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Appendix § 39

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Appendix § 39 § 39 Relation to other algebras 139 is an extension of the product mo of ml , m 2• Let B n E SB be the sequence defined in example C). By definition, n ~;;;;n<oo B n = /1, B n +l C B n , and lim mo (Bn ) > O. On the other hand, lim mo (Bn ) = lim m (Bn ) = 0 since 1t~OO n"""""-TOO n~oo n r;;;; n < 00 B n = /1 and m is a a-measure on Q{. Contradiction. Replacing a by any infinite cardinal m everywhere in the definition of a-product, we obtain the analogous notion of Boolean m-product of an indexed set {SBthE T of non-degenerate Boolean m-algebras. Since no representation theorem analogous to 29.1 is true for Boolean m-algebras, the theory of general Boolean m-products is more complicated than in the case m = a. One can prove that m-products exist always, it is also possible to introduce a not ion analogous to the maximal a-productl. Appendix § 39. Relation to other algebras Boolean algebras are a special case of abstract algebras (with a finite number of finite operations). Many notions introduced in Chapter I belong to the general theory of abstract algebras. We quote here such notions as homomorphism, isomorphism, subalgebra, generator, free algebra etc. Boolean m-algebras investigated in Chapter II can be also interpreted as a special case of abstract algebras but with some infinite operations, viz. with the complementation -A, the infinite join U tETA t and the infinite meet n tE TAt where T is a fixed set of cardinal m. Thus such notions as m-subalgebra, m-homomorphism (between two m-algebras), m-generator, free Boolean m-algebra etc. belongs also to the general theory of abstract algebras. Many remarks in Chapters land II and also some theorems belong to the theory of abstract algebras. We mention here, for instance, theorems 12.1 and 23.3 which are a particular case of a general theorem on abstract algebras. The notions of field product and m-product, and of Boolean product and maximal a-product are particular cases of a general notion of product of abstract algebras 2 • Also some other general algebraic notions not examined in Chapters I and II can be applied to the theory of Boolean algebras. As an example we quote here the notion of inverse and direct systems 3 • It follows immediately from the set ofaxioms assumed in § 1 that Boolean algebras are distributive lattices. More exactly, the notion of Boolean algebra coincides with the notion of distributive complemented lattice 4• Some theorems proved here for Boolean algebras can be 1 See SIKORSKI [191 2 See SlKORSKl [19]. 3 See HAlMO [IJ and WALLACE [1]. 4 For other characterizations of Boolean algebras among lattices. see BALA­ CHANDRAN [IJ. BIRKHOFFand WARD [IJ. DlLWORTH [2J. MlCHIURA [IJ. NACHBlN[I]. 140 Appendix generalized to distributive lattices. In particular, the fundamental representation theorem 8.2 can be considered as a particular case of a representation theorem for distributive latticesl . As we have observed in § 17, the theory of Boolean algebras coincides (if only finite joins and meets are taken into consideration) with apart of the theory of algebraic rings. The term "ideal" is taken from this theory. In fact, it is easy to verify that a set S of elements of a Boolean algebra Q( is an ideal in the sense defined in § 3 if and only if it is an ideal of the Boolean ring Ql in the sense assumed in the general theory of algebraic rings. The forming of factor algebras described in § 10 is a particular case of forming algebraic factor rings. It is also a particular case of forming factor algebras modulo a congruence relation in thc general theory of abstract algebras. It is worth noticing that the two-valued homomorphisms method used in the proof of the fundamental representation theorem 2 8.2 is a particular case of a general method of investigation of more complicated algebras by means of homomorphisms into certain special simple algebras with the same operations. An example of the use of this method is given by the notion of the space dual to a given Banach space. This space is the set of an linear continuous functionals, i.e. homomorphisms (of the theory of Banach spaces) into the simplest Banach space, viz. the field of scalars (real or complex). Another example of this method is given by the theory of normed rings in Functional Analysis. Here the dual space Ql* of a given normed ring Ql is the set of an continuous multiplicative linear functionals, i.e. homomorphisms X into the simplest normed ring, viz. the ring of scalars. If