By Roman Sikorski
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Extra resources for Boolean Algebras
N and A (\ B; = /1 for j = 1, ... , m . By (7) and (8), I(A) e I(Ai) for i = 1, ... , nand I(A) (\ - I(B;) = /1 for j = 1, ... , m. Since I (B) =f: /1, we infer that I (Al) (\ ... (\ I (An) (\ - I (Bl) (\ ... (\ - I (Bm) =f: /1 . (10) Similarly we prove that (10) implies (9). Examples. A) A Boolean algebra is atomic if and only if the set of all atoms is dense. B) Any dense set €I of non-zero elements of a Boolean algebra Q( is partially ordered by the Boolean inclusion e and satisfies the following condition: (a) If A, BEe and A C[ B, then there exists an element C e A, C E €I, such that no element DEe satisfies simultaneously Dec and DeB.
In fact, every automorphism of the field tr (of all openclosed subsets of any compact totally disconnected space X) onto itself is induced by a homeomorphism of X onto X, and conversely. 1 An example of such a space was given by KAT:ihov  by means of the p-compactification technique. Two similar examples of linearly ordered compact spaces with this property were given independently and simultaneously by J 6NSSON  and RIEGER . § 12. Theorems on extending to homomorphisms 35 D) A Boolean algebra Q( is said to be superatomic1 if every homomorphic image of Q( is atomic.
Induced homomorphisms between quotient algebras 49 h' of CU into CU' such that (8) h(A) = [h'(A)JLI' for every A ECU. To give the answer to this question we introduce the following definition which is an algebraic analogue of the topological notion of retract examined above: a subalgebra CUo of a Boolean algebra CU is said to be retract of CU provided there exists a homomorphism g (called a retract homomorphism) of CU onto CUo such that g(A) = A for A ECU o. 3. 3 can be deduced from the equivalence of (a), (b), (c) just proved since we can restrict our investigation to the case where CU is a perfect field and CU' is a field of sets.