Chapter 31: Boolean algebras

311
Boolean algebras

Boolean rings and algebras; ideals and ring homomorphisms to
**Z**_{2}; 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.

312
Homomorphisms

Subalgebras; ideals; Boolean homomorphisms; the ordering
determines the ring structure; quotient algebras; extension of
homomorphisms; homomorphisms and Stone spaces.

313
Order-continuity

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.

314
Order-completeness

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.

315
Products and free products

Simple product of Boolean algebras; free product of Boolean
algebras; algebras of sets and their quotients; projective and
inductive limits.

316
Further topics

The countable chain condition; weak
(σ,∞)-distributivity; Stone spaces; atomic and atomless
Boolean algebras; homogeneous Boolean algebras.

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Revised 18.11.12