Every $m$-permutable variety satisfies the congruence identity $\alpha\beta_h= \alpha \gamma_h$

Abstract

It is known that congruence lattices of algebras in $ m$-permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest. We show here that every $ m$-permutable variety satisfies the well-known and well-studied congruence identity $