Submission date: 10 May 2022

Abstract:

We show that under certain conditions, well-studied algebraic properties transfer from the class V_{FSI} of finitely subdirectly irreducible members of a variety V to the whole variety, and, in certain cases, back again. First, we prove that a congruence-distributive variety V has the congruence extension property if and only if V_{FSI} has the congruence extension property. We then prove that for a variety V with the congruence extension property such that V_{FSI} is closed under subalgebras, V has a one-sided amalgamation property (equivalently, since V is a variety, the amalgamation property) if and only if V_{FSI} has this property. We also establish similar results for the transferable injections and strong amalgamation properties, and prove that possession of all these properties is decidable for finitely generated varieties satisfying certain conditions. Finally, as a case study, we describe the subvarieties of a notable variety of BL-algebras that have the amalgamation property.

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Full text arXiv 2205.05148: pdf, ps.