Algebra univers. 39 (1998) 57–70 0002–5240/98/020057–14 $ 1.50+0.20/0 © Birkha¨user Verlag, Basel, 1998
a-cut-complete Boolean algebras A. W. HAGER
Abstract. Let A be a Boolean algebra, and a an infinite cardinal number or the symbol . An a-cut in A is an ordered pair (F, H) of subsets of A, each of power Ba, with F 5H elementwise, with 0 as the meet of differences h− f (h H, f F). A is called a-cut-complete if for each a-cut (F, H) there is a A with F 5 a5 H elementwise. We describe the simply-constructed a-cut-completion A a, show that a-cut-completeness solves a natural a-injectivity problem, determine when A a is the a-completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for a =v1 , has received considerable study.
1. Preliminaries We generally use [17] as a reference for Boolean algebras, though we do not always follow the notation. If A is a Boolean algebra, with elements a1 , a2 A, the complement of a1 is a%1 , the join (resp., meet) of a1 and a2 is a1a2 (resp., a1 a2 ), and a2 − a1 is a2 a%1 . For F ¤ A, the join (resp., meet) of F is, when it exists, 0F (resp., / F). If A is a Boolean subalgebra of B, we write A5 B. For F¤ A5 B, we may have to write 0A F or 0B F. A is dense in B if A 5B, and for each b B there is F¤ A with 0F = b (and dually); we write A 5d B. Then (1) if G¤ A with 0A G existing, then 0A G= 0B G ([17], p. 93); (2) if 8: B C is a Boolean homomorphism with 8 A one-to-one, then 8 is one-to-one (for, if b \ 0 there is a A with 0Ba5 b, so 0B 8(a)5 8(b)).
Presented by M. Henriksen. Received June 25, 1997; accepted in final form March 2, 1998. 1991 Mathematics Subject Classification: Primary 06E10, 06E15, 54G05; Secondary 06F20, 46A40, 54C45, 54C50. Key words: Boolean algebra, a-cut-completion, a-injective, a-cloz space, quasi-Fa space. 57
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A is complete if 0F exists for each F¤ A. Each Boolean algebra A has a completion A , by definition: A 5d A , and A is complete. We note ([17], §35) that A 5d B iff there is a one-to-one Boolean homomorphism 8: B A which is the identity on A; we say, B embeds into A over A, and just write A 5B5 A . (This last can be proved from Sikorski’s Injectivity Theorem ([17], §33), but a more elementary argument suffices; see §3 here.)
2. a-cut-completeness and -completion For the rest of the paper, a denotes a regular cardinal number or the symbol , thought of as larger than any cardinal. The meanings of various notions involving should be clear from context, and will occasionally be remarked on, e.g., for the cardinality F of the set F, F B holds vacuously. DEFINITION 2.1. An a-cut in the Boolean algebra A is an ordered pair (F, H), where: F, H¤ A; F , H B a; f5 h for each f F and h H; /{h − f h H, f F} = 0. A is a-cut-complete if, whenever (F, H) is an a-cut in A, then there is a A with f 5 a 5 h for each f F and h H (equivalently, 0F exists; equivalently, /H exists). A is a-dense in B if A 5B and for each b B there is F¤ A with F B a and 0F = b (and dually); we write A 5ad B. Note that, when A 5ad B, an element b B produces an a-cut in A: simply choose F, H¤ A with F , H B a, and 0F = b = /H. It is easy to see that -cut-completeness is completeness, and that -density is density. (But an a-cut is not a ‘‘cut’’ in the sense of [17], p. 155, for cuts have a maximality property.) It is now easy to see that, in the following definition when a = , the two meanings of A coincide. The construct there, A a, will be called the a-cut-completion of A, which term is justified by the various results of this section. DEFINITION 2.2. Let A be a Boolean algebra. Let A a dfn
{b A 0F = b= /H for some a-cut (F, H) in A}.
