Jónsson's theorem in non-modular varieties. (Q1771886)

Jónsson's Lemma says that if \(HSP(\mathcal K)\) is congruence distributive and \(B\in HSP(\mathcal K)\) is subdirectly irreducible then \(B\in HSP_u(\mathcal K)\). It was generalized in 80s by Hagemann, Hermann, Freese, McKenzie and Hrushovski: If \(HSP(\mathcal K)\) is congruence modular and \(B\in HSP(\mathcal K)\) is subdirectly irreducible and \(\alpha \) is the centralizer of the monolith of \(B\), then \(B/\alpha \in HSP_u(\mathcal K)\). The present author dropped out the assumption of congruence modularity and proves: Let \(B\) be a subdirectly irreducible algebra in \(HSP(\mathcal K)\) and let \(\alpha \) be a maximal congruence that laxly centralizes the monolith of \(B\). Then \(B/\alpha \in HSP_u(\mathcal K)\).