Posets of \({\mathcal C}\)-congruences (Q1272233)

Let \({\mathcal C}\) be a non-empty class of algebras of the same type \(\tau \) closed under isomorphisms. Let \(A\) be an algebra of type \(\tau .\) A congruence \(\theta \in \operatorname { Con} A\) is called a \({\mathcal C}\)-congruence if \(A/\theta \in {\mathcal C}.\) Denote by \(\operatorname {Con}_{\mathcal C} A\) the poset of all \({\mathcal C}\)-congruences on \(A\) ordered by inclusion. Theorem: \({\mathcal C}\) is a variety if and only if \(\operatorname {Con}_{\mathcal C} A\) is a complete sublattice of \(\operatorname { Con}A\) for each \(A\) of type \(\tau .\)