The following shows that A a generalizes some of the features of A mentioned in §1. THEOREM 2.3. For any Boolean algebra A, (a) A a is a Boolean algebra,
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(b) A a is a-cut-complete, and whene6er A5 B5 A with B a-cut-complete, then A a 5B, (c) A 5ad A a, and whene6er A 5ad B, then B5 A a. Proof. (a) For M, N¤ A, let M% {m% m M}, and MN {mn m M, n N}. Then, it’s easy using the laws in [17], pp. 59, 60, that (1) if (F, H) is an a-cut, then so is (H%, F%), and (2) if (Fi , Hi ) (i= 1, 2) are a-cuts, then so is (F1F2 , H1H2 ). Then, A a is closed under complements within A (using (1)), and finite joins within A (using (2)), and this suffices. (b) Let (F, H) be an a-cut in A a. Then, in A , 0F = /H since /{h − f h H, f F} = 0. (The meet in A a is assumed 0, hence the meet in A is 0, since A a 5 d A (§1). Let b denote 0F=/H. We show b A a (and it follows that b=0F in A a). For each f F, f = 0Ff (0 in A ) for an Ff which is the left half of an a-cut in A, and likewise, for each h H, h=/Hh for an Hh which is the right half of an a-cut in A. Let F0 =.{Ff f F} and H0 = .{Hh h H}. Since a is regular, or , F0 , H0 Ba. Note that 0F0 = 0{0Ff f F}=0F =b, and /H0 = /{/Hh h H} = /H = b, using the associative laws in [17], §19. It now follows that (F0 , H0 ) is an a-cut in A, so that b= 0F0 A a, so A a is a-cut-complete. Now suppose A 5 B 5A , with B a-cut-complete. Let x A a, so that x= 0F = /H in A , for an a-cut (F, H) of A. Since B 5d A , it follows that /B {h − f h H, f F} =0, and (F, H) is an a-cut in B. Since B is a-cutcomplete, 0B F exists. Again, this is also the join in A , i.e., x. So x B, and A a 5 B. (c) It is obvious that A 5ad A a. Suppose that A 5ad B. Since A 5d B, we have A5 B 5 A (as noted in §1). If x B, then there are F, H¤ A with 0F = x= /H, by a-density. Thus x = 0F in A , and (F, H) is an a-cut in A (again, since B 5d A ). So x A a, and B 5 A a. The two corollaries following are immediate from 2.3. COROLLARY 2.4. B is isomorphic to A a o6er A iff A 5ad B and B is a-cutcomplete. COROLLARY 2.5. The following are equi6alent: A is a-cut-complete; A= A a; if A 5ad B, then A=B. We must note that both [1] and [12] present constructions of the l-group or vector lattice version of A a, neither method being that of 2.2. For vector lattices in the case a =v1 , the method in 2.2 was used in [16] and another intrinsic
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method appears in the seminal [15]. See 3.5 for a more extensive discussion of the connections between the Boolean, and, say, the vector lattice theories.
3. Topological interpretation A Boolean space is a topological space which is compact Hausdorff and zero-dimensional. By Stone duality, for a Boolean algebra A, there is an essentially unique Boolean space SA (the Stone space of A) for which A is isomorphic to the Boolean algebra of clopen subsets of SA, denoted clop SA. We shall simply identify A and clop SA when convenient. For example, PROPOSITION 3.1. Let A be a Boolean algebra. (a) For F¤A = clop SA, 0A F exists iff .F is open, and then .F =0A F. (b) The Boolean algebra of regular open subsets of SA, RSA, with the inclusion embedding A= clop SA5 RSA, is a model of the completion A5 A . (One may see [17], §8 and §35, and [6], §21.) In these topological terms, we shall briefly describe a-cut-completeness, and the a-cut-completion. Let X be a Boolean space. A subset of the form .F, for some F¤ clop X with F B a, is called an a-cozero set. (These coincide with the unions of Ba cozero sets, which sets are of interest in non-Boolean spaces; thus the name.) Note that -cozero means open. Let u be open in X. An open complement for u is an open set 6 with uS6= ¥ and u @ 6 dense in X. (6= X − u) is an example.) We say that u and 6 are a-complements if they are open complements, and each is a-cozero; in this case, u is called an a-complemented a-cozero set. (Not every a-cozero set is a-complemented.) PROPOSITION 3.2. Let X be a Boolean space. (a) For F, G ¤ clop X with F , G B a, .F and .G are a-complements iff (F, G%) is an a-cut in the Boolean algebra clop X. (b) For u¤ X, u is an a-complemented a-cozero set iff there is an a-cut (F, H) in clop X with u= .F. Proof. It will suffice to show that, for F, G ¤clop X, .F and .G are open complements iff (F, G%) is an -cut. (The rest follows easily.) Let F, G¤ clop X.
