Lo spazio duale di un prodotto di algebre di Boole e le compattificazioni di Stone

Abstract

This paper is concerned with the Stone space X of a direct product $$B = \prod\limits_{i \in I} {B_i }$$ of infinitely many Boolean algebras. In paragraph 2, after recalling that X is the Stone-Čech compactification of the sum (disjoint union) $$\sum\limits_{i \in I} {X_i }$$ of the Stone spaces of the algebras Bi, we exhibit a compactification of $$\sum\limits_{i \in I} {X_i }$$ which is not a Stone space and we give a method to construct all the «Stone compactifications» of $$\sum\limits_{i \in I} {X_i }$$ (the corresponding Boolean algebras are easily characterised). In paragraph 3, a set of ultrafilters of B (the «decomposable» ultrafilters) are introduced: this set properly contains $$\sum\limits_{i \in I} {X_i }$$ , but, as is shown in paragraph 5, there are direct products that admit nondeeomposable ultrafilters (this is the case iff the set {Card Bi: i ε I } is not bounded by a natural number). In paragraph 4, among other things, we prove, for the set of decomposable ultrafilters, a weak form of countable compactness, in the sense that every countable clopen cover has a finite subcover; then, we deduce that the set of decomposable ultrafilters is pseudocompact, while obviously $$\sum\limits_{i \in I} {X_i }$$ is not. Lastly, in paragraph 6, we give a second characterisation of the Stone space of B, showing that every ultrafilter of B can be obtained by iterating in a suitable way the procedure which leads to the construction of decomposable ultrafilters.