boolean powers over incomplete boolean algebras

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Zeigchr. 1: math Logik und Grundhgen d Math Bd 36. S. 431 -440 (1 990) BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS by WOJCECH SACHWANOWICZ in Torun (Poland) 1. Introduction and preliminaries The Boolean power construction of structures was introduced by A. L. FOSTER [7] to pro- vide a transparent structure theory of certain classes of algebras. From the beginning it suf- fered from the defect (S. BURRIS [2]) - it can be carried out if the Boolean algebra applied to the construction was complete. To overcome this difficulty the bounded Boolean power con- struction was introduced by FOSTER, some remedy, via a topological approach, was proposed by B. BANASCHEWSKI and E. NELSON [l], but really this difficulty was not overcome. The aim of this paper is to introduce the extension of FOSTER’S construction which is appli- cable to any structure and any Boolean algebra and, nevertheless, preserves all desirable pro- perties of the original construction. To this end we remind the definition of a complete partition in a Boolean algebra intro- duced by W. SACHWANOWICZ [lo]. Complete partitions are in ample supply in each Boolean algebra since they are “coding” the direct product representations of the algebra (The- orem 1.1), moreover they “carry” as much completeness as it is needed to accomplish the construction. In the first section the main definition is introduced (Definition 2.1). Then it is proved that the Boolean power construction is a bifunctor from the category of all Boolean al- gebras (with homomorphisms preserving complete partitions) and the category of all struc- tures of some language. This bifunctor preserves infinite direct products, injections and sur- jections on the first coordinate and preserves products and surjections on the second coordinate (Theorem 2.3). In the third section we are concerned with the logical properties of the construction, it is shown that a kind of Feferman-Vaught theorem (Theorem’3.4) holds and it is proved that disjunctions of the Horn sentences are preserved. The main result in the last section is the topological representation of the construction by the set of all perfect map- pings which domain is an extremally dense subset of the Stone space of the Boolean algebra (Theorems 4.8 and 4.9). It occurs that this representation generalizes and unifies the repre- sentations proposed by B. JONSSON, A. DAIGNEAULT [4] and P. RIBENBOIM [9]. We use bold capital letters A, B, ... to denote Boolean algebras (B.a.3 for short), letters A, B, ... to denote the underlying sets of these algebras, the symbols +, ., -, 0, 1 to denote the operations and the constants of the respective B.a. By sup C we denote the least upper bound of the subset C of a B.a. The symbols a, 0, y, ... and 41, v, .. . are used to denote cardinals and ordinals, respectively. By dom f, mg f and f”X we denote the domain and range off and the image of the set X and f, respectively. A subset C of a B.a. B is said to be a partition in B if a. b = 0 for any different a, b of C. The symbol n{Ai: i E I) denotes the direct product of the B.a.’s Ai. If A, C are subsets of a B.a. B, then A * C denotes the set {a * c: a E A and c E C} .

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Page 1: Boolean powers over incomplete boolean algebras

Zeigchr. 1: math Logik und Grundhgen d Math Bd 36. S. 431 -440 (1 990)

BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS

by WOJCECH SACHWANOWICZ in Torun (Poland)

1. Introduction and preliminaries

The Boolean power construction of structures was introduced by A. L. FOSTER [7] to pro- vide a transparent structure theory of certain classes of algebras. From the beginning it suf- fered from the defect (S. BURRIS [2]) - it can be carried out if the Boolean algebra applied to the construction was complete. To overcome this difficulty the bounded Boolean power con- struction was introduced by FOSTER, some remedy, via a topological approach, was proposed by B. BANASCHEWSKI and E. NELSON [l], but really this difficulty was not overcome.

The aim of this paper is to introduce the extension of FOSTER’S construction which is appli- cable to any structure and any Boolean algebra and, nevertheless, preserves all desirable pro- perties of the original construction.

