics of Boolean Valued Analysis 575 Let x E Y and x C y(B)j i.e., let x be some set composed of B-valued sets or, in other words, x E 9I'(y(B». Put 0i := 0 and dom(xt) = x, im(xt) = {1} if x =I=- 0. The element xi (of the separated universe y(B), i.e., the distinguished representative of the class {y E y(B) I [y = xi] = 1}) is called the ascent of x. (1) The following equalities hold for every x E 9I'(y(B» and every formula f.{J: [(V'z E xi) f.{J(z)] = 1\1'P(Y)], yEx [(3z E xi) f.{J(zH = V['P(y)]. yEx Introducing the ascent of a correspondence 4> C X x Y, we have to bear in mind a possible difference between the domain of departure X and the domain dom( Y1 E (X1) -4 [Xl = X2] ~ V [Y1 = Y2] Y2E~(X2) for Xl,X2 E dom( y(B) 1= (w 0 )t = wi 0 i. Note that if and -1 are extensional then (1)-1 = (-1)1. However, in general, the extensionality of in no way guarantees the extensionality of 4>-1. (4) It is worth mentioning that if an extensional correspondence f is a function from X to Y then its ascent fi is a function from Xi to Vi. Moreover, the exten• sionality property can be stated as follows: [Xl = X2) ~ [f(X1) = f(X2)] (X1,X2 EX). Given a set X C y(B), we denote by the symbol mixX the set of all mixings of the form mix(bexe), where (xe) C X and (be) is an arbitrary partition of unity. The following propositions are referred to as the rules faT cancelling arrows or the "descending-ascending" and "ascending-descending" rules. (5) Let X and X' be subsets of y(B) and f : X -4 X' be an extensional mapping. Suppose that Y, Y',g E V(B) are such that [Y =I=- 0) = [g: Y -4 V'] = 1. Then Xil = mix X, Y!j = Yj fll = f, g!i =g. 576 BOOLEAN ALGEBRAS IN ANALYSIS (6) From (6) follows the useful relation: Suppose that X E y, X oj:: 0; i.e., X is a nonempty set. Let the letter L denote the standard name embedding x I-> x" (x E X). Then L(X)l = X" and X = L-I(X" 1). Using the above relations, we may extend the descent and ascent operations to the case in which is a correspondence from X to Y 1 and [w is a correspondence from X" to Y] = 1, where Y E y(B). Namely, we put 1:= (oL)l and wl:= WloL. In this case, 1 is called the modified ascent of the correspondence and WI is called the modified descent of the correspondence W. (If the context excludes ambiguity then we simply speak of ascents and descents using simple arrows.) It is easy to see that wI is a unique correspondence inside y(B) satisfying the relation [1(x") = (x)ll = 1 (x EX). Similarly, wI is a unique correspondence from X to Y 1 satisfying the eqUality wl(x) = w(x")l (x EX): If := I and W := 9 are functions then these relations take the form [fl(x") = I(x)] = 1, gl(x) = g(x") (x EX). (1) A Boolean set or a set with B-structure or just a B-set is a pair (X,d), where X E y, X oj:: 0, and d is a mapping from X x X to the Boolean algebra B such that for all x, y, z E X the following hold: (a) d(x, y) = 0 ..... x = y; (b) d(x, y) = dey, x); (c) d(x, y) :S d(x, z) V d(z, y). An example of a B-set is given by each 0 oj:: X C y(B) if we put d(x,y) := [x oj:: y] = C[x = y] (x, y EX). Another example is a nonempty X with the "discrete B-metric" d; i.e., d(x,y) = 1 if x oj:: y and d(x, y) = (l if x = y. (2) Let (X, d) be some B-set. There exist an element te E y(B) and an injection L : X -+ Xl := tel such that d(x,y) = [LX oj:: LY] (x,y E X) and every element Xl E Xl admits the representation Xl = mix~E8(b~LXd, where (Xd~E8 C X and (b~)~E8 is a partition of unity in B. The element te E y(B) is referred to as the Boolean valued realization of the B-set X. If X is a discrete B-set then te = X" and LX = x" (x E X). If Xc y(B) then Ll is an injection from Xl to te (inside y(B». A mapping I from a B-set (X,d) to a B-set (X/,dl) is said to be nonexpanding if d(x, y) ~ dl(f(x), I(y» for all x, y E X. (3) Let X and Y be some B-sets, te and ?J/ be their Boolean valued realizations, and L and x be the corresponding injections X -+ tel and Y -+ ?J/ 1. If I : X -+ Y is a nonexpanding mapping then there is a unique element 9 E y(B) such that [g : te -+ ?J/] = 1 and I = X-I 0 gl 0 L. We also accept the notations te := $~(X) := X~ and g:= $~(f):= r· (4) Moreover, the following are valid: (1) y(B) F= I(A)~ = r(A~) for A c X; APPENDIX B: Basics of Boolean Valued Analysis 577 (2) If 9 : Y -+ Z is a contraction then go I is a contraction and y(B) 1== (g 0 f)~ = g~ O/~j (3) y(B) 1== "r is injective" if and only if I is a B-isometryj (4) V(B) 1== lOr is surjective" if and only if V{d(f(x),y) 1 x E X} = 1I. for all yEY. R.ecall that a signature is a 3-tuple u := (F, P, a), where F and P are some (pos• sibly, empty) sets and a is a mapping from F U P to w. If the sets F and P are finite then u is a finite signature. In applications we usually deal with algebraic systems of finite signature. An n-ary operation and an n-ary predicate on a B-set A are contractive mappings I : An -+ A and P : An -+ B respectively. By definition, I and p are contractive mappings provided that n-l d(f(ao, .. . ,an-l),j(a~, . .. ,a~-l» :5 V d(ak' a~), k=O n-l ds(p(ao, ... ,an-l),p(a~, ... ,a~_l»):5 V d(ak,a~) k=O for all ao, a~, ... , an-l, a~-l E A, where d is the B-metric of A, and ds is the symmetric difference on Bj i.e., ds(bl, b2) := bl6.b2. Clearly, the above definitions depend on B and it would be cleaner to speak of B-operations, B-predicates, etc. We adhere to a simpler practice whenever it entails no confusion. An aJ.gebraic B-system 21 of signature