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Then, .F S .G = ¥ iff f S g = ¥ Öf F, g G iff f¤ g% Öf, g. And, .F @.G is dense in X iff Öclopen c" ¥, either fSc " ¥ for some f F or gSc" ¥ for some g G iff [c clopen with c S( f @ g)= (gSc)@( fS c)= ¥ Öf, g] implies c= ¥. Since cS(g @f ) =¥ iff c¤ X −(g @ f )=g% Sf %, the last condition holds iff [c clopen, c¤ g% Sf % Ög%, f implies c =¥] iff /{h f % h G%, f F}= 0 (meet in the Boolean algebra clop X). COROLLARY 3.2. Let X be a Boolean space (or, let A be a Boolean algebra, with X =SA, so A =clop X). Then, clop X is a-cut-complete iff each a-complemented a-cozero set has open closure. (Then we say ‘‘X is a-cloz’’). This is more or less immediate from 3.1(a) and 3.2. We now consider A a, within A = RSA. Note that for F¤ clop SA, the join in RSA is 0F =int .F. For A a, we just adjoint to A all 0F, where F is the left half of an a-cut. Using 3.2(b) then, COROLLARY 3.4. Let A be a Boolean algebra. Then, A ] A a is modeled by RSA] {int u) u is an a-complemented a-cozero set in SA}. COMMENTS 3.5. The term ‘‘a-cloz’’ is inherited from [8], where the property v1 -cloz, there called just cloz, is defined and discussed. (1) For a Boolean space X, v1 -cloz is equivalent to the property ‘‘quasi-F’’: each dense cozero set is C*-embedded (though not for non-Boolean X). See [8], 3.6, which proof easily generalizes to show that, for Boolean X, a-cloz is equivalent to (2): Each dense a-cozero set is C*-embedded. Therein lies an issue we have not resolved: The quasi-F spaces arose from a consideration of v1 -cut-completeness of the vector lattices C(X). (See [3], where the concept is called ‘‘o-Cauchy completeness’’.) It can be argued that, by virtue of the proposition (1) above, many of our results for the case a= v1 follow, or at least are fairly clear, from [3] (though, I would claim, the separate exposition for Boolean algebra still has value). Now, [1] and [12] have considered the cardinal generalization of [3] to archimedean l-groups and vector lattices, which involves a-cut-completeness (called a-jamd-completeness in [1], and a-Cauchy completeness in [12]), and quasi-Fa spaces, defined as (3): Each dense a-Lindelo¨f subset is C*-embedded, or equivalently, each dense countable intersection of a-cozero sets is C*-embedded. Of course, (3) implies (2), and, as shown in [3], for a= v1 , (2) implies (3); that is crucial to the development in [3], and use of (3) rather than (2) is crucial to [1]. But for general a, we do not know if (2) implies (3) (and, to me, it seems rather
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unlikely). So, while this paper clearly owes a huge debt to [1] (and to the related [12] and [13], which will be mentioned later), the logical connection is unclear. (Again, even if (2) implies (3) always, a separate exposition for Boolean algebras would seem to have value.) These papers [3], [1], [7] heavily involve ‘‘covers’’ of topological spaces. That theme arises here from the observation: the Stone dual of A5 A a is a surjection of Stone spaces SA SA a, which is a minimum a-cloz cover in Boolean spaces. Beyond this, we shall be silent above covering theory: there are many technical problems, and it seems expeditious to confine this paper to Boolean theory.