To this end we remind the definition of a complete partition in a Boolean algebra intro- duced by W. SACHWANOWICZ [lo]. Complete partitions are in ample supply in each Boolean algebra since they are “coding” the direct product representations of the algebra (The- orem 1.1), moreover they “carry” as much completeness as it is needed to accomplish the construction. In the first section the main definition is introduced (Definition 2.1). Then it is proved that the Boolean power construction is a bifunctor from the category of all Boolean al- gebras (with homomorphisms preserving complete partitions) and the category of all struc- tures of some language. This bifunctor preserves infinite direct products, injections and sur- jections on the first coordinate and preserves products and surjections on the second coordinate (Theorem 2.3). In the third section we are concerned with the logical properties of the construction, it is shown that a kind of Feferman-Vaught theorem (Theorem’3.4) holds and it is proved that disjunctions of the Horn sentences are preserved. The main result in the last section is the topological representation of the construction by the set of all perfect map- pings which domain is an extremally dense subset of the Stone space of the Boolean algebra (Theorems 4.8 and 4.9). It occurs that this representation generalizes and unifies the repre- sentations proposed by B. JONSSON, A. DAIGNEAULT [4] and P. RIBENBOIM [9].

We use bold capital letters A , B, . . . to denote Boolean algebras (B.a.3 for short), letters A , B, ... to denote the underlying sets of these algebras, the symbols +, ., -, 0, 1 to denote the operations and the constants of the respective B.a.

By sup C we denote the least upper bound of the subset C of a B.a. The symbols a, 0, y, . . . and 41, v, . . . are used to denote cardinals and ordinals, respectively. By dom f, mg f and f ” X we denote the domain and range off and the image of the set X and f, respectively.

A subset C of a B.a. B is said to be a partition in B if a . b = 0 for any different a, b of C. The symbol n { A i : i E I ) denotes the direct product of the B.a.’s Ai. If A , C are subsets of a B.a. B, then A * C denotes the set { a * c : a E A and c E C} .

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432 W. SACHWANOWICZ

In this paper we use without reference the well-known "conditional" equations (cf. e.g. [12])

( 1 ) ( 2 ) (3) for any subsets Bi ( i E I ) , C, D and any element a of a B.a.A.

We need the following notions, which were first introduced in [ lo ] . Let B be a B.a. and let P, Q be any partitions in B, P is said to be a narrowing of Q (denoted by P 5 Q), if each ele- ment of P is under some element of Q and under each element of Q there is at most one non-zero element of P. A partition Q is said to be a complete partition (cpartition, for short) if sup P exists for each narrowing P ef Q . A c.partition P is said to be a c.partition of an ele- ment a if sup P = a. Cl(B) denotes the set of all c.partitions of the unity in B. One should see that each finite partition is complete, and any partition in a complete B.a. is complete, more- over, if every partition in a B.a. B is complete, then B is complete. Let B)b denote the restric- tion of B to b E B .

Theorem 1.1. (a) Lei P be a c.partition of the unity in B. Then B is isomorphic to n { B Ib : b E P } . (b) I f B = n{Ai: i E I } , for some B.a.'s Ai, then there is a c.partition {ai: i E I } in B such that Ai is isomorphic to Blai for each i E I . (We shall say that the c.partition {ai: i E I } corresponds to the direct product representation of B.)

a . sup c = sup(a. C), sup{sup Bi: i E I } = sup u { B i : i E I } , sup C. sup D = SUP(C. D),

We say that c.partitions P, Q are disjoint if sup P. sup Q = 0.

Lemma 1 . 2 . (a) For any set X of disjoint c.partitions in B the set U X is a c.partition if and only if so is {sup P: P E X } . (b) r f P1, . . . , p n are c.partitions, then PI * . .. * P, is a c.partition and sup(P. ... . P ) = supP. ... .sup P.

2. The definition and basic properties

In this chapter the notion of Boolean power construction over an arbitrary Boolean algebra is introduced. It is shown that this construction is a bifunctor from the category of B.a.'s and c.partitions preserving homomorphisms into the category of L-structures. We prove that this functor preserves finite direct products (surjections, injections) on the first and arbitrary pro- ducts (surjections) on the second coordinate, respectively.

First we specify the well-known definition of Boolean powers [7] (B.p.'s for short) to make it working in the case of incomplete B.p.'s.

Defini t ion 2 . 1 . Let B be a B.a., let M be a structure of a language L, and let a be an infinite cardinal. By an a-bounded Boolean power (B.p., for short) of M to B we mean a struc- ture M[B]" of L such that (a) the universe M[B]" of M[B]" is the set of all mappings f : M-+ B with

(i) f m - f n = 0, whenever m =k n , (ii) rng f is a c.partition of the unity in B, (iii) card(rng f ) < a.