u is a pair (A, v), where A is a nonempty B-set, the underlying set or carner or universe of 21, and v is a mapping such that (a) dom(v) = FUPj (b) v(f) is an a(f)-ary operation on A for all I E Fj and (c) v(p) is an a(p)-ary predicate on A for all pEP. It is in common parlance to call v the interpretation of 2l in which case the notation r and pV are common substitutes for v(f) and v(P). The signature of an algebraic B-system 21:= (A, v) is often denoted by u(2l)j while the carrier A of 2l, by 12l1. Since AO = {0}, the nullary operations and predicates on A are mappings from {0} to the set A and to the algebra B respectively. We agree to identify a mapping g : {0} -+ Au B with the element g(0). Each nullary operation on A thus transforms into a unique member of A. Analogously, the set of all nullary predicates on A turns into the Boolean algebra B. If F := {h, ... , In} and P := {Pl, ... , Pm} then an algebraic B-system of signature u is often written down as (A, v(h), ... , v(fn) , V(Pl), ... , v(Pm» or even (A, /1, ... , In, Pl, ... ,p",,). In this event, the expression u = (/1, ... ,In, Pl, ... ,Pm) is substituted foru = (F, P, a). We now address the B-valued interpretation of a first-order language. Consider an algebraic B-system 2l:= (A, v) of signature u := u(21) := (F, P, a). Let l l 1--''PI!ll (ao, ... ,an-d:= CI'PI!ll(ao, ... ,an-l)j I(V xO)'PI!ll (al, . .. , an-d:= A 1'P1!ll (ao,. " , an-dj aoEA aoEA Now, the case of atomic formulas is in order. Suppose that pEP symbolizes an m-ary predicate, q E P is a nullary predicate, and to, ... , tm-l are terms of signa• ture u assuming values bo, . .. , bm-l at the given values ao, . .. , an-l of the variables Xo, ... , Xn-I. By definition, we let l'PI!ll(ao, ... ,an-d := v(q), if'P = qVj l'PI!ll(ao, ... , an-I) := Cd(bo, bl), if'P = (to = tl)j l'PI!ll(ao, ... , an-d := pV (bo, . .. , bm-l), if'P = pV (to, . .. , tm-l), where d is a B-metric on A. Say that 'P(xo, ... , Xn-l) is valid in 2( at the given values ao, ... , an-l E A of Xo, ... , Xn-I and write 2( 1= 'P(ao, ... , an-I) provided that l'PI!ll(ao, ... , an-I) = 1I.B. Alternative expressions are as follows: ao, ... , an-l E A satisfies 'P(xo"", Xn-l)j or 'P(ao, ... ,an-I) holds true in 2(. In case B:= {V,1I.}, we arrive at the conventional definition of the validity of a formula in an algebraic system. Recall that a closed formula 'P of signature u is a tautology if 'P is valid on every algebraic 2-system of signature u. Consider algebraic B-systems 2( := (A, v) and X) := (D,I-') ofthe same signature u. The mapping h : A -+ D is a homomorphism of 2( to X) provided that, for all ao, ... ,an-l E A, the following are valid: (1) dB(h(ad, h(a2» :5 dACal, a2)j (2) her) = r if aU) = OJ (3) h(J"(ao, ... , an-I» = r(h(ao), ... , h(an-l» if 0 of- n := aU)j (4) p"(ao, ... , an-d :5 pl-'(h(ao), ... , h(an-l», with n := a(p). A homomorphism h is called strong if (5) a(p) := n of- 0 for pEP, and, for all do, .. . , dn- l E D the following inequality holds: pl-'(do, . .. , dn- l ) v {pV (ao, . .. , an-I) /\ dD(do, h(ao» /\ ... /\ dD(dn-l, h(an-l))}' ao,···,an_1EA If a homomorphism h is injective and (1) and (4) are fulfilled with equality hold• ingj then h is said to be a isomorphism from 2( to X). Undoubtedly, each surjective isomorphism h and, in particular, the identity mapping IdA: A -+ A are strong homomorphisms. The composite of (strong) homomorphisms is a (strong) homomor• phism. Clearly, if h is a homomorphism and h-l is a homomorphism too, then his an isomorphism. Note again that in the case of the two-element Boolean algebra B : = {(), 11.} we come to the conventional concepts of homomorphism, strong homomorphism, and isomorphism. Before giving a general definition of the descent of an algebraic system, consider the descent of the two-element Boolean algebra. Choose two arbitrary elements, 0, APPENDIX B: Basics of Boolean Valued Analysis 579 1 E V CB ), satisfying [0 i' III = 1B. We may for instance assume that °:= Gl~ and 1:= 1~. (1) The descent D ofthe two-element Boolean algebra {a, l}B E VCB) is a complete Boolean algebra isomorphic to B. The formulas [X(b) = I) = b, [x(b) = 0) = Cb (b E B) defines an isomorphism X : B --> D. Consider now an algebraic system 21 of signature (TA inside V CB ), and let [21 = (A, v)Bll = 1 for some A, v E V CB ). The descent of 21 is the pair 211 := (At J.L), where J.L is the function determined from the formulas: J.L: f t-> (v IU))l U E F), J.L : p t-> X -1 0 (vl(P))l (p E P). Here X is the above isomorphism of the Boolean algebras Band {a, I}B 1. In more detail, the modified descent v I is the mapping with domain dom(v 1) = FUP. Given pEP, observe [21(p)" = 21A(pA)J = 1, [vl(p) = v(pA)ll = 1 and so VCB) F vl(p) : ACl(J)A --> {a, I}B. It is now obvious that (vl(p))l : (Al)Cl(J) --> D := {a, 1}B 1 and we may put J.L(p) := x-1 0 (v I(P))l. Let r.p(xo, . .. , Xn-1) be a fixed formula of signature (T in n free variables. Write down the formula (xo, . .. ,xn -1,21) in the language of set theory which formalizes the proposition 21 F r.p(xo, ... , Xn -1). Recall that the formula 21 F r.p(xo, .. . , Xn -1) determines an n-ary predicate on A or, which is the same, a mapping from An to {a, 1}. By the maximum and transfer principles, there is a unique element Ir.pl' VCB) F "r.p(ao, ... , an -1) is valid in 21" holds true if and only if [(ao, . .. ,an-l, 21)) = 1. (2) Let 21 be an algebraic system of signature (TA inside V CB). Then 211 is a uni• versally complete algebraic B-system of signature (T. In this event, for each formula r.p of signature (T. (3) Let 21 and 18 be algebraic systems of the same signature (TA inside V CB ). Put 21' := 211 and 18' := 181. Then, if h is a homomorphism (strong homomorphism) inside VCB) from 21 to 18 then h' := hl is a homomorphism (strong homomorphism) of the B-systems 21' and 18'. Conversely, if h' : 21' -+ 18' is a homomorphism (strong homomorphism) of alge• braic B-systems then h := h't is a homomorphism (strong homomorphism) from 21 to 18 inside VCB) . 