Mal'cev conditions for Horn sentences with congruence permutability (Q794686)

Let \({\mathcal V}\) be a variety and \({\mathcal S}{\mathcal C}{\mathcal V}\) the class of lattices embeddable in congruence lattices of members of \({\mathcal V}\). Then the first proposition of this paper is: ''For any n-permutable variety \({\mathcal V}\), \({\mathcal S}{\mathcal C}{\mathcal V}\) is a quasivariety, that is, a class of lattices definable by a set of Horn sentences.'' Although it is known that satisfaction of a Horn sentence in an n- permutable variety can be characterized by Mal'cev conditions [see \textit{W. D. Neumann}, J. Aust. Math. Soc. 17, 376-384 (1974; Zbl 0294.08004) and \textit{W. Taylor}, Algebra Univers. 3, 351-397 (1973; Zbl 0304.08003) previously no concrete Mal'cev conditions have been known in general. The main aim of this paper is to give an algorithm (too technical to state here) which associates a suitable concrete Mal'cev condition with any (lattice) Horn sentence.