4. a-injectivity The Boolean homomorphism 8: B A is called an a-homomorphism if F¤ B, F Ba, and b = 0B F implies 8(b)= 0A 8(F). (Or dually; or, just if F¤ B, F B a and 0=/B F implies 0= /A 8(F).) (See [17], §22.). The -complete homomorphisms are called complete. One may call the property of A in the next theorem ‘‘a-injectivity’’ (and compare Monk’s theorem that, for a uncountable, in the category of a-complete Boolean algebras with a-homomorphisms, there do not exist injectives in the usual sense [14]). THEOREM 4.1. A is a-cut-complete iff whene6er 8: B A is an a-homomorphism, and B 5ad C, then there is a Boolean homomorphism 8) : C A with 8) B =8. Proof. Assume the condition, and extend the identity A A to 8) : A a A. Of course 8) is onto, and is one-to-one by §1. So A a = A, and A is a-cut-complete (2.4). For the converse, we first note (1) if 8: B A is an a-homomorphism, and (F, H) is an a-cut in B, then (8(F), 8(H)) is an a-cut in A (since /B {h−f h H, f F} = 0 implies /A {8(h) −8(f) h H, f F}= 0). Now suppose A is a-cut-complete, B 5ad C, and 8: B A is an a-homomorphism. We define 8) (c), for c C: Since B 5ad C, there is an a-cut (F, H) in B with 0F = c= /H. Then, by (1) above, (8(F), 8(H)) is an a-cut in A, so 08(F)= x= /8(H) for some x A. Define 8) (c) x. We must show that this is a proper definition: Let (F1 , H1 ) be another a-cut in B with 0F1 = c= /H1 , and let x1 be the element of A with 08(F1 ) =x1 = /8(H1 ). We want to see that x1 = x. Here,
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the pair (FF1 , H H1 ) is an a-cut, because ff1 5 c5 h h1 for elements f F, f1 F1 , h H, h1 H1 . Thus 08(H H1 )= z= /8(H H1 ) for some z A. But clearly (using [17], §19), x = 08(F) 5 08(FF1 ) =z= /8(HH1 )5 /8(H1 )= x1 , and likewise x1 5x, so that x1 =x. So we have a function 8) : B C which we want to be a Boolean homomorphism extending 8. The freedom of choice of the previous paragraph makes this easy: If a A, then ({a}, {a}) is an a-cut with 0{a}= a= /{a}, so 8) (a)= 08({a}) =8(a). So 8) extends 8. If c C, and (F, H) is an a-cut in B with 0F = c, then (H%, F%) is an a-cut ((1) in the proof of 2.3) with 0H%= c%, so that 8) (c%)= 08(H%)= (/8(H))% = 8) (c)%. If for i= 1, 2, ci C, and (Fi , Hi ) is an a-cut in B with 0Fi = ci , then (F1F2 , H1H2 ) is an a-cut ((2 in the proof of 2.3) with 0(F1F2 )= c1c2 , so that 8) (c1c2 ) = 08(F1F2 ) = 08(F1 )08(F2 )= 8) (c1 )8) (c2 ). Thus 8 is a Boolean homomorphism. More features of the 8) ’s, and of A a, come from the following. The proofs are just like the proofs for vector lattices presented in [12], 2.9 and 2.8. LEMMA 4.2. Suppose A 5ad B. (a) If 8) : B C is a homomorphism for which 8) A is an a-homomorphism, then 8) is an a-homomorphism. (b) If 81 , 82 : B C are a-homomorphisms for which 81 A= 82 A, then 81 = 82 . COROLLARY 4.3. Each extension 8) in 4.1 is unique, and an a-homomorphism. In the category of Boolean algebras with a-homomorphisms, for each A, the embedding A5 A a is an a-cut-complete monoreflection of A. Proof. The first sentence follows from 4.2. The meaning of the second sentence (see [9]) is that each a-homomorphism 8: AB, with B a-cut-complete, has a unique extension to an a-homomorphism 8) : A a B; A5 A a is an a-homomorphism; A a is a-cut-complete. These properties follow from 4.1 and 4.2, 2.3, and §1. For the case a = , 4.3 is known ([11], p. 84, Ex. 2). Theorem 4.1 also affords another characterization of A a (and see §6). COROLLARY 4.4. Let A5B. Then, B is isomorphic to A a over A iff B is a-cut-complete, the embedding A 5B is an a-homomorphism, and whenever A5 B%5 B with B% a-cut-complete, then B% =B.