580 BOOLEAN ALGEBRAS IN ANALYSIS Let 2( := (A, l.I) be an algebraic B-system of signature u. Then there are d and J.L E V(B) such that the following are fulfilled: (1) V(B) 1= 'Xd, J.L) is an algebraic system of signature U""j (2) If 2(' := (A', l.I') is the descent of (d, J.L) then 2(' is a universally complete algebraic B-system of signature Uj (3) There is an isomorphism ~ from 2( to 2(' such that A' = mix(~(A»j (4) For every formula 1 = X-l 0 (1 x + y = z +-+ [x + y = z) = 1, xy = z +-+ [xy = z] = 1, x ~ y +-+ [x ~ y] = 1, Ax = y +-+ [A"x = y] = 1 (x, y, z E !Ji!, A E lR). Gordon Theorem. Let !Ji be the reals in V(B). Assume further that !Ji ! stands for the descent 1!Ji1! of the underlying set of!Ji equipped with the descended operations and order. Then the algebraic system!Ji is a universaJly complete K-space. Moreover, there is a (canonical) isomorphism X from the Boolean algebra B onto the base H9L of!Ji! such that the following hold: X(b)x = X(b)y +-+ b ~ [x = y), X(b)x ~ X(b)y +-+ b ~ [x ~ yJ for all x, y E !Ji! and b E B. References Aleksandroff P. S. 1 An Introduction to the General Theory of Sets and Functions [in Russian], Gostekhizdat, Moscow (1948). Adel'son-Vel'skiT G. M. and ShreTder Yu. A. 1 "The Banach mean on groups," Uspekhi Mat. Nauk, 12, No.6, 131-136 (1957). Akilov G. P. and Kutateladze S. S. 1 Ordered Vector Spaces [in Russian], Nauka, Novosibirsk (1978) AntonovsiT M. Ya., Boltyanskil V. G., and Sarymsakov T. A. 1 Topological Boolean Algebras [in Russian], Izdat. Akad. Nauk UzSSR, Tashkent (1963). Araki H. and Woods E. J. 1 "Complete Boolean algebras of type I factors," Publ. Res. Inst. Math. Sci. Ser. A, 2, No.2, 157-242 (1966). Bazhenov I. I. 1 "On a theorem of A. A. Lyapunov on the range of a vector mea• sure," in: Optimizatsiya [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1986, No. 38(55), pp. 135-144. 2 "On convexity of the range of a vector measure," Vestnik Lenin• grad. Univ. Mat. Mekh. Astronom. Ser. 1, No.3, 93-95 (1986). 581 582 BOOLEAN ALGEBRAS IN ANALYSIS Balcar B. and Franek F. 1 "Independent families in complete Boolean algebras," Trans. Amer. Math. Soc., 274, No.2, 607-618 (1982). Balcar B. and Stepanek P. 1 "Boolean matrices, subalgebras and automorphisms of complete Boolean algebras," Fund. Math., 96, 211-229 (1977). Banach S. 1 "Sur la probleme de la mesure," Fund. Math., 4, 7-33 (1923). 2 TMorie des Operations Lineaires, Warsaw (1932). (Monografie Matematyczne. ) Bernstein S. N. 1 "An attempt of axiomatic foundations of probability theory," So• obshch. Khar'kovsk. Mat. Obshch. Ser. 2,15,209-274 (1917). Biktasheva V. A. 1 "Rotations of the spaces A(w)," Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 13, No.3, 116-118 (1979). Birkhoff G. 1 "Moore-Smith convergence in general topology," Ann. of Math., 38, 39-56 (1937). 2 Lattice Theory [Russian translation], Izdat. Inostr. Lit., Moscow (1952). 3 Lattice Theory [Russian translation], Nauka, Moscow (1984). Bogolyubov N. N. and Krylov N. M. 1 "La tMorie generale de la mesure dans son application a l'etude des systemes dynamiques de la mecanique non linearie," Ann. of Math., 38, 65-113 (1937). Boole G. 1 The Mathematical Analysis of Logic, Cambridge (1847). 2 An Investigation of the Laws of Thought, Walton and Maberly, London (1854). Borovkov A. A. 1 Probability [in Russian], Nauka, Moscow (1976). References 583 Bourbaki N. 1 General Topology. Basic Structures [Russian translation], Fiz• matgiz, Moscow (1958). 2 Topological Vector Spaces [Russian translation], Izdat. Inostr. Lit., Moscow (1959). Caratheodory C. 1 Mass und Integral und ihre Algebraisierung, Birkhauser Verlag, Basel and Stuttgart (1956). 2 "Die Homomorphieen von Somen und die multiplikation von in• haltsfunktionen," Ann. Scuola Norm. Sup. Pisa Cl. ScL, No.8, 105-130 (1939). Chang C.-C. 1 "On the representation of a-complete Boolean algebras," Trans. Amer. Math. Soc., 85, No.1, 208-218 (1957). Christensen J. P. R. and Herer W. 1 "On the existence of pathological submeasures and the construc• tion of exotic topological groups," Math. Ann., 213, 203-210 (1975). Darmois G. 1 "Analyse generale des liaisons stochastiques. Etude particuliere de l'analyse factorfelle lineaire," Rev. Inst. Internat. Statistique, No. 21, 2-8 (1953). Davis M. 1 Applied Nonstandard Analysis, John Wiley & Sons, New York etc. (1977). De Marr R. 1 "A martingale convergence theorem in vector lattices," Canad. J. Math., 18,424-432 (1966). Dikanova Z. T. 1 "On some properties of the partially ordered space of continuous functions on a bicompactum," Uspekhi Mat. Nauk, 14, No.6, 165-172 (1959). Doob J. L. 1 Stochastic Processes [Russian translation], Izdat. Inostr. Lit., Moscow (1956). 584 BOOLEAN ALGEBRAS IN ANALYSIS Dunford N. and Schwartz J. T. 1 Linear Operators. Vol. 1 [Russian translation], Izdat. Inostr. Lit., Moscow (1962). 