The operations and relations on M[B]" are defined as usual, and we remind: (b) for an k-ary operation symbol F of L:

F(f,, .. ., fk) m = sup{f,m,. . .. . fkmk: F'(m,, . .., mk) = m } , forf,, ..., f k ~ M[B]"and m E M .

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BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS 433

(c) for an k-ary relation symbol R of L :

R ( f 1 , ..., fk) iff SUp{flm,. ...’ fkmk: R M ( m l ,..., mk)} = 1 , for fi,..., f k ~ M [ B l ~ .

The correctness of the Definition 2.1 follows immediately from Lemma 1.2.

If we drop the condition (iii)a of Definition 2.1 we get an (unbounded) Boolean power of M to B (we use M[B] to denote it). However, it is obvious that M[B] is equal to M[BIa for any 01 2 max{w, min{card M, satc B}} (where satc B denote the least cardinal which bounds the cardinalities of c.partitions in B). From Definition 2.1 it follows immediately that the well- known B.p. constructions due to FOSTER [7] are special cases of our construction, namely, the bounded B.P. M[B]* is M[BIw in our notation and the both constructions of unbounded B.p.’s over a complete B.a. coincide exactly.

Let A, B be B.a.3. We say that a homomorphism h : A -+ B preserves complete partitions if, for each c.partition P of A, h”P is a c.partition of B and h sup P = sup h”P.

Since the B.a. P(o) of all subsets of w cannot be, injected into any free (or superatomic) B.a. A , any homomorphism defined on A preserves c.partitions. It is also obvious that each complete (i.e. sup’s preserving) homomorphism into a complete B.a. preserves c.partitions. But the injection h from the B.a. P(w) into P(w) [F(w)] (where P(w) denote the free w-gen- erated B.a.), given by XH f x , where fxX = 1 and f x Y = 0 for any subset Y * X of w , is com- plete and does not preserve c.partitions ([ll]). However, we can state

Lemma 2 . 2 . If h : A -+ B is a surjection and if h preserves sup’s of complete partitions of A , then h preserves complete partitions of A.

Proof. Assume that P E C , ( A ) and Q S h”P. Then for some S = {ab: b E P } E A , h’S = Q and hab 5 hb. Hence there are (for b E P ) dbE h- ’ { l } Such that ab.db 5 b ’ d b , SO

{ab ’ db: b E P } 2 { b ’ db: b E P } 2 P . Thus {ab. db: b E P } E C,(A) and, by the assumptions, we have h sup{ab‘ db: b E P } = SUp{h(ab. db): b E P } = SUp{hab: b E P } = SUP h”S= SUP Q. Hence h”P is a c.partition. 0

As an immediate corollary of Lemma 2.2 we see that each complete surjection of B.a.’s preserves c.partitions.

Let Lstr denote the category of all structures for a language L with homomorphisms as morphisms and let Boolc denote the category of all B.a.’s with c.partitions preserving hom- omorphisms as morphisms. For each structure M for L we define a mapping M[?] from the category Boolc, which to every B.a. A assigns the B.p. M[A] and to each morphism h : A + B of Boolc assigns the homomorphism M [ h ] : M[A] + M[B] of L-structures defined by

(4) M [ h ] f = h o f , for f e M[A].

For any B.a. A we define a mapping ?[A] from the category Lstr into Lstr in the following way: to each L-structure M the B.p. M[A] is assigned, and to each morphism g : M + Nthere is assigned the mapping g[A]: M[A] -+“A] given by

( 5 ) (g[A] f ) n = sup{ f m : gm = n } , for f~ M [ A ] and n E N .

The o r e m 2 . 3 (see [ 11). The Boolean power construction is a bifinctor from the category Lstr x Boolc into the category Lstr, which has the following properties: (a) For each L-structure M the functor M[?] preserves products and injections. (b) For each Boolean algebra B the functor ?[B] preservesfinite products, injections and surjections.