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Proof. The conditions are necessary by 2.3 (since A5 A a is even an -homomorphism by §1). For the sufficiency: Label the embedding A5 B as 8, and use 4.1 to extend 8 to 8) : A a B. Since A 5d A a, 8) is one-to-one (§1), thus an isomorphism onto 8) (A a). So 8(A a) is a-cut-complete and coincides with B. In the case a = , 4.4 is a known description of the completion ([17], p. 154, condition (c1 )). However, in this case, the condition that A5 B be an -homomorphism can be dropped, as we now show (since I cannot find the result in the literature). PROPOSITION 4.5. Let A5 B. Then, B is isomorphic to A o6er A iff B is complete, and whene6er A5 B%5B with B% complete, then B%= B. Proof. The necessity is clear. For the converse, label the embedding A5 B as 8, and use the Sikorski Extension Theorem ([17], §33) to extend 8 to 8) : A B. Then, 8) is one-to-one (§1), thus an isomorphism onto 8) (A ). So 8) (A ) is complete, and coincides with B. A vector lattice version of 4.3 is earlier due to Macula ([12], 2.16). Our proof resembles his (though the logical connection is not clear (3.5)).
5. Preservation of a-ideals (This section derives from the analogous development for archimedean l-groups in [1]. The logical connection is unclear (3.5), and the Boolean theory seems simpler.) In the Boolean algebra A, a subset I is an a-ideal if it is an ideal (closed under sub-element formulation, and finite joins) for which, whenever, F¤ I, F B a, and 0A F exists, then 0A F I. (These are the kernels of a-homomorphisms ([17], §22).) The collection of a-ideals is denoted aIdlA. When A 5B, we define the extension map e: aIdlAaIdlB by: e(I) is the least a-ideal of B containing I. (This exists, since the intersection of a-ideals is an a-ideal.) We say that A 5B is a-ideal preser6ing if e is a bijection. After some preliminary calculation, we shall prove the following. THEOREM 5.1. A 5 B is a-ideal preser6ing iff A 5ad B. COROLLARY 5.2. A 5B is a-ideal preser6ing iff B embeds into A a o6er A. So A is the maximum a-ideal preser6ing extension of A. a
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Of course, 5.2 follows from 5.1 and 2.3. We turn to the proof of 5.1. PROPOSITION 5.3. (cf. [1], 3.2) Let B be a Boolean algebra, and S¤ B. Then, dfn the least a-ideal of B containing S is S* {b b= 0{b f f F}, for an F ¤S with F B a}. Proof. Clearly, S* is contained in the least a-ideal containing S, so it suffices to show that S* is an a-ideal. If c5 b =0{b f f F}, then c= cb = c 0{b f f F}= 0{c b f f F}= 0{c f f F}. Suppose b =0bi , for an index set of cardinal B a with bi = 0{bi f f Fi }, for Fi ¤S with Fi Ba. By regularity of a, .i Fi B a, and we have b= 0{b f f .i Fi }. (Proof. Suppose c ]b f for each f .i Fi . Then, fixing i, for each f Fi , c] b f hence c ]bi f hence c] bi . Since c] bi for each i, c] 0bi =b.) COROLLARY 5.4. Let A5B, with associated extension map e: aIdlA aIdlB. (a) For each I aIdlA, e(I) SA= I. So e is one-to-one. (b) If e is onto then for each J aIdlB, (JSA)*= J. (c) Assume that whene6er G ¤ A, G Ba, and 0A G exists, then 0A G= 0B G. Then, the con6erse to (b) holds. Proof. (a) ± is clear. For ¤: Note that e(I)= the I* of 5.3. So, if x e(I)SA, then x = 0B {x f f F} for F ¤I with F B a. Since x A and F¤ I, each x f I, and since 0B {x f } A, it is also 0A {xf }, and thus lies in I, since I is an a-ideal. That is, x I. (b) Note this: (a) says (e(I) SA)*= e(I) for each I aIdlA. Thus, if e(I)= J, then (J SA)* =(e(I) S A)* =e(I)= J. (c) The hypothesis, and a glance at 5.3, guarantee that, for any S¤A, the a-ideal in A generated by S is contained in the a-ideal in B generated by S. Now let J aIdlB; we suppose that (JSA)*= J. Let I be the a-ideal in A generated by JS A. We then have J SA ¤I ¤ (JSA)*=J. Thus I*= J, i.e., e(I)=J. So e is onto. We now prove 5.1. Suppose A 5 B is a-ideal preserving. Then e is onto, and 5.4(b) holds. Let b B. The principal ideal [b] ={c c 5b} is even an -ideal, so ([b] SA)*±[b]. By 5.3, b= 0{b f f F}, for some F ¤ [b] SA with F B a. Thus, each f5 b and b f= f, so b = 0F as desired.
