# Download Boolean Algebras by Roman Sikorski PDF

By Roman Sikorski

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**Sample text**

There exists a reduced field'J of subsets of To such that {§ = m. For every set A E'J, let aA = {at}tET. be a point in EC2m defined as follows: at = 1 if tEA, and at = - 1 if t E - A . The set X of all the points aA, where A E'J. has power m. The field 'J e e e', e' e e e 1 This theorem is a particular case of a general theorem (due to HEWITT [2J) on Cartesian products of topological spaces. The case m = Xo of this theorem was independently found by MARCZEWSKI [9J. § 15. Induced homomorphisms between quotient algebras 45 being reduced, for any distinct points tv ...

Under what condition has a given Boolean algebra Q{ the following property? (a /) For every homomorphism h of Q{ into any algebra Q{ljiJ ' (where Q{' is a Boolean algebra and iJ I is an ideal of Q{/) there exists a homomorphism rr rr rr rr rr rr rr Err. § 15. 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.

This will be done often in the sequel. Here we shall give a topological interpretation of ideals and filters. Let X be the Stone space of a Boolean algebra Qt, and let h be the isomorphism of Qt onto the field of all open-closed subsets of X. For every open set G C X, the class of all A E Qt such that h (A) C G is an ideal called the ideal corresponding to G. Conversely, every idealLJ corresponds to an open subset of X, viz. to the union of all sets h (A) where A ELJ. For every closed set F C X, the class of all A E Qt such that F C h (A) is a filter called the filter corresponding to F.