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434 W. SACHWANOWICZ

Proof. The fact that ?,[?4 is a bifunctor can be proved by a standard argument. For (a) let M be any L-structure. First we show that M[?] preserves products. Let

B = n{Bi: i E I } . We define the mapping h : M [ B ] + n { M [ B i ] : i E I } by

(6) (pi(hf)) m = p i ( f m ) , for i E I , f~ M[B] and m E M ,

where pi denote the projection onto the ith coordinate (we use the same symbol pi for diffe- rent projections but we hope that the domain and the range of pi will be clear from the con- text). First we check that h f ~ N = n { M [ B i ] : i E I } , i.e. (pi(hf)) m E M[Bi] , for each i E I . It is obvious that (pi(hf)) m, * (pi(hf)) m2 = 0 for different m , and m2 of M , and by Lemma 2.2 mg(pi(hf)) = pi”(rngf) is a c.partition of the unity, hence the defrnition (6) of h is correct. To check that h is a homomorphism, let F be any (binary, for convenience) operation symbol of L , and let fi, fi E M [ B ] , let m E M and let i E I . Then by (6), Lemma 2.2 and Definition 2.1

(Pi(h(Fhf2)) m =~i((Ffifi) m ) =pi(sup{f,m, ‘fzmz: FMmlmz = m } ) = suP{pi(flmJ .P i ( f imz) : FMm1mz = m } = Pi((Fhf1h.L) m ) .

The proof of the fact that homomorphism h preserves relations is similar. It is obvious that h is an injection, so we have to prove that it is a surjection. Let g E N . Define f: M + B by

pi(fm)=(pig)m,for i E Z and m E M .

It can be easily proved that f E M [ B ] and that hg =f. Hence h is an isomorphism. The fact that the functor M[?] preserves injections is obvious. To prove (b) let B be any B.a. and let M = n { M i : i E I } for a finite set I . We define the

mapping h from M[B] into f l {Mi[B]: i E I } by

(p i (hf ) ) m = sup{fn : pin = m } , for each i E I, m E Mi, and f e M[B] .

It is a standard calculation which shows that the mapping h is well defined and an isomor- phism. The proof that ?[B] preserves injections and surjections is also easy. 0

Generally the functor M[?] does not preserve surjections as the following example shows. Let M be any infinite structure and let g : F(2”) + P ( w ) be any surjection. Then it is easy to see that g preserves c.partitions and for each f E M[F(2”)] the set rng(M[g]f ) is finite; how- ever, there are elements of M[P(w)] with infinite range.

Remarks. (a) The category Bool, of all B.a.’s with the B.homomorphisms which preserve c.partitions

(b) One can see that Lemma 2.2 has an “a-bounded” version. (c) It can be noticed that

of cardinalities less than a as morphisms can be defined.

is a functor from the category Bool, x Lstr to the category Lstr . Moreover, (i) M[?]“ preserves injections and direct products of less than a B.a.’s if a is a regular cardi-

nal; (ii) ?[B]” preserves injections and surjections, moreover it preserves products of /3 L-struc-

tures whenever in a B.a. B for any set X of at most B c.partitions of the unity, each of cardinality less than a , there is a common narrowing into a c.partition (any pa-complete and a-distributive B.a. has this property).

(d) In fact it can be proved (compare with [ l ] ) that the functor ?[A] preserves pure embed- ding and M[?] is the only functor from the category Boolc into the category Lstr which pre- serves infinite products and c.partitions reflecting direct limits.

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BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS 43 5

3. Logical properties of the Boolean power construction

In this chapter we investigate the logical properties of the B.p. construction. We define the valuation function (which is a generalization of the valuation introduced by R. MANSFIELD [8]) and prove that it has the maximality (Proposition 3.2) and the patchwork property (Lemma 3.1). As a consequence we get a kind of the Feferman-Vaught theorem (The- orem 3.4) and the preservation of elementary equivalence and substructure by the B.p. con- struction, and it is also proved that for any structure M and any filter U on a B.a. B the struc- tures M[B,] and M[BIv are elementarily equivalent (Corollary 3.5).

Let @(F) be a formula of a language L, let M be an L-structure and let B be a B.a. Then for any tuple Jof elements of M[B] define (see [S])

(7) [@(n] = sup{fm: M k @[GI}. It is obvious, by Lemma 1.2, that the definition (7) is correct. (@(fiD is called the valued of @ on f in W B ] .

The following lemma, called the patchwork lemma, is well known for bounded B.p.’s and it appears in [ll]. Here we can state it for arbitrary B.p.’s.

Lemma 3 . 1 (Patchwork Lemma). Let P be a c.partition in Band let If,: a E P} be a subset of

Proof. Without loss of generality we can assume that supP = 1 . It is obvious that = {a . f , m : a E P} are c.partitions. Hence we can

M[B]. Then there exha an element of M[B] such that if= f , D h a, for any a E P.

