Boolean Algebras by Roman Sikorski


By Roman Sikorski

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We shall prove that 5= is perfect. Let V' be a maximal filter of 5=. By isomorphism, the class V of all elements A EQt such that h (A) EV' is a maximal filter in Qt. e. B = h(A) for an element A EQt. e. by (1) B EV' if and only if V EB . e. that 5= is perfect. 1. A compact totally disconnected space X is said to be the Stone space of a Boolean algebra Qt provided Qt is isomorphic to the (perfect reduced) field of all open-closed subsets of X. It follows from the remarks at the end of § 7 that all Stone spaces of Qt are homeomorphic.

E. there are points Xj E A j • Every point x = {Xt} E X such that x t, = Xj belongs to At (\ ... (\ A!. Observe that 'iJt is isomorphic to'iJt. Viz. the mapping (3) gt(A) = A* for A E'iJt is an isomorphism of 'iJ tonto 'iJt· The above consideration suggests the following generalization of the notion of the field products. Let {Q(thET be an indexed set of non-degenerate Boolean algebras. Bya Boolean productl (or simply product) of {Q(t}tET we shall understand any pair (4) {{ithE T, Q3} such that (a) Q3 is a Boolean algebra; (b) for every t E T, it is an isomorphism of Q(t into Q3; (c) the indexed set {it(Q(t)}tET of sub algebras of Q3 is independent; (d) the union of all the subalgebras ie(Q(t), t E T, generates Q3.

D) Every finite Boolean algebra is atomic. If it has 2n elements [see § 8 B)], then n is the number of atoms. g. KURATOWSKI [4], p. 58. 29 § 10. Quotient algebras § 10. Quotient algebras Let ,1 be an ideal of a Boolean algebra A, B EQ{ we write Q{. For arbitrary elements A",B if and only if A - B E,1 and B - A E ,1 . e. it is reflexive, symmetrical and transitive: A",A; (1) (2) if A '" B, (3) if A '" Band then B '" A ; B '" C, then A '" C . Property (1) follows from the fact that ,1 contains the zero element /I.

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