Joins and subdirect products of varieties

Abstract

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and Płonka proved that independent subvarieties ${\mathcal{V}_{1}, \mathcal{V}_{2}}$ of a variety ${\mathcal{V}}$ are disjoint and such that their join ${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$ (in the lattice of subvarieties of ${\mathcal{V}}$) is their direct product ${\mathcal{V}_{1} \times \mathcal{V}_{2}}$ . Jónsson and Tsinakis provided a partial converse to this result: if ${\mathcal{V}}$ is congruence permutable and ${\mathcal{V}_{1}, \mathcal{V}_{2}}$ are disjoint, then they are independent (and so ${\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}$). We show that (i) if ${\mathcal{V}}$ is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if ${\mathcal{V}}$ satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and ${\mathcal{V}_{1} \vee \mathcal{V}_{2}}$ is the subdirect product of ${\mathcal{V}_{1}}$ and ${\mathcal{V}_2}$ ; (iii) the same holds if ${\mathcal{V}}$ is congruence 3-permutable.