Coproducts of distributive lattice-based algebras

Abstract

This paper presents a systematic study of coproducts. This is carried out principally, but not exclusively, for finitely generated quasivarieties ${{\bf \mathcal{A}}}$ that admit a (term) reduct in the variety ${{\bf \mathfrak{D}}}$ of bounded distributive lattices. In this setting, we present necessary and sufficient conditions on ${{\bf \mathcal{A}}}$ for the forgetful functor ${U_{{\bf \mathcal{A}}}}$ from ${{\bf \mathcal{A}}}$ to ${{\bf \mathfrak{D}}}$ to preserve coproducts. We also investigate the possible behaviours of ${U_{{\bf \mathcal{A}}}}$ as regards coproducts in ${{\bf \mathcal{A}}}$ under weaker assumptions. Depending on the properties exhibited by the functor, different procedures are then available for describing these coproducts. We classify a selection of well–known varieties within our scheme, thereby unifying earlier results and obtaining some new ones.

The paper’s methodology draws heavily on duality theory. We use Priestley duality as a tool and our descriptions of coproducts are given in terms of this duality. We also exploit natural duality theory, specifically multisorted piggyback dualities, in our analysis of the behaviour of the forgetful functor into ${{\bf \mathfrak{D}}}$. In the opposite direction, we reveal that the type of natural duality that the class ${{\bf \mathcal{A}}}$ can possess is governed by properties of coproducts in ${{\bf \mathcal{A}}}$ and the way in which the classes ${{\bf \mathcal{A}}}$ and ${U_{{\bf \mathcal{A}}}}$ (${{\bf \mathcal{A}}}$) interact.