Transfer theorems for finitely subdirectly irreducible algebras

Abstract

We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole quasivariety, and, in certain cases, back again. First, we prove that if $\mathcal{Q}$ is relatively congruence-distributive, then it has the $\mathcal{Q}$-congruence extension property if and only if $\mathcal{Q}_{_\text{RFSI}}$ has this property. We then prove that if $\mathcal{Q}$ has the $\mathcal{Q}$-congruence extension property and $\mathcal{Q}_{_\text{RFSI}}$ is closed under subalgebras, then $\mathcal{Q}$ has a one-sided amalgamation property (equivalently, for $\mathcal{Q}$, the amalgamation property) if and only if $\mathcal{Q}_{_\text{RFSI}}$ 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 $\mathcal{Q}$ is a variety -- so that $\mathcal{Q}_{_\text{RFSI}}$ is the class of finitely subdirectly irreducible members of $\mathcal{Q}$ and the $\mathcal{Q}$-congruence extension property is the usual congruence extension property -- and prove that when $\mathcal{Q}$ is finitely generated and congruence-distributive, and $\mathcal{Q}_{_\text{RFSI}}$ 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.