2 Linear Operators. Vol. 2 [Russian translation], Mir, Moscow (1966). Efimov B. A. 1 "On extremally disconnected bicompacta," Dokl. Akad. Nauk SSSR, 172,771-774 (1967). 2 "On extremally disconnected compact spaces and absolutes," Trudy Moskov. Mat. Obshch., 23, 243-283 (1970). Erdos P. and Tarski A. 1 "On families of mutually exclusive sets," Ann. of Math., 44, 315-329 (1943). Feller W. 1 An Introduction to Probability Theory and Its Applications. Vol. 2 [Russian translation], Mir, Moscow (1967). Fikhtengol'ts G. M. and Kantorovich L. V. 1 "Sur les operations lineaires dans l'espace des fonctions bornees," Studia Math., 5, 69-98 (1935). Floyd E. E. 1 "Boolean algebras with pathological order topologies," Pacific J. Math., 5, 687---689 (1955). Freudenthal H. 1 "Teilweise geordnete Moduln," Proc. Acad. Amsterdam, 39, 641-651 (1936). Gaifman H. 1 "Concerning measures on Boolean algebras," Pacific J. Math., 14, No.1, 61-73 (1964). Gelfand 1. M., Kolmogorov A. N., and Yaglom A. M. 1 "On the general definition of the amount of information," Dokl. Akad. Nauk SSSR, 111, 745-748 (1956). Glivenko V. 1. 1 A Course in Probability Theory [in Russian], GONTI, Moscow and Leningrad (1939). References 585 2 Theorie Generale des Structures, Herman and Cie, Paris (1938). 3 "Fundamentals of the general theory of lattices," Uchen. Zap. Leningrad. Gos. Ped. Inst. Ser. Fiz.-Mat., 1, 3-33 (1937). Gordon E. 1. and Morozov S. F. 1 Boolean-Valued Models of Set Theory [in Russian], Gor'kovsk. Univ., Gor'ki! (1982). Greenleaf F. P. 1 Invariant Means on Topological Groups and Their Applications [in Russian], Mir, Moscow (1973). Grigorchuk R. 1. 1 "Degrees of growth of finitely generated groups and the theory of invariant means," Izv. Akad. Nauk SSSR Ser. Mat., 48, No.5, 939-985 (1984). Hajian A. B. and Kakutani S. 1 "Weakly wandering sets and invariant measures," Trans. Amer. Math. Soc., 110, 136-151 (1964). Hales A. 1 "On the non-existence of free-complete Boolean algebras," Fund. Math., 54,44-66. Halmos P. R. 1 Lectures on Boolean Algebras, Van Nostrand, Toronto (1963). 2 Lectures on Ergodic Theory [Russian translation], Izdat. Inostr. Lit., Moscow (1959). 3 Measure Theory, Springer-Verlag, New York (1974). Halmos P. R. and von Neumann J. 1 "Operator methods in classical mechanics. II," Ann. of Math., 43, 332-350 (1942). Hodges J. L. and Horn A. 1 "On Maharam's conditions for measure," Trans. Amer. Math. Soc., 64, 594-595 (1948). 586 BOOLEAN ALGEBRAS IN ANALYSIS Hopf E. 1 "Theory of measure and invariant integrals," Trans. Amer.Math. Soc., 34, 373-393 (1937). Horn A. and Tarski A. 1 "Measures on Boolean algebras," Trans. Amer. Math. Soc., 64, 467-497 (1948). Ionescu Tu1cea A. and Ionescu Tulcea C. 1 Topics in the Theory of Lifting, Springer-Verlag, Berlin etc. (1969). Jonsson B. 1 "A Boolean algebra without proper automorphisms," Proceed• ings Amer. Math. Soc., 2, 766-770 (1951). Jech T. J. 1 Lectures in Set Theory with Particular Emphasis on the Method of Forcing [Russian translation], Mir, Moscow, (1973). Kal'manovich V. A. and Vershik A. M. 1 "Random walks on discrete groups: boundary and entropy," Ann. Probab., 11, No.3, 457-490 (1983). Kakutani S. 1 "On equivalence of infinite product measures," Ann. of Math., 49, 214-224 (1948). 2 "Ueber die Metrization der topologischen Gruppen," Proc. Imp. Acad. Japan, No. 12,82-84 (1936). Kakutani S. and Oxtoby J. C. 1 "Construction of a non-separable invariant extension of the Lebe• sgue measure space," Ann. of Math., 52, 580-590 (1950). Kalinin V. V. 1 "Dimension functions on Boolean logics. I," Izv. Akad. Nauk SSSR Ser. Mat., 160, No.9, 89-90 (1975). 2 "Dimension functions on Boolean logics. II," Izv. Akad. Nauk SSSR Ser. Mat., 161, No. 10,92-93 (1975). References 587 Kalton N. J. and Roberts J. W. 1 "Uniformly exhaustive submeasures and nearly additive set func• tions," Trans. Amer. Math. Soc., 278, 803-816 (1983). Kantorovich L. V. 1 "Some theorems on almost everywhere convergence," Dokl. Akad. Nauk SSSR, 14, 537-540 (1937). 2 "Sur les proprietes des espaces semiordonnes lineaires," C. R. Acad. ScL, 202, 813-816 (1936). Kantorovich L. V., Vulikh B. Z., and Pinsker A. G. 1 Functional Analysis in Partially Ordered Spaces [in Russian], Gostekhizdat, Moscow and Leningrad (1950). Kappos D. 1 Strukturtheorie der Wahrscheinlichkeitsfelder und Raume, Sprin• ger Verlag, Berlin (1960). Karp C. R. 1 "A note on representation of a-complete Boolean algebras," Proc. Amer. Math. Soc., 14, 705-757 (1963). Katetov M. 1 "Remarks on Boolean algebras," Colloq. Math., 2, 229-235 (1952). Kellerer H. G. 1 "Funktionen auf Produktraumen mit vorgegebenen Marginal-Fu• nktionen," Math. Ann., No. 144, 323-344 (1961). Kelley J. L. 1 General Topology [Russian translation], Nauka, Moscow (1968). 2 "Measures on Boolean algebras," Pacific J. Math., 9, 1165-1177 (1959). Kharazishvili A. B. 1 Invariant Extension of Lebesgue Measure [in Russian], Tbiliss. Univ., Tbilisi (1983). 588 BOOLEAN ALGEBRAS IN ANALYSIS Kingman J. F. C. and Robertson A. P. 1 "On a theorem of Lyapunov," J. Lond. Math. Soc. 43, 347-351 (1968). Kiseleva T. G. 1 "Partially ordered sets endowed with a uniform structure," Vest• nik Leningrad. Univ. Mat. Mekh. Astronom. Ser. 1, No. 13, 51-57 (1967). Kislyakov S. V. 1 "Free subalgebras of complete Boolean algebras and the spaces of continuous functions," Sibirsk. Mat. Zh., 14, No.3, 569-581 (1973). Kluvanek I. and Knowles G. 1 "Liapunov decomposition of a vector measure." Math. Ann., 210, 123-127 (1974). Knowles G. 1 "On Liapunov vector measures," in: Measure Theory, Proceed• ings Conf. Oberwolfach 1975, Lect. Notes Math. 541, 61-67 (1976). 