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Another characterization of congruence distributive varieties. (English) Zbl 1474.08013

By definition, an algebra \(A\) is congruence distributive if and only if, for all congruences \(\alpha\), \(\beta\), \(gamma\) of \(A\) and for every \(h\), the inclusion \[(1)\quad \alpha(\beta\circ_h\gamma)\subseteq\alpha\beta+\gamma\] holds (juxtaposition denotes intersection, \(+\) is join and \(\beta\circ_h\gamma:=\beta\circ\gamma\circ\beta\circ\gamma\dots\) with \(h-1\) occurences of \(\circ\)).
Let \(\mathcal{V}\) be a variety. It is known that \(\mathcal{V}\) is congruence distributive if and only if there is \(k\) such that (1) \(\alpha(\beta\circ_h\gamma)\subseteq \alpha\beta\circ_k\gamma\) holds in \(\mathcal{V}\). K. A. Kearnes and E. W. Kiss [The shape of congruence lattices. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1294.08002)] showed that (1) with \(h=2\) is valid if and only if \(\mathcal{V}\) is join congruence semidistributive.
The main result of the paper is the following
Theorem: A variety \(\mathcal{V}\) is congruence distributive if and only if there is \(k\) such that \(\alpha(\beta\circ\gamma\circ\beta)\subseteq \alpha\beta\circ_k\gamma\) holds in \(\mathcal{V}\).
The result of the theorem is also expressed in terms of a Maltsev conditions. A sharper boundary is described in Theorem 5.

MSC:

08B10 Congruence modularity, congruence distributivity
08B05 Equational logic, Mal’tsev conditions

Citations:

Zbl 1294.08002
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