A characterization of varieties. (Q1771852)

Let \(\mathcal L\) be a language of algebras and \(K\) a quasivariety of \(\mathcal L\)-algebras. Let \(\mathcal A\) be an \(\mathcal L\)-algebra. A congruence \(\Theta \in \text{Con} \mathcal A\) is called a \(K\)-congruence if \(\mathcal A / \Theta \in K\). The set \(\text{Con}_K \mathcal A\) of all \(K\)-congruences forms an algebraic lattice which is a meet-complete subsemilattice of \(\text{Con} \mathcal A\). The author proves that \(K\) is a variety if \(\text{Con}_K \mathcal A\) is a sublattice of \(\text{Con} \mathcal A\) for all \(\mathcal A \in K\).