2 "Lyapunov vector measures," SIAM J. Control, 13, 294-303 (1975). Kolmogorov A. N. 1 Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin (1933). The Principal Ideas of Probability [in Russian], ONTI, Moscow and Leningrad (1936). 2 "Structure of complete metric Boolean algebras," Uspekhi Mat. Nauk, 3, No. 1(23), 212 (1948). 3 "Algebres'de Boole metriques completes. VI," Zjazd Mat. Pols• kich., 22-30 (1948). Kolmogorov A. N. and Fomin S. V. 1 Elements of FUnction Theory and Functional Analysis [in Rus• sian], Nauka, Moscow (1989). Koppelberg S. 1 "Free subalgebras of complete Boolean algebras," Notices Amer. Math. Soc., 20, No.4, 1-18 (1973). References 589 Krylov A. N. 1 Lectures on Approximate Calculations [in Russian], Izdat. Tekh• nich. Teor. Lit., Moscow and Leningrad (1950). Kulakova V. G. 1 "Positive projections and conditional expectations," Zap. Nauch• nych Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), No. 47, 172-174 (1974). Kuratowski K. 1 "Une methode d'elimination des nombres transfinis des raisonne• ments matMmatiques," Fund. Math., 5, No.3, 76-108 (1922). 2 Topology. Vol. 1 [Russian translation], Mir, Moscow (1966). Kurosh A. G. 1 Lectures on General Algebra [in Russian], Nauka, Moscow (1962). Kusraev A. G. 1 Elements of Boolean-Valued Analysis [in Russian], Inst. Mat. (N ovosi birsk), Novosi birsk (1987). 2 Vector Duality and Its Applications [in Russian], Nauka, Novosi• birsk (1985). Kusraev A. G. and Kutateladze S. S. 1 Nonstandard Methods of Analysis [in Russian], Nauka, Novosi• birsk (1990). 2 Boolean Valued Analysis, Kluwer Academic Piblishers, Dordh• recht (1999). Kusraev A. G. and Malyugin S. A. 1 Some Questions of Vector Measure Theory [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk (1988). 2 "A product and projective limit of vector measures," in: Contem• porary Problems of Algebra and Analysis. Vol. 14 [in Russian], Trudy Inst. Mat. (Novosibirsk), Novosibirsk, 1987, pp. 132-152. Liptser R. Sh. and Shiryaev A. N. 1 The Theory of Martingales [in Russian], Nauka, Moscow (1986). 590 BOOLEAN ALGEBRAS IN ANALYSIS Lorch E. R. 1 "Functions of self-adjoint transformations in Hilbert space," Acta Sci. Math. (Szeged), 7, 136-146 (1935). Lozanovskil' G. Va. "On the functions of elements of vector lattices" [in Russian], Izv. Vyssh. Uchebn. Zaved., Mat. 131, No.4, 45-54 (1973). Lyapunov A. A. 1 "On totally additive vector-functions. I," Izv. Akad. Nauk SSSR Ser. Mat., 4, No.6, 465-478 (1940). Lyubetskil V. A. and Gordon E. 1. 1 "Boolean extensions of uniform structures," in: Studies on Non• classical Logics and Formal Systems [in Russian], Moscow, 1983, pp.82-153. MacNeille H. M. 1 "Partially ordered sets," Trans. Amer. Math. Soc., 42, 416-460 (1937). Mackey G. W. 1 Lectures on Mathematical Foundations of Quantum Mechanics [Russian translation], Mir, Moscow (1965). Maharam D. 1 "An algebraic characterization of measure algebras," Ann. Math., 48, 154-167 (1947). 2 "On homogeneous measure algebras," Proc. Nat. Acad. Sci. U. S. A., 28, 108-111 (1942). 3 "Automorphisms of products of measure spaces," Proc. Amer. Math. Soc., 9, 702-707 (1958). Malyugin S. A. 1 "On sequential order topologies," Sibirsk. Mat. Zh., 31, No.1, 194-197 (1990). Marczewski E. and Sikorski R. 1 "On isomorphism types of measure algebras," Fund. Math., 38, 92-98 (1951). References 591 Markov A. A. 1 "On existence of an integral invariant," Dokl. Akad. Nauk SSSR, 17, 459-462 (1937). Matthes K. 1 "Uber die Ausdehnung von x-Homomorphismen Boolescher AI• gebren. I and II," Z. Math. Logik Grundlag. Math., I: 6, 97-105 (1960); II: 7, 16-19 (1961). 2 "Uber eine Schar von Regularitatsbedingungen fur Verbande," Math. Nachr., No. 22, 93-128 (1960). McKenzie R. 1 "Automorphism groups of denumerable Boolean algebras," Ca• nad. J. Math., 26,466-471 (1977). McAlloon K. 1 "Consistency results about ordinal definability," Ann. Math. Lo• gic, 2, No.4, 449--467 (1971). Monk J. D. and Bonnet R. (eds.) 1 Handbook of Boolean Algebras, North-Holland, Amsterdam etc. (1987). Monk J. D. and Rassbach W. 1 "The number of rigid Boolean algebras," Algebra Universalis, 9, No.2, 207-210 (1979). Moore E. H. and Smith H. L. 1 "A general theory of limits," Amer. J. Math., 44,102-121 (1922). Natanson 1. P. 1 Theory of Functions of a Real Variable [in RUSSian], Nauka, Mo• scow (1974). Neumann J., von 1 "Zur Allgemeiner Theorie des Mass," Fund. Math., 13, 73-116 (1929). 2 "Algebraische Reprasentanten der Functionen Ibis auf eine Menge von Masse Null'" J. Reine Angew. Math., 165, 109-115 (1931). 592 BOOLEAN ALGEBRAS IN ANALYSIS Neveu J. 1 Mathematical Foundations of Probability Theory [in French], Masson et Cie, Paris (1964). Ogasawara T. 1 "Theory of vector lattices. I and II," J. Sci. Hiroshima Univ. Ser. A, I: 12,37-100 (1942); II: 13,41-161 (1944). Ol'shanskil A. Yu. 1 "To the question of existence of an invariant mean on a group," Uspekhi Mat. Nauk, 35, No.4, 199-200 (1980). Ore O. 1 "On the foundation of abstract algebra. I," Ann. of Math., 36, 406-437 (1935). Parthasarathy K. R. 1 Introduction to Probability and Measure [Russian translation], Mir, Moscow (1983). Pashenkov A. V. 1 "Representations of topological spaces on Boolean algebras of reg• ular open sets," Mat. Sb., 91(133), No. 3(7), 293-309 (1973). Pierce R. S. 1 "Some questions about complete Boolean algebras," Proc. Sym• pos. Pure Math., 2, 129-140 (1961). Pinsker A. G. 1 "Lattices equivalent to K-spaces," Dokl. Akad. Nauk SSSR, 99, 503-505 (1954). 