Transfer theorems for finitely subdirectly irreducible algebras

Abstract

We show that under certain conditions, well-studied algebraic properties transfer from the class QRFSI of the relatively finitely subdirectly irreducible members of a quasivariety Q to the whole quasivariety, and, in certain cases, back again. First, we prove that if Q is relatively congruence-distributive, then it has the Q-congruence extension property (Q-CEP) if and only if QRFSI has this property. We then prove that if Q has the Q-CEP and QRFSI is closed under subalgebras, then Q has a one-sided amalgamation property (for quasivarieties, equivalent to the amalgamation property) if and only if QRFSI has this property. We also establish similar results for the transferable injections property and strong amalgamation property. For each property considered, we specialize our results to the case where Q is a variety — so that QRFSI is the class of finitely subdirectly irreducible members of Q and the Q-CEP is the usual congruence extension property — and prove that when Q is finitely generated and congruence-distributive, and QRFSI is closed under subalgebras, possession of the property is decidable. Finally, as a case study, we provide a complete description of the subvarieties of a notable variety of BL-algebras that have the amalgamation property.