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Suppose A 5ad B. Then A is dense in B, (1) of §1 holds, so 5.4(c) obtains. Now let J aIdlB. Toward showing (J SA)*±J, let x J. Now, x= 0F for some F ¤ A with F B a. Since f 5 x for each f F, F¤J, so F¤ J SA. Finally, x= 0F = 0{f f F} =0{x f f F}, so x (JSA)*. Thus e is onto (5.4(c)), and a bijection (5.4(a)). The proof of 5.1 is complete. Note that 5.1, and the proof, are valid for the case a= v0 . Of course, an v0 -dense is just an ideal, and A 5v 0 d B implies A= B. Thus COROLLARY 5.5. A Boolean algebra has no proper ideal preser6ing extension.
6. Comparison with a-completeness The boolean algebra A is called a-complete if for each F¤ A, F B a, 0F exists. The topological space X is called a-disconnected if for each a-cozero set u in X, u) is open. Then ([17], p. 86), A is a-complete iff SA is a-disconnected. ([17] uses ‘‘5a’’ in the definition of a-complete. Our ‘‘Ba’’ causes no problem.) Of course, each a-complete A is a-cut-complete, and each a-disconnected X is a-cloz. To reverse these implications, we have the following obvious definitions and proposition. DEFINITION 6.1. The Boolean algebra A is a-complemented if for each F¤ A with F Ba, there is H ¤ A for which (F, H) is an a-cut. The space X is a-cozero-complemented (abbreviated acc) if each a-cozero set has an a-complement. PROPOSITION 6.2. (a) The Boolean algebra A is a-complemented iff SA is acc. (b) The Boolean algebra A is a-complete iff A is a-cut-complete and a-complemented. (c) The Boolean space X is a-disconnected iff X is a-cloz and acc. EXAMPLES 6.3. Boolean spaces which are a-cloz, not a-disconnected. Let D be a set with D ] a, and let X be the one-point compactification of discrete D. It is not hard to see that the only cozero sets of X are (i) the countable subsets of D, or (ii) the complements of finite subsets of D. It follows that the only a-complemented a-cozero sets are the clopen sets. So X is a-cloz by de-
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fault. Since no infinite countable subset of D has open closure in X, X is not even v1 -disconnected. (Or, no infinite countable subset of D has an a-complement, so X fails acc.) We now compare the a-cut-completion A a, with the a-completion, for the nonce denoted Aa . A characterization of Aa is as the intersection of all the a-complete subalgebras of A which contain A ([17], p. 156). Thus A a 5 Aa , by 2.3(b). PROPOSITION 6.4. Suppose that, A 5ad B. Then, A is a-complemented iff B is a-complemented. Proof. Suppose A is a-complemented, and F¤B with F B a. Then, for each f F, f =0B Ff for some Ff ¤A with Ff B a. Let F0 = .{Ff f F}; so F0 B a since a is regular. Since A is a-complemented, there is H0 ¤ A such that (F0 , H0) is an a-cut in A. A routine calculation shows (F, H0) is an a-cut in B. Suppose B is a-complemented, and F¤A with F B a. Then, there is H¤B so that (F, H) is a-cut in B. For each h H, h= /Hh for some Hh ¤ A with Hh B a. Let H0 =.{Hh h H}, so H0 Ba. A routine calculation shows (F, H0 ) is an a-cut in A. PROPOSITION 6.5. The following are equi6alent for the Boolean algebra A. (a) A is a-complemented. (b) A a is a-complemented. (c) A a is a-complete. (d) A a =Aa (e) A 5ad Aa . Proof. (a) [ (b): 6.4. (b) [ (c): 6.3. (c) [ (d): always A a ¤ Aa , and when A a is a-complete, A a ]Aa , by the description of Aa quoted above. (d) [ (e): since A 5ad A a. (e) [ (a): by 6.2, Aa is a-complemented, so also A by 6.4. We consider briefly the a-extensions of a Boolean algebra A. Let us say that B% is an a-replete subalgebra of B if [F¤B%, F B a, and 0B F exists] implies 0B F B%; we indicate this by B% 5arep B. (The terminology of [17], p. 91, differs.) An a-extension of A ([17], p. 165) is an extension A5 B, with (i) B a-complete, (ii) A a-embedded in B, and (iii) whenever A5 B% 5arep B, then B%=B. (In the presence of (i), (iii) is equivalent to (iii)%: whenever A5 B% 5B, with B% a-complete, then B% =B.) Of course, Aa is an a-extension of A, but generally there are others ([17], p. 172).
