Chapter 31: Boolean algebras
Boolean rings and algebras; ideals and ring homomorphisms to Z2; Stone's theorem; the operations È, Ç, D, \ and the relation Í; partitions of unity; topology of the Stone space; Boolean algebras as complemented distributive lattices.
Subalgebras; ideals; Boolean homomorphisms; the ordering determines the ring structure; quotient algebras; extension of homomorphisms; homomorphisms and Stone spaces.
General distributive laws; order-closed sets; order-closures; Monotone Class Theorem; order-preserving functions; order-continuity; order-dense sets; order-continuous Boolean homomorphisms; and Stone spaces; regularly embedded subalgebras; upper envelopes.
Dedekind completeness and σ-completeness; quotients, subalgebras, principal ideals; order-continuous homomorphisms; extension of homomorphisms; Loomis-Sikorski representation of a σ-complete algebra as a quotient of a σ-algebra of sets; regular open algebras; Stone spaces; Dedekind completion of a Boolean algebra.
Products and free products
Simple product of Boolean algebras; free product of Boolean algebras; algebras of sets and their quotients; projective and inductive limits.
The countable chain condition; weak (σ,∞)-distributivity; Stone spaces; atomic and atomless Boolean algebras; homogeneous Boolean algebras.
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