Locally Boolean spectra
This paper is a continuation of the author's earlier paper [ibid. 29, 232-272 (1992; Zbl 0772.08005)]. He uses the same approach to study now locally Boolean representability. The generality of the results obtained in both works put into perspective the power of universal algebra in the study of sheaf representations of algebras (in all cases the results are obtained by combining a minimal amount of particular algebra with strong general universal algebra theorems). The main results of this paper are found in Sections 6, 7 and 8. In section 6, the author defines locally Boolean spectra and locally projectives and proves two important theorems characterizing them. In section 7, the author introduces the SUP (spectra uniqueness property) and uses Beth's definability theorem to prove that for every \(\forall \exists\) class \({\mathcal M}\) with the SUP, the class of (locally) Boolean \({\mathcal M}\)-representable algebras is elementary. Also he proves that every class of finitely subdirectly irreducible algebras in a congruence distributive variety has the SUP and that the class of domains and the class of simple rings with identity (in the language of rings and rings with identity, respectively) have the SUP. In section 8 the author uses the characterization theorems proved in Section 6 to study the locally Boolean representation of semi-irreducible algebras by finitely subdirectly irreducibles.
scientific article; zbMATH DE number 759385
scientific article