A E Ql is fixed, then the formula (X E Q(*) defines a continuous mapping from Ql* to scalars. Under some additional hypothesis, the mapping h defined by the formula (1) h(A) = <PA is one-to-one. In this case Ql can be represented as a ring of continuous functions on the dual space Q(*. The last example has an analogue in the theory of Boolean algebras. Every Boolean algebra Ql is not only an algebraic ring but also a linear ring over the two-element algebraic field 9\ (see § 17, p. 47). According to the remarks on p. 16-17 and the definition in the proof of 8.2, the Stone space X of a given Boolean algebra Ql can be interpreted as the set of an two-valued homomorphisms of Ql into 9\, i.e. the set of all multiplicative 1 Proved by BIRKHOFF [IJ and STONE [9J. See also RIEGER [2J. 2 In the proof of 8.2 we used the notion of maximal filter instead of two-valued homomorphisms but these notions are equivalent (see p. 16-17). § 39 Relation to other algebras 141 linear functionals with values in the field ~ of sealars. The dass of an continuous mappings from the Stone space X into the two-element Hausdorff space ~ can be indentified with the dass of all both open and dosed subsets of Q( sinee every mapping of this type is uniquely deter­ mined by the (both open and dosed) set of points where it assumes the unit as its value. Thus the isomorphism h in the proof of the fundamental representation theorem 8.2 can be considered as a particularcase of (1). Consequently the Stone space of Q( is sometimes called dual space 0/ Q( 1. The analogy between the representation theory of normed rings and Boolean algebras is so deep that it is possible to develop a general theory of maximal ideals which gives, as partieular cases, the both representation theorems 2. The notion of Boolean a-algebras is a partieular case of the notion of cardinal algebras 3• Cardinal algebras are abstract algebras with an infinite operation whieh is a common generalization of the Boolean join of an enumerable sequenee of elements and of the sum of an enumerable sequence of cardinal numbers. The axioms eharacterizing this operation are such that their consequences form a large part of the additive arithmetic of cardinal numbers. However they are also satisfied by Boolean a-algebras, types of isomorphisms for Boolean a-algebras, the set of non-negative integers or reals (with + 00), the set of an non-negative functions on a set, and other systems. Theorems on cardinal algebras are also theorems on Boolean algebras but they were not quoted in this book. It is worth notieing that some of these theorems on Boolean algebras are generalizations of some wen known fundamental theorems on cardinals. For instanee, the fundamental Cantor-Bernstein theorem on cardinals is a particular case of the following theorem on Boolean a-algebras Q(, Q3: if Q3 is isomorphie to Q( IA (A E Q() and Q( is isomorphie to Q3 IB (B E Q3), then Q( and Q3 are isomorphie 4 . To obtain the Cantor-Bernstein theorem it suffices to assurne that Q{ and Q3 are fields of an subsets of some sets X, Y with cardinals n1> n 2 respecti vel y. By the same method we infer that the Bernstein theorem 1 See HALMOS [4]. 2 SLOWIKOWSKI and ZAWADOWSKI [1]. 3 The notion and the theory of cardinal algebras are due to TARSKI [8J. See also JONSSON and TARSKI [2]. 4 See SIKORSKI [lJ and TARSKI [8J. The hypothesis thatm and mare a-complete is essential. See KINOSHITA [lJ and HANF [1]. 142 Appendix § 40 is a partieular case of the following theorem valid for every Boolean a-algebra Q{: if A, B E Q{, Q{ lAis isomorphie to Q{ 1-A, and 211 B is isomorphie to Q{ 1-B, then Q{ IA and Q{ IBare isomorphiel . Many papers are devoted to study Boolean algebras with some additional operations and to various generalizations of Boolean algebras and rings2• § 40. Applications to Mathematical Logic. Classical calculi The most important applications of the theory of Boolean algebras are those to Mathematical Logic. That is not surprising because the notion of Boolean algebras was created as the result of Boole's investigation of the algebraic structure of the "laws of thought"3. In the first part of its development, the theory of Boolean algebras was also called the Algebra 01 Logic. Consider first the case of the (two-valued) proposition al ca1culus. Denote the logical connective "or", "and", "not", "if ... , then ... " by U, n, -, -)0 respectively. The set of all formulas of the propositional ca1culus becomes a Boolean algebra after identifieation of equivalent formulas (see an analogous remark in § 1 D)).
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