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Appendix § 39 § 39. Relation to other algebras 191 Appendix § 39. Relation to other algebras Boolean algebras are a special case of universal algebras (with a finite number of finite operations). Many notions introduced in Chapter I belong to the general theory of universal algebras. We quote here such notions as homomorphism, isomorphism, subalgebra, generator, free algebra etc. Boolean m-algebras investigated in Chapter II can also be interpreted as a special case of universal algebras but with some infinite operations, viz. the complementation -A, the infinite join U tETAt and the infinite meet ntETA t where T is a fixed set of cardinality m. Thus such notions as m-subalgebra, m-homomorphism (between two m-algebras), m-generator, free Boolean m-algebra etc. also belong to the general theory of universal algebras. Many remarks in Chapter I and II and also some theorems belong to the theory of universal algebras. We mention here, for instance, theorems 12.1 and 23.3 which are particular cases of a general theorem on universal algebras. The notions of field product and m-product, and of Boolean product and maximal m-product, and maximal representable m-product are particular cases of a general notion of product of universal algebras1. 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 2 and the notion of projectivity and injectivity 3 from the general theory of categories. It follows immediately from the set of axioms 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 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 lattices 5. As we have observed in § 17, the theory of Boolean algebras coincides (if only finite joins and meets are taken into consideration) with a part of the theory of algebraic rings. The term "ideal" is taken from this 1 SIKORSKI [19]. See also CHRISTENSEN and PIERCE [1]. From the point of view of the general theory of categories Boolean products and maximal m-products are free joins in the category of Boolean algebras or of Boolean m-algebras (see e.g. KUROS, LIVSIC and SULGEIFER [1], SEMADENI [4,5]). 2 See ENGELKING and KURATOWSKI [1], HAIMO [1], WALLACE [1]. 3 For an investigation of projective and injective Boolean algebras, see HALMOS [8, 10]. 4 For other characterizations of Boolean algebras among lattices, see BALA­ CHANDRAN [1,2,3], BIRKHOFF and WARD [1], DILWORTH [2], MICHIURA [1], NACHBIN [1]. 5 Proved by BIRKHOFF [1] and STONE [9]. See also RIEGER [2]. 192 Appendix theory. In fact, it is easy to verify that a set L1 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 Q( in the sense assumed in the general theory of algebraic rings. The forming of quotient algebras described in § 10 is a particular case of forming algebraic quotient rings. It is also a particular case of forming quotient algebras modulo a congruence relation in the general theory of universal algebras. It is worth noticing that the method of two-valued homomorphisms used in the proof of the fundamental representation theoreml 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 dual space to a given Banach space. This space is the set of all continuous linear 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 Q(* of a given normed ring Q( is the set of all continuous multiplicative linear functionals, i.e. homomorphisms X into the simplest normed ring, VIZ. the ring of scalars. If A EQ( is fixed, then the formula (X (Q(*) defines a continuous mapping from Q(* to the scalars. Under some additional conditions, the mapping h defined by the formula (1) is one-to-one. In this case Q( 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 Q( is not only an algebraic ring but also a linear ring over the two-element algebraic field 2\ (see § 17, p. 52). According to the remark on p. 18 and the definition in the proof of 8.2, the Stone space X of a given Boolean algebra Q( can be interpreted as the set of all two-valued homomorphisms of Q( into 2\, i.e. the set of all multi­ plicative linear functionals with values in the field 2\ of scalars. The class of all continuous mappings from the Stone space X into the two-element Hausdorff space 2\ can be identified with the class of all open-closed subsets of X since every mapping of this type is uniquely determined by the (open-closed) set of points where it assumes the unit as its value. Thus the isomorphism h in the proof of the fundamental representation 1 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. 18). § 39. Relation to other algebras 193 theorem 8.2 can be considered as a particular case of (1). Consequently the Stone space of Q{ is sometimes called dual space of Q{l. 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 particular cases, the two representation theorems 2• The notion of Boolean a-algebras is a particular case of the notion of cardinal algebras 3• Cardinal algebras are abstract algebras with an infinite operation which is a common generalization of the Boolean join of an enumerable sequence of elements and of the sum of an enumerable sequence of cardinal numbers. The axioms characterizing this operation are such that from them one may deduce 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 + (0), the set of all 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 noticing that some of these theorems on Boolean algebras are generalizations of some well-known fundamental theorems on cardinals. For instance, the fundamental Cantor-Bernstein theorem on cardinals is a particular case of the following theorem on Boolean a-algebras Q{, Q3 which is another formulation of theorem 22.4: if Q3 is isomorphic to Q{ 1A (A EQ{) and Q{ is isomorphic to Q31 B (B EQ3), then Q{ and Q3 are isomorphic 4• To obtain the Cantor-Bernstein theorem it suffices to assume that Q{ and Q3 are fields of all subsets of some sets X, Y with cardinality nl> n 2 respectively. By the same method we infer that the Bernstein theorem if 2nl = 2n2, then nl = n 2 is a particular case of the following theorem valid for every Boolean a-algebra Q{: if A, B EQ{, Q{ 1A is isomorphic to Q{ 1-A, and Q{ 1B is isomorphic to Q{ 1-B, then Q{ 1A and Q{ 1B are isomorphic 5. 1 See HALMOS [4], [8]. 2 SLOWIKOWSKI and ZAWADOWSKI [1]. 8 The notion and the theory of cardinal algebras are due to TARSKI [8]. See also J6NSSON and TARSKI [2]. 4 SIKORSKI [1] and TARSKI [8]. See also BRUNS and SCHMIDT [2]. 6 TARSKI [8]. The hypothesis that Q( is a-complete is essential. See HANF [1]. Ergebn. d. Mathern. N.F. Bd. 25, Sikorski, 2. Aufl. 13 194 Appendix Many papers are devoted to the study of Boolean algebras with some additional operations and of various generalizations of Boolean algebras and rings!. § 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 a Boolean algebra was created as the result of Boole's investigation of the algebraic structure of the "laws of thought" 2. In the first part of its development, the theory of Boolean algebras was also called the algebra of logic. Consider first the case of the (two-valued) propositional calculus. Denote the propositional connectives "or", "and", "not", "if ... , then ... " by v, n, -, -+ respectively. The set of all formulas of the propositional calculus becomes a Boolean algebra after identification of equivalent formulas (see an analogous remark in § 1 D)). We recall that formulas a, f3 are said to be equivalent if both the implications a -+ f3 and f3 -+ a are derivable. The Boolean algebra Qt so obtained will be called the Lindenbaum-Tarski algebra of the propositional calculus in question. Let lal denote the element of Qt determined by a formula a. We have the fundamental identities.
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