\(n\)-permutable varieties satisfy non trivial congruence identities
A variety \({\mathcal V}\) is called \(n\)-permutable if, for every algebra \(A\in {\mathcal V}\) and congruences \(\alpha, \beta\) on \(A\), \(\alpha\circ \beta\circ \alpha\circ \ldots= \beta\circ \alpha\circ \beta\circ \ldots\) with \(n-1\) \(\circ\)'s on each side. The author proves that, for each \(n\geq 2\), there is a nontrivial identity holding in every congruence lattice of algebras in \(n\)-permutable varieties. Under the presence of a semilattice operation, a similar result was obtained by A. Day [see \textit{B. Jónsson}, Algebra Univers. 10, 355-394 (1980; Zbl 0438.08003)]. \textit{D. Hobby} and \textit{R. McKenzie} proved the result for locally finite varieties [The structure of finite algebra (Contemp. Math. 76) (1988; Zbl 0721.08001), Theorem 9.19], but their identity depends also on the variety, not just on \(n\). The paper contains some other results and several open problems.
scientific article; zbMATH DE number 750785