2 "A latticial characterization of function spaces," Uspekhi Mat. Nauk, 12, No. 1(73), 226-229 (1957). 3 "On completing partially ordered spaces," Dokl. Akad. Nauk SSSR, 21, 6-9 (1938). 4 "On some properties of universally complete K-spaces," Dokl. Akad. Nauk SSSR, 22, 216-219 (1939). Pkhakadze Sh. S. 1 "To the theory of Lebesgue measure," Trudy Tbiliss. Razmadze Inst. Mat., 25 (1958). References 593 Plesner A. I. 1 Spectral Theory of Linear Operators [in Russian], N auka, Moscow (1965). Plesner A. I. and Rokhlin V. A. 1 "Spectral theory of linear operators. II," Uspekhi Mat. Nauk, 1, No. 1(11), 71-191 (1946). Pontryagin L. S. 1 Continuous Groups [in Russian], GITTL, Moscow (1954). Popov V. A. 1 "Additive and subadditive functions on Boolean algebras," Sib• irsk. Mat. Zh., 17, No.2, 331-355 (1976). Potepun A. V. 1 On Order Convergence and Topology in Boolean Algebras and Vector Lattices [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Lenin• grad (1975) 2 "Generalized regularity of K-lineals and the properties of order topology," Dokl. Akad. Nauk SSSR, 207, No.3, 541-543 (1972). 3 "On equivalence relations on Boolean algebras and vector lat• tices," Vestnik Leningrad. Univ. Mat. Mekh. Astronom. Ser. 1, No.1, 140-142 (1973). 4 "Some forms of order convergence and topologies on partially ordered spaces," Sibirsk. Mat. Zh., 12, No.4, 819-827 (1971). 5 "Some properties for existence of measure on a Boolean algebra," submitted to VINITI on August, 29, 1986, No. 6265-1. Purtseladze R. R. 1 "On open mappings of extremally disconnected compact sets," Vestnik Leningrad. Univ. Mat. Mekh. Astronom., No.1, 117- 118 (1981). Rasiowa H. and Sikorski R. 1 The Mathematics of Metamathematics [Russian translation], Mo• scow (1972). Rieger L. S. 1 "On free Xe-complete Boolean algebras. (With an application to logic)," Fund. Math., 38, 35-52 (1951). 594 BOOLEAN ALGEBRAS IN ANALYSIS 2 "Some remarks on automorphisms of Boolean algebras," Fund. Math., 38, 209-216 (1951). Riesz F. and Sz.-Nagy B. 1 Lectures on FUnctional Analysis [Russian translation], Izdat. In• ostr. Lit., Moscow (1954). Rokhlin V. A. 1 "On the principal ideas of measure theory," Mat. Sb., 25, No.1, 107-150 (1949). 2 "The metric classification of measurable functions," Uspekhi Mat. Nauk, 12, No. 2(74), 169-174 (1957). Romanovskil 1. V. and Sudakov V. N. 1 "On existence of independent partitions," Trudy Mat. Inst. Stek• lov., 79, 5-10 (1965). Sacks S. 1 Theory of the Integral [Russian translation], Izdat. Inostr. Lit., Moscow (1949). Samorodnitskil A. A. 1 "Boolean exhaustion and the structure of a measure space," Vest• nik Syktyvkarsk. Univ., No.1, 89-100 (1995). 2 Measure Theory [in Russian], Leningrad University Press, Lenin• grad (1990). Sarymsakov T. A. 1 "Topological semi fields and the principal ideas of probability," Dokl. Akad. Nauk UzSSSR Ser. Fiz.-Mat. Nauk, No.8, 5-9 (1965). Sarymsakov T. A. et al. 1 Ordered Algebras [in Russian], Fan, Tashkent (1983). Savel'ev L. Ya. 1 "Extension of measures by continuity," Sibirsk. Mat. Zh., 5, No.3, 639-650 (1964). 2 "On continuous measures," Dokl. Akad. Nauk SSSR, 160,44-45 (1965). References 595 3 "On count ably additive vector measures," Dokl. Akad. Nauk SSSR, 167, 758-759 (1966). 4 "On order topologies and continuous measures," Sibirsk. Mat. Zh., 6, No.6, 1357-1364 (1965). Schubert H. 1 Kategorien. Vol. 1 and 2, Springer Verlag, Berlin and New York (1970). Semadeni Z. 1 Banach Spaces of Continuous Functions. Vol. 1, PWN-Polish Sci• entific Publishers, Warsaw (1971). (Monografie Matematyczne.) Shamaev I. 1. 1 "Countable extension of a vector-Iattice-valued measure," Sibirsk. Mat. Zh., 22, No.3, 197-203 (1981). Shatunovski'l S. O. 1 An Introduction to Analysis [in Russian], Odessa (1923). Sheffer H. M. 1 "A set of five independent postulates for Boolean algebras with application to logical constants," Trans. Amer. Math. Soc., 14, 481-488 (1913). Shelah S. 1 "Why there are many non-isometric models for unsuperstable the• ories," in: Proceedings of the International Congress of Mathe• matics, 1974, Vancouver, pp. 85-90. Sikorski R. 1 Boolean Algebras [Russian translation], Mir, Moscow (1969). Sirvint Yu. F. 1 "Convex sets and linear functionals in abstract space. I: Convex sets. II: Linear functionals," Izv. Akad. Nauk SSSR Ser. Mat., 6, I: 143-170; II: 189-226 (1942). Skitovich V. P. 1 "On a certain property of the normal distribution," Dokl. Akad. Nauk SSSR, 89, 217-219 (1953). 596 BOOLEAN ALGEBRAS IN ANALYSIS Sobolev V. I. 1 "On a partially ordered measure of sets, measurable functions, and certain abstract integrals," Dokl. Akad. Nauk SSSR, 91, 23-26 (1953). Soler J.-L. 1 Notion de Liberte en Statistique MatMmatique [Russian transla• tion], Mir, Moscow (1972). Solovay R. M. and Tennenbaum S. 1 "Iterated Cohen extensions and Suslin's problem," Ann. of Math., 94, No.2, 201-245 (1972). Stone M. H. 1 "Applications of the theory of Boolean rings to general topology," Trans. Amer. Math. Soc., 41, 375-481 (1937). 2 "Algebraic characterizations of special Boolean rings," Fundam. Math., 29, 223-303 (1937). 