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PROPOSITION 6.6. A is a-complemented iff, whene6er A5 B satisfies (ii) and (iii) abo6e, then A 5ad B. Proof. Since A5 Aa satisfies (ii) and (iii), if the condition holds then A 5ad Aa , and by 6.5, A is a-complemented. Suppose A is a-complemented, and A5 B satisfies (ii) and (iii). Since B 5Ba satisfies (ii) and (iii), it follows that A5 Ba satisfies (ii) and (iii). Also, if we can show A 5ad Ba , then A 5ad B follows. So we can assume B is a-complete. Now label the embedding A 5 B as 8. By 6.5, A 5ad Aa , so 3.3 applies to give an a-homomorphism 8) : Aa B extending 8. It follows that A 5ad 8) (Aa ). By §1, 8) is one-to-one, so 8) (Aa ) is a-complete, thus 8) (Aa ) 5arep B, and thus 8) (Aa )= B. So A 5ad B. COROLLARY 6.7. If A is a-complemented, then Aa (=A a) is the only a-extension of A. I don’t know if the converse to 6.7 holds (cf., [13], 4.1). Analogues of the issues of this section have been considered for vector lattices and dual topology in [2] and [13] with some analogous conclusions. (Keep the remarks of 3.5 in mind here.) [7], 2.16 is a topological version of 6.5 for a =v1 .
7. Comparison with completeness We consider those Boolean algebras A (and Boolean spaces SA) for which A a = A . The space X is called a-fraction-dense, abbreviated aFD, if for each open g there is an a-cozero set u with u) = g) . (In [5], v1 FD is called ‘‘fraction-dense,’’ and the terminology justified.) For a Boolean algebra A, with F¤A, let ubF= {a A a] f for each f F}. The proof of the following is simple, and omitted. LEMMA 7.1. Let A be a Boolean algebra, with F, F1 ¤ A. With indicated joins and meets meant in A , (a) (F, ubF) is an -cut in A (and dually). (b) 0F =/ubF (and dually). (c) 0F 5 0Fi iff ubF±ubF1 (and dually). (d) 0F = 0F1 iff ubF=ubF1 (and dually).
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A. W. HAGER
ALGEBRA UNIVERS.
Acknowledgement The author thanks the referee for pointing out an error in the earlier version of the paper. REFERENCES [1] BALL, R. N., HAGER, A. W. and NEVILLE, C. W., The quasi-Fk co6er of a compact Hausdorff space and the k-ideal completion of an archimedean l-group, in Dekker Notes 123, General Topology and its Applications (R. M. Shortt, Ed.), pp. 7 – 50 (Dekker, 1990). [2] COHEN, H., The k-extremally disconnected spaces as projecti6es, Canad. J. Math. 16 (1964), 253–260. [3] DASHIELL, F., HAGER, A. W. and HENRIKSEN, M., Order-Cauchy completions of rings and 6ector lattices of continuous functions, Canad. J. Math. 32 (1980), 657 – 685. [4] GILLMAN, L. and JERISON, M., Rings of Continuous Functions (Van Nostrand, 1960). [5] HAGER, A. W. and MARTINEZ, J., Fraction-dense algebras and spaces, Canad. J. Math. 45 (1993), 977–996. [6] HALMOS, P., Lectures on Boolean Algebras (Van Nostrand, 1963). [7] HENRIKSEN, M., VERMEER, J. and WOODS, R. G., Quasi-F co6ers of Tychonoff spaces, Trans. Amer. Math. Soc. 303 (1987), 779–803. [8] HENRIKSEN, M., VERMEER, J. and WOODS, R. G., Wallman co6ers of compact spaces, Diss. Math. 280 (1989). [9] HERRLICH, H. and STRECKER, G., Category Theory (Allyn and Bacon, 1973). [10] HUIJSMANS, C. B. and DE PAGTER, B., Maximal d-ideals in a Riesz space, Canad. J. Math. 35 (1983), 1010–1029. [11] KOPPELBERG, S., Handbook of Boolean Algebras, 6ol. 1 (J. Monk and R. Bonnet, Eds.), (NorthHolland, 1989). [12] MACULA, A. J., a-quasi-F spaces, Top. and its Applic. 