Congruence properties in congruence permutable and in ideal determined varieties, with applications.

Abstract

We define a weak version of EDPC (equationally definable principal congruences), called EDPC*, that is shown to be preserved under varietal closure in congruence permutable varieties. We show that if \(\mathcal{V}\) is a congruence permutable variety generated by a class \(\mathcal{K},\) then \(\mathcal{V}\) has EDPC iff \(\mathcal{V}\) has EDPC* iff \(\mathcal{K}\) has EDPC*. An equational condition is given which, if satisfied by \(\mathcal{K},\) implies that \(\mathcal{V}\) has the CEP (congruence extension property). Similar results are proved for ideal determined varieties. These results are applied to the variety of residuated lattices, with examples.