3 "The theory of representations for Boolean algebras," Trans. Amer. Math. Soc., 40, 37-111 (1936). Sucheston L. 1 "On existence of finite invariant measures," Math. Z., 86, 327- 336 (1964). Sudakov V. N. 1 "Geometric problems of the theory of infinite-dimensional prob• ability distributions," Trudy Mat. Inst. Steklov., 141 (1976). Sukonkin V. I. 1 "A certain example in the theory of sets," Sibirsk. Mat. Zh., 7, No.6, 1435 (1966). Talagrand M. 1 "A simple example of pathological submeasure," Math. Ann., 252, No.2, 97-102 (1980). Tsalenko M. Sh. and Shul'gelfer E. G. 1 Fundamentals of the Theory of Categories [in RUSSian], Nauka, Moscow (1974). References 597 Vallee-Poussin Ch. 1 "Sur l'integrale de Lebesgue," Trans. Amer. Math. Soc., 16 (1915). Veksler A. I. 1 "On topological completeness of Boolean algebras in the sequen• tial order topology," Sibirsk. Mat. Zh., 14, No.4, 726-737 (1973). Vershik A. M. 1 "Decreasing sequences of measurable decompositions and their applications," Dokl. Akad. Nauk SSSR, 193, No.4, 748-751 (1970). Vinokurov V. G. 1 "On additional representations of measure algebras," in: Proba• bility and Statistics [in Russian], Tashkent, 1964, No.1, pp. 126- 129. 2 "On infinite products of Lebesgue spaces," Dokl. Akad. Nauk SSSR, 158, 1247-1249 (1964). 3 "Representations of Boolean algebras and measure spaces," Mat. Sb., 56(98), No.3, 374-391 (1962). Vinokurov V. G., Rubshteln B. A., and Fooorov A. L. 1 The Lebesgue Space and Its Measurable Partitions [in Russian], Tashkent (1985). Vladimirov D. A. 1 Boolean Algebras [in RUSSian], Nauka, Moscow (1969). 2 "Invariant measures on Boolean algebras," Mat. Sb., 67(109), No.3, 440-460 (1965). 3 "On completeness of a partially ordered space," Uspekhi Mat. Nauk, 15, No. 2(92),165-172 (1960). 4 "On existence of invariant measures on Boolean algebras," Dokl. Akad. Nauk SSSR, 157, 764-766 (1964). 5 "On independence of subalgebras and partitions," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 85, 30-38 (1979). 6 "On normability of a Boolean algebra," Dokl. Akad. Nauk SSSR, 146, 987-990 (1962). 598 BOOLEAN ALGEBRAS IN ANALYSIS 7 "On the countable additivity of a Boolean measure," Vestnik Le• ningrad. Univ. Mat. Mekh. Astronom., 16, No. 19, 5~15 (1961). 8 "Structure of complete Boolean algebras," in: Abstracts: The International Mathematical Congress, Moscow, 1966, No.2, p. 36. 9 "Isomorphic subalgebras of probability Boolean algebras," Jour• nal Math. Sci., New York 75, No.5, 1900~1902 (1995); translation from Zap. Nauchn. Semin. POMI 194, 44~47 (1992). Vladimirov D. A. and Efimov B. A. 1 "On the cardinalities of extremally disconnected spaces and com• plete Boolean algebras," Dokl. Akad. Nauk SSSR, 194, No.6, 1247~1250 (1970). Vladimirov D. A. and Samorodnitskil A. A. 1 "The metric classification of measurable functions in nonsepara• ble measure spaces," in: Optimizatsiya [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1986, No. 37(54), pp. 17~27. Vladimirov D. A. and Shergin Yu. V. 1 "Characterization of £P spaces in the class of normed vector lat• tices by means of the group of their isometric order automor• phisms," Vestnik Leningrad. Univ. Mat. Mekh. Astronom. Ser. 1, No. 1, 118~120 (1991). Vladimirov D. A. and Zenf P. 1 "Partitions of totally disconnected bicompacta," Z. Anal. An• wendungen, 2, No. 3, 281~286 (1983). Vulikh B. Z. 1 Introduction to the Theory of Partially Ordered Spaces [in Rus• sian], Fizmatgiz, Moscow (1961). 2 "Concrete representations of partially ordered linear spaces," Dokl. Akad. Nauk SSSR, 58, 733~736 (1947). 3 "On a Boolean measure," Uchen. Zap. Leningrad. Gos. Ped. Inst., 125, 95~114 (1956). Weber H. 1 "Die atomare Struktur topologischer Boolescher Ringe und s-be• schrankter Inhalte," Studia Math., 74, No. 1, 57~81 (1982). 2 "Topological Boolean rings. Decomposition of finitely additive set functions," Pacific J. Math., 110, No. 2, 471~495 (1984). References 599 Wright J. D. M. 1 "Stone-algebra-valued measures and integrals," Proc. London Math. Soc., 19, No.1, 107-122 (1969). 2 "The solution of a problem of Sikorski and Matthes," Bull. Lon• don Math. Soc., 3, 52-54 (1971). Yablonskil S. V., Gavrilov G. P., and Kudryavtsev V. B. 1 Boolean Functions and Post Classes [in Russian], Nauka, Moscow (1966). Yaglom I. M. 1 A Boolean Lattice and Its Models [in Russian], Sov. Radio, Mos• cow (1980). Yosida K. and Hewitt E. 1 "Finitely additive measures," Trans. Amer. Math. Soc., 72, 4(H)6 (1952). Yudin A.I. 1 "On completing partially ordered linear spaces," Uchen. Zap. Leningrad Univ., 12, 57-61 (1941). Zaldenberg M. G. 1 "On the isometric classification of rearrangement-invariant spac• es," Dokl. Akad. Nauk SSSR, 234, No.2, 283-286 (1977). Zhegalkin I. I. 1 "On the technique of computing sentences in symbolic logic," Mat. Sb., 34, 9-28 (1927). Zorn M. 1 "A remark on a method in transfinite algebra," Bull. Amer. Math. Soc., 41, 667-670 (1935). Index (E)-canonical elementary polynomial, 45 Bijective mapping, 333 (o)-closure, 195 Birkhoff-Ulam homomorphism, 253 (o)-limit, 283 Boolean algebra, 10, 70 (o)-topology, 185 Boolean function, 18, 71 (Os)-cI08ure, 195 Boolean isomorphism, 324, 464 (os)-topology, 185 Boolean partition, 143 (!U)-topology, 203 Boolean ring, 74 B-valued set, 570 Boolean set, 576 d-regular set, 90 Boolean valued realization, 576 F+-set, 272 Boolean valued universe, 570 F;t-set, 272 Boolean, 15 a+-set, 272 Borel algebra, 83 at-set, 272 Borel subalgebra u-generated by a set, 86 K-lineal, 277 Bounded formula, 573 u-algebra of sets, 