Algèbres de Boole primitives (Q2266034)

The collection of isomorphism classes of countable Boolean algebras has the structure of a commutative semiring \({\mathcal B}\) with addition induced by the categorical product and multiplication by the coproduct. The additive structure of \({\mathcal B}\) is very complicated. However, there are two important subsemirings \({\mathcal P}\) and \({\mathcal F}\) in \({\mathcal B}\) whose additive monoids are fairly tame. They correspond to the quasi-primitive algebras introduced by \textit{W. Hanf} [Proc. Symp. Pure Math. 25, 75-90 (1974; Zbl 0344.02041)] and the smaller collection of finitary algebras that were defined by the reviewer [Compact zero-dimensional metric spaces of finite type, Mem. Am. Math. Soc. 130 (1972; Zbl 0253.54028)]. In this paper, the author shows how to recover the semiring \({\mathcal P}\) from the collection \({\mathcal V}\) of isomorphism classes of primitive algebras, that is, pseudo-indecomposable algebras all of whose principal ideals are products of pseudo-indecomposable algebras. The collection \({\mathcal V}\) is characterized as a particular commutative monoid with distinguished antisymmetric, transitive relation \(\triangleleft\). The semiring \({\mathcal P}\) is obtained in the form \(\omega\) [\({\mathcal V}]/\theta\), where \(\omega\) [\({\mathcal V}]\) is the semiring analogue of the group algebra of (\({\mathcal V},\cdot)\), and \(\theta\) is a congruence relation on \(\omega\) [\({\mathcal V}]\) that is described in terms of \(\triangleleft\). This construction generalizes the corresponding description that was given by the reviewer [loc.cit.]. The author introduces an important class of Boolean algebras (well founded algebras) whose semiring \({\mathcal W}\) of isomorphism types is between \({\mathcal F}\) and \({\mathcal P}\). He shows that \({\mathcal W}\) enjoys most of the good properties of \({\mathcal F}\). In particular, the system \({\mathcal F}\cap {\mathcal V}\) can be characterized by simple arithmetical properties.