Algebra universalis

, Volume 65, Issue 4, pp 371–391 | Cite as

Joins and subdirect products of varieties

  • Tomasz Kowalski
  • Francesco Paoli


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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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Department of PhilosophyUniversity of CagliariCagliariItaly

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