44 (1992), 217 – 234. [13] MACULA, A. J., Monic sometimes means a-irreducible, in Dekker Notes 134, General Topology and its Applications (S. J. Andima et al., Eds.), pp. 239 – 260 (Dekker, 1991). [14] MONK, J. D., Nontri6ial m-injecti6e Boolean algebras do not exist, Bull. Amer. Math. Soc. 73 (1967), 526–527. [15] PAPANGELOU, F., Order con6ergence and topological completion of commutati6e lattice-groups, Math. Annalen 155 (1964), 81–107. [16] QUINN, J., Intermediate Riesz spaces, Pac. J. Math. 56 (1975), 225 – 263. [17] SIKORSKI, R., Boolean algebras, Third edition (Springer-Verlag, 1969). Department of Mathematics Wesleyan Uni6ersity Middletown, CT 06459 U.S.A. e-mail:
[email protected]
Vol. 39, 1998
a-cut-complete Boolean algebras
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THEOREM 7.2. For a Boolean algebra A, these are equi6alent. (a) A a =A . (b) SA is aFD. (c) For each F¤A there is F1 ¤ A with F1 B a, for which ubF1 = ubF. Proof. By 3.4, (a) means (b1 ) for each open g in SA, g) = u) for some a-complemented a-cozero set u. This clearly implies (b). Conversely, if (b) holds, then SA is a-cozero-complemented: if 6 is a-cozero, then g= SA− 6) is open, and g) = u) for another a-cozero u, which must be an a-complement for 6. Thus (b) implies (b1 ). Directly from the definitions, (a) also means (c1 ): For each F¤A there is an a-cut (F1 , H1 ) with 0F1 =0F. Using 7.1(d) then, (c1 ) implies (c). Conversely, suppose that (c) holds. Let F ¤A. Then 0F = /ubF (7.1(a)). By (c) applied to F, and the dual of (c) applied to ubF, there are F1 , H1 ¤ A with F1 , H1 B a, and ubF1 =ubF (so 0F1 =0F, by 7.1(d)) and lbH1 = lb(ubF) (so /H1 = /ubF, by the dual of 7.1(d)). Since /H1 =0F1 , (F1 , H1 ) is an a-cut. Thus (c1 ) holds. Since A a 5Aa 5A always, it follows from 7.2 that, if A satisfies 7.2, then Aa = A . I don’t know if the converse holds; it seems unlikely. REMARK 7.3. We note a number of relevant topological propositions. The proofs are not difficult, and omitted. For simplicity, we consider just compact Hausdorff spaces. Various pieces of the following are true under weaker hypotheses. (a) X is extremally disconnected ( -disconnected) iff X is a-cloz and aFD. (b) Each of these conditions implies the next: X is a-separable ( dfn
there is a dense set of power B a); X has the a-chain condition dfn ( each pairwise disjoint family of open sets has powerB a); X is aFD; X is a-cozero-complemented. (c) X has the a-chain condition iff for each open g, there is an a-cozero set u with u ¤ g and u) =g) iff each open set is weakly a-Lindelo¨f. (A space is weakly a-Lindelo¨f if each open cover has a subfamily of power B a with dense union.) (d) X is aFD ( dfn
for each open g, there is an a-cozero set u with u) = g) ) iff each regular open set is weakly a-Lindelo¨f. Archimedean vector lattices and l-groups which satisfy the analogue of A v1 = A have been considered in [16], [10], and [5]. The first two call this property ‘‘almost-complete’’ for vector lattices, and the last ‘‘absolute’’ for l-groups and ‘‘fraction-dense’’ for f-rings.