Q = {a * A m : m E M and a E P } and define:

f m = s u p { a . f , m : a E P } , for m E M .

From this definition it follows that f m . f n = 0, whenever m * n , and since 1 = sup Q = sup(rngA, from Lemma 1.2 we have that f E M[B] . Now,

( f = f.]= sup{ f m . f , m : m E M} = sup{ b .fbm . f , m : m E M and b E P }

2 sup{ a . f, m : m E M } = a sup{& m : m E M } = a . 0

The following facts are well known in the “world” of bounded B.p.’s (see [2]); we show that

Proposition 3 . 2 . Let @(a, Y(5) be any L-fonnulas and let e,f,g, h E M [ B ] , then

they hold for arbitrary B.p.’s.

(a) 0) [@ L?L u? tfiD = (@tfiD. ( WfiD, (ii) 0 1 WfiD = - U@(fiD, (iii) (Maximality) if @ = 3,@(y, F), then

U3,@(fiD=s~{il@(g,fiD: gEM[BI}=SuP{BO(fm,fiD: m E M } ,

where f , denotes an element of M[B] such that f,m = 1 and fmn = 0 for m * n, i.e.

(3,@(fi] = lO(g,fi], for some g E M [ B ] .

B f = f D = 1 , (iii) (f= g] - (g = e] 6 ( f = eD,

(b) 0) (ii) B f = gD = (g = f D , (iv) if= gD U@U 6 1 5 U@(& KID.

Proof. The proofs of (a) (i), (ii) and (b) (i)-(iv) are quite standard so we prove here only (a) (iii). Let p be a Skolem function of @, i.e. M C @[r/M, a] whenever M e 3,0[m]. We de- fine:

gn = sup{frii: prfi = n } , for n E M.

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436 w. SACHWANOWICZ

The definition is correct, .by Lemma 1.2. We show that g E M [ B ] . If m +; n, then gm . gn = 0, since y is a function; moreover,

supign : n E M I = sup{sup{fm : y m = n } : n E M } = s u p ( 7 ~ : I,, ,,,ye = n ) = sup{ffi : m E M } = 1.

From Lemma 1.2 it follows that rngg is a c.partition. It remains to check that g is the maxi- mal element:

(3,@(f)D = sup{fm: MI= 3 , 0 [ f i ] } =sup{ffi: M t O [ n , f i i ] , for some n E M } = sup{fm : M + @[@, m ] } = sup{g(ym) s f m : M + @[pi, M I } = sup{gn . f f i : M t O[pfi, m ] } = ( O ( g , f ) D .

The other equation in (a) (iii) follows from (a) (i), (ii), the tautology @(y)*3,@(y) and the obvious fact 1 = sup{(g = f,): m E M } . 0

Y , k

Proposition 3.2 (a) (i) can be stated for formulas with infinite conjunctions.

Proposi t ion 3 . 3 . vZ(T) is a set of L-formulas with afinite tuple X offree variables, then for f= WBI,

(A\c(f)D = inf{@(fl: @ E Z}.

Proof. From Proposition 3.2 it follows that [A\c(f)) 5 (@(f)), for all @ E Z. To show that (AZ(f)D is the greatest lower bound suppose that a s Q @ ( f ) ) , for all @E Z, hence a sup{fm : M + @(m)}, for @ E C. Inasmuch as VrTi : m E M } is a complete partition of the unity, we have that for @ , Y ‘ Y Z , if a . f m 1 0 and M k @(f i ) , then Mk= Y(m); so we have a ~ { ~ ~ : M ~ @ ( r i i ) , f o r a l l @ ~ E ] ~ s u p { f ~ : M ~ ~ Z [ t T i ] } = ( ~ E ( f ) ~ .

For Lemma 3.1 , Proposition 3.2 and 3.3 “@-bounded” versions can be given easily.

Let U be a filter on a B.a. B. We define the relation (v) on a B.p. M[B] by

(8) f ( U ) g iff sup(fm.grn: m E M } E U .

Using this relation we define in a standard way the reduced by U Boolean power of M to B and denote it by M[B], .

Since B.p.’s and reduced B.p.’s have the maximality and the patchwork property, a kind of Feferman-Vaught theorem can be proved (cf. [ 1 3 ] and [15] ) . Let d@ = : (d@*, dQ0,. . ., d a m ) be the companion sequence of an L-formula @.

