The Jónsson Class of Boolean Algebras

Abstract

The general results of the theory of Jonsson classes, developed in §4, are applied here to the Jonsson class of (proper) Boolean algebras. The α-homogeneous-universal Boolean algebra ℭ α of cardinality a and its Stone space S α , which exist if and only if \(\alpha = {\alpha ^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\alpha } }}\), are studied in considerable detail and characterized by properties given in terms of the partial order of a Boolean algebra; these properties are similar to the η α -property of ordered sets (studied in §5). We study the Jonsson class of Boolean algebras not mainly to illustrate the general theory of Jonsson classes, but rather in order to describe the space S α itself; in later sections some of its properties will be set in analogy or contrast to those of certain spaces of ultrafilters. There are intriguing parallels between some rather refined properties of S α and corresponding properties of the space U(α + ) (to be defined in § 7) of uniform ultrafilters on α+ the particular case α = ω+=2 ω the space S ω + indeed is homeomorphic to the space (of non-principal ultrafilters on ω) s(ω)\ ω (cf. the introduction and results of § 14 and § 15).