84 Canonical embedding, 97, 286, 302 u-complete Boolean algebra, 83 Canonical epimorphism, 58 u-ideal,84 Canonical extension, 435 601 602 BOOLEAN ALGEBRAS IN ANALYSIS Composition, 151 Filter, 49 Congruent elements, 401 Finite set, 573 Continuity point, 287 Finitely equipartite elements, 402 Continuous Boolean algebra, 93 Free Boolean algebra, 68 Continuous extension condition, 246 Fully homogeneous BA, 410 Continuous homomorphism, 88 General duality principle, 6 Convergence with regulator, 284 Generalized sequence, 181 Coproduct, 158 Generators, 36 Coretract, 163 Gordon's Theorem, 580 Coretraction, 163 Greatest lower bound, 4 Countable chain condition, 24 Hellinger type, 21 Countable type, 24 Homogeneous nondegenerate Boolean Countably additive function, 26 algebra, 119 Covariant functor, 152 Homogeneously-primitive subalgebra, 504 Cover, 244 Homomorphism, 57 Decomposable element, 489 Ideal, 48, 209 Decomposition, 54 Indicator, 7 Degenerate algebra, 10 Individual isomorphism, 475 Degree of nonsaturation, 116 Induced homomorphism, 60 Density, 347 Infimum, 4 Descent, 574 Injective object, 163 Diagonal principle, 223 Internal element, 483 Direct product, 471 Isomorphic cones, 157 Direct sum, 54 Isomorphic mapping, 3 Directed upward (downward) partially Isomorphic normed BAs, 464 ordered set, 181 Isomorphic objects, 154 Direction, 181 Isomorphic subobjects, 162 Discrete Boolean algebra, 93 Isomorphism of cocones, 157 Disjoint complement, 278 Isomorphism, 3, 154 Disjoint decomposition, 283 Isotonic mapping, 4 Disjoint elements, 9 Isotonicity property, 58 Disjoint sum, 13 Join,8 Distribution function, 304 Jordan decomposition, 26 Dual category, 153 Kernel,58 Dual isomorphism, 4 Lattice, 8 Dual table, 254 Least upper bound, 4 Element of maximal type, 522 Lebesgue algebra, 61 Element of rank 8,462 Lebesgue-Rokhlin space, 337 Elementary polynomial, 40 Left continuity, 287 Endpoint, 417 Length,459 Entry, 181 Lifting, 179, 234, 238 Epimorphism, 58, 154 Limit of a generalized sequence, 182 Equipartite elements, 401 Liouville automorphism group, 407 Equivalent partitions, 379 Liouville automorphism, 407 Ergodic automorphism, 30 Lower and upper limits, 186 Ergodic group, 30 Lower semicontinuous partition, 143 Essential positivity band, 352 Lyapunav Theorem, 388 Essentially positive quasimeasure, 24 Martingale, 364 Event, 70, 84 Maximal chain, 418 Excrescence, 136 Maximal ruter, 126 Expectation, 348 Maximal ideal, 125 Extended space, 286 Maximum principle, 572 Extension condition, 246 Measurable partition, 367 External element, 483 Measure preserving mapping, 322 Extremal compact space, 137-138 Measure, 26 Extremally disconnected compact space, Meet, 8 137-138 Metric homomorphism, 333 INDEX 603 Metric independence, 73 Right continuity, 287 Metric isomorphism, 324, 333 Rotation, 435 Metrically independent family, 470 Saturated element, 480 Minorant, 89 Section, 163 Mixing, 572 Separable Boolean algebra, 108 Modulus, 278 Separated Boolean valued universe, 571 Monomorphism, 58, 154 Sequence, 181 Monotonicity, 202 Sheffer stroke, 13 Morphism, 150 Sikorski Theorem, 230 Natural homomorphism, 58 Simple subalgebra, 35 Negative part, 278 Size, 27 Negative variation, 25 Skeleton, 154 Net, 181 Solid core, 91 Nonstandard analysis, 136 Solid set, 48 Normable complete BA, 321 Solid subspace, 278 Object, 150 Solidly-embedded set, 48 Order topologies, 185 Space of bounded elements, 301 Outer quasimeasure, 206 Spectral family, 287 Partition, 37 Spectral function, 287 Passport, 466 Spectral integral, 299 Planar set, 418 Spectral measure, 290 Polynomial, 41 Spectral type, 21 Positive element, 278 Spectrum, 288 Positive part, 278 Standard half-space, 294 Positive variation, 25 Standard name, 572 Preproduct, 76 Standard pair, 538 Primitive element, 505 Step function, 300 Primitive regular subalgebra, 504 Stone representation, 131 Primitively-discrete element, 505 Stone space, 130 Principal band projection, 53 Stone-Cech compactification, 136 Principal ideal, 49 Strong homomorphism, 578 Principal pair, 538 Subalgebra generated by a set, 36 Probability algebra, 317 Subalgebra, 35 Probability measure, 27 Subcategory, 151 Product, 61, 157 Support of a quasimeasure, 24 Projection, 53, 70 Support, 504 Projective object, 164 Supremum, 4 Proper ideal, 49 Symmetric difference, 12 Quasi-inverse functor, 156 System of generators, 36 Quasimeasure, 23 Term, 181 Quotient algebra, 57 Thin set, 418 Quotient object, 162 Topologies of ordering, 185 Quotient space, 368 Total variation, 26 Radon-Nikod;Ym derivative, 347 Totally additive real function, 26 Random variable, 348 Totally disconnected space, 127 Regular K-space, 285 Totally distributive Boolean algebra, 226 Regular Boolean algebra, 223, 266 Trace, 53, 87 Regular ideal, 273 Trivial subalgebra, 35 Regular open set, 103 Trivial ultrafilter, 135 Regular subspace, 278 Trivially-primitive subalgebra, 504 Regularity, 223 Truth-value, 570 Regulator of convergence, 284 Type, 498 Relatively uniform convergence, 284 Ultrafilter, 126 Representation, 130 Uniform topology, 202'· Resolution of the identity, 288 Uniformity base,203" Restricted formula, 573 Unit element, 279 Retract, 163 Unity, 8 604 BOOLEAN ALGEBRAS IN ANALYSIS Universally complete space, 286 Weak IT-distributive law, 227 Upper semicontinuous partition, 143 Weak unit, 279 Weight function, 509 Upper table, 254 Weight, 108 Valid formula, 578 Wide set, 418 Vector lattice, 277 Zero, 8