Algebra universalis

, Volume 65, Issue 4, pp 371–391

# Joins and subdirect products of varieties

• Tomasz Kowalski
• Francesco Paoli
Article

## 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.

## 2000 Mathematics Subject Classification

Primary 08B26 Secondary 08B05 03C05

## Keywords and phrases

disjoint varieties joins of varieties direct and subdirect products of varieties weakened congruence permutability

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