Theorem 3 . 4 . Let M be an L-structure, let A be a Boolean algebra and let U be afilter on A and let 01 be an infinite cardinal. Then for a first order formula @(Z) of L and for a tuple f of M [ A]“ the following are equivalent:

(a) M [ A I ~ + W ~ I , (b) &~d@*[Ud@df)D., ...,Gd@m<f)noI.

Now we can state

Corollary 3 . 5 . Let M and N be L-structures, let A , B be Boolean algebras and let U be afilter on A . Then: (a) For any infinite cardinals 01 5 p, M [ A ] ” , < M [ A ] $ . (b) If M = N and A = B, then M [ B ] = “ A ] (see [14 ] ) .

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BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS 437

(c) If M < N and A i c B , then M [ B ] i “ A ] , where the subscript “c” under i denotes that

(d) M [ A l u M[AuI. c.partitions are preserved by the set inclusion.

Now since every B.a. is elementarily equivalent to a reduct of P(w) from Corollary 3.5 and KEISLER’S theorem (see [3]) on preservations of sentences by the reduced power construction we get

C o r o 11 a r y 3 . 6 , A first order sentence @ of L is preserved by the Boolean power construction if and only if @ is logically equivalent to a finite dkjunction of first-order Horn sentences.

Theorem 3 . I . Let M be an L-structure and let U be a filter on a Boolean algebra B. Then for any L-fonnula @(Z) and for all TE M [ B ] the following hold: (a) If@($ is a Horn formula, then M[B] ,+ @,[&I, whenever (@(y)j E U. (b) (LoS’s theorem) i’f U is an ultrafiller, then M[BIui= @[.&,I (c) I f U is an ultrajilter, then M is an elementary substructure of M[B]..

The following proposition shows some connections between B.p.’s constructed on a B.a. B and on its completion BC. An B.a. B is called separable, if the set of all atoms of B has a sup. It was noticed in [l] that B is separable if and only if B can be elementarily injected into BC.

Now, from the above and Corollary 3.5 it follows a generalization of Proposition 24 in [I].

Proposit ion 3 . 8 . For any Boolean algebra B and for any infinite cardinal GC the following are

iff (@(f)) E U.

equivalent: (a) B is a separable Boolean algebra. (b) M[B]“ < M[Bc]“ , for any L-structure M .

4. Topological representation of Boolean powers

It is well known that the categories Boo1 of B.a.’s and TBool of Boolean spaces (i.e. com- pact, zero-dimensional topological spaces) are dually isomorphic and this dual isomorphism is provided by the functors CO and S, where the object part of CO associates with every Boo- lean space X the B.a. of clopen (i.e. closed and open) subsets of X and S(B) is the Stone space of all ultrafilters of the B.a. B.

Proposit ion 4 . 1 (P.DWINGER [ 5 ] ) . For any Boolean algebras B and Ai ( i E I ) , if B = n { A i : i E I } , then S(B) = B(@{S(A,): i E I } ) , where @ denotes the topological sum and B is the Cech-Stone compactiJication operator.

Later we shall use the same notation for an element of a B.a. B and its counterpart the clopen subset of S(B). From Theorem 1.1 and Proposition 4.1 it follows

Corollary 4 . 2 . A partition P in B is complete iff P is a disjoint family of clopen subsets of S(B) such that b ( @ P ) = S ( B ) .

First we remind some definitions and facts from the general topology (for unexplained no- tions the reader should consult e.g. the monograph by R.ENGELKING [6]). A mapping of Hausdorff spaces is called p e ~ e c t if it is continuous, closed and has compact fibres. It is known that each completely regular topological space has the Cech-Stone compactification.

Lemma 4 . 3 . Let X be a completely regular topological space. Then we have: (a) The following conditions are equivalent:

(i) X is a locally compact space, (ii) X is open in bX.

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438 w. SACHWANOWICZ

(b) Every pair of disjoint closed Gd-sets has disjoint closures in BX. (c) Every clopen subset of X has a clopen closure in BX. (d) Zf X is normal and Z is a closed subset of X, then cl,Z is equivalent to BZ (i.e. clBXZ and g Z

are homeomorphic over Z) and we write claxZ = BZ. (e) Z f X E YE PX, then g X = BY. (0 If Y is a completely regular space and p : X -+ Y is a continuous mapping, then p is perfect iff p

cannot be extended to a continuous mapping on a Hausdorff space Z which contains X as a dense and proper subspace.

For proofs of these facts the reader should consult [6], where (a) is an immediate corollary from 3.5.8, @) follows from 3.6.2 and (c), (d), (e), (f) are 3.6.5, 3.6.8, 3.6.9, and 3.7.16, respec- tively.

We have to prove some more facts from topology.

Lemma 4.4. Let {Xi: i E Z} be a closed cover of a normal topological space X and let Y be a closed subset of PX. Then

(a) j9X = BU{BXi: i E I } ,

Proof. For (a): By Lemma 4.3(d) for ~ E Z we have fixi= clsxXiS/3X, hence U{BX;: i E I } E BX. But X = U { X i : i E I } E u { B X i : i E I } . So by Lemma 4.3 (e), pX = BU{BXi: i E I } .

For (b): Immediately we see that U ( X i n Y ) = Y n UXi = Y n X is a closed subset of X , hence, by Lemma 4.3(d), since U ( X i n Y ) is a dense subset of the closed set r, BU(Xi n Y ) = Y. 0

(b) BU{Xi n Y : i E Z} = Y.

Lemma 4.5. Let X and Y be subspaces of a topological space Z such that (a) X, Y, X n Yare normal subspaces, (b) there is a cover {x: i E I } of Y such that

Then @ ( X n Y ) = j9Y. (i) ( i E Z) are closed subsets of BX, (ii) X n ( i E Z) are closed subsets of X n Y.

Proof. Follows immediately from Lemma 4.4. 0

Now we are ready to define the topological power. First the universe of the topological power of a structure will be defined. Let X be a compact and Hausdorff topological space and let M be a discrete space. Then by P ( X , M) we denote the set of all perfect mappings u de- fined on a subspace of X such that Bdom u = X .

(a) For each u of P ( X , M), dom u is a locally compact, dense and open subspace of X . (b) For any u l , . . ., uk o f P ( x , M), fi(dom u1 n .. . n dom uk) = X .

(c) Any two elements of P(X , M ) equal on a dense subset of X are equal.

Lemma 4.6.

Proof. (a) follows immediately from Lemma 4.3. For (b) it suffices to give a proof for two mappings u1 and u2 of P ( X , M ) . Since ui are perfect and M is discrete, we see that dom ui = @{u;’{m}: m E M } . By (a) u;’{m} is a clopen subset of X . Now from Lemma 4.5 it follows that B(dom u1 n dom u 3 = B(dom u2) = X . For (c) let u l , u2 E P ( X M ) be equal on a dense subset of X . Then u1 and u2 are equal on dom u1 n dom u2. Now, from Lemma 4.3 we have u1 = u2.

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BOOLEAN POWERS OVER INCOMPLETE BOOLEAN ALGEBRAS 439

Let a structure M of a language L be given. For any operation symbol F of L (we assume,

Step 1. For any u, u E P(X, M), let F(u, u ) be the mapping from dom u n dom u into M de- for convenience, that F is binary) the operation Fx on P(X, M ) can be defined in two steps:

fined by

F( u, u ) x = FM( ux, ux ) . It is obvious that F(u,u) is a continuous and closed mapping, but it may not be perfect. Hence we have to do the second step.

Step 2 . Since {F(u, u)-'m: m E M} is a family of clopen and disjoint subsets of dom u n dom u, so, by Lemma 4.3 (b) (c), the set Z = {clx(F(u, u)-'m): m E M} is a family of clopen and disjoint subsets of X . Thus we can define the mapping Fx(u, u ) on @ Z by

Fx(u, u ) x = m iff x E clx(F(u, u ) - ' m ) .

It is immediate that Fx(u, u ) is a perfect mapping from @Z into M. To finish the definition notice that dom u n dom u @ Z E X , so, by Lemma 4.3 and 4.6 (b), B @ Z = X . Hence Fx(u, u ) E P(X, M) and that is the unique extension of F(u, u) .

In this way we can associate with each k-ary operation FM of M the k-ary operation Fx on P(X, M ) . To each relation RM on M a relation RX on P(X, M ) can be associated by

R X ( u l , . .., uk) iff p{x E dom u1 n .. . n dom l ( k : R'(ulx, . .., ukx)} = x. Definit ion 4 .7 . For any L-structure Mand a compact topological space X the structure

P(X, M) of L can be associated by the above process; we call this structure the topological power of M to X .

The topological power construction provides us with a representation of the B.p. construc- tion.

Theorem 4 . 8 . For any B.a. B and any L-structure M the structure M[B] is komolphic to P(S(B), M).

Proof. By standard calculations it can be proved that the mapping h : M [ B ] - P(S(B) , M ) defined by

( h f ) x = m iff x E f m , for any x E S(B), f E M[B] and m E M, provides the isomorphism. 0

The topological power is not the essentially new construction as the following theorem shows.

Theorem 4.9 . For any compact topological space X and any L-structure M , P(X , M ) b bo- motphic to M[CO(X)] .

Proof. By Corollary 4.2 it is routine to show that the mapping g : P(X, M ) +M[CO(X)] defined by (gu) m = u-lm, for u E P(X, M) and m E M, is the isomorphism. 0

Remarks. (a) Definition 4.7 and Theorems 4.8 and 4.9 can be easily modified to provide a topological representation for a-bounded B.p.'s of structures. Indeed, let X be a compact topological space, let M be an L-structure and let a be an infinite cardinal. Denote by P(X, M)" the set of all mappings u of P(X, M) with card(mg u ) < a. Restricting the relations and operations of P(X, M ) to this set one obtains an L-structure P ( X , M)" the a-bounded to- pological power of M to X . The analog of Theorems 4.8 and 4.9 can be proved.

Page 10: Boolean powers over incomplete boolean algebras

440 w. SACHWANOWICZ

(b) It can be noticed that an element of P(X, M)”, i.e. an element of P(X, M ) with finite range, has as domain the whole space X . Thus P(X, M)” is exactly the Jonsson representation of bounded Boolean powers.

(c) Let B be a complete B.a. It is known that S(B) is an extremally disconnected (i.e. the closure of an open subset of S(B) is open) and compact space. In such spaces exactly the dense and open subsets have the cech-Stone compactification (in fact this property charac- terizes the extremally disconnected compact spaces). Since by Lemma 4.3 (f) the continuous and partial mappings from S(B) into M , which cannot be extended, are exactly the perfect mappings, we see that P(S(B) , M ) is exactly the topological representation of B.p.’s intro- duced by P. RIBENBOIM [9].

There is another representation of B.p.’s introduced by DAIGNEAULT [4] (see also [l]): Let E(X, M ) = D(X, M ) / - , where X is a topological space, D(X, M ) is the structure of all map- pings v: X - t M continuous on some dense and open subset of X with the pointwise defined operations and relations, and where - is the equivalence relation on D(X, M ) identifying any two mappings which agree on some dense and open set. Then it can be shown ([l]) that for any complete B.a B and any L-structure M , E ( S ( B ) , M ) is isomorphic to M[B] . DAIGNEAULT’S representation can be generalized to cover the case of the powers to incom- plete B.a’s. Moreover, it can be proved that in each equivalence class there is exactly one per- fect mapping of P ( X , M) .

(d) It is known that the B.p. construction for complete B.a.’s can be stated for incomplete, B.a.’s introducing some modifications to this representation.

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ENGELKING, R., General Topology. PWN (Polish Scientific Publishers), Warsaw 1977. FOSTER, A. L., Generalized “Boolean” theory of universal algebras. Part 11: Identities and subdirect sums of functionally complete algebras. Math. 2. 59 (1953), 191-199. MANSFIELD, R., The theory of Boolean ultrapowers. Ann. Math. Logic 2 (1971), 297-325. RIBENBOIM, P., Boolean powers. Fund. Math. 65 (1969), 243-267. SACHWANOWICZ, W., A note on complete partitions in Boolean algebras. This Zeitschrift 36 (1990),

SACHWANOWICZ, W., Boolean powers of Boolean algebras. In preparation. SIKORSKI, R., Boolean Algebras. Springer-Verlag, Berlin-Heidelberg-New York 1964. VOLGER, H., The Feferman-Vaught theorem revised. Colloquium Mathematicum 36 (1976), 1- 11. WASZKIEWICZ, J. , and B. WF~LORZ, Some models of theories of reduced powers. Bull. Acad. Polon. Sci., Set. Sci. Math. Astronom. Phys., 16 (1968), 683-685. WOJCIECHOWSKA, A,, Generalized limit powers. Bull. Acad. Polon. Sci., SBr. Sci. Math. Astronorn.

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Poland 87-100 Tonth

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