Lipparini, Paolo Another characterization of congruence distributive varieties. (English) Zbl 1474.08013 Stud. Sci. Math. Hung. 57, No. 3, 284-289 (2020). 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. Reviewer: Danica Jakubiková-Studenovská (Košice) MSC: 08B10 Congruence modularity, congruence distributivity 08B05 Equational logic, Mal’tsev conditions Keywords:congruence distributive variety; Maltsev condition; congruence identity Citations:Zbl 1294.08002 PDFBibTeX XMLCite \textit{P. Lipparini}, Stud. Sci. Math. Hung. 57, No. 3, 284--289 (2020; Zbl 1474.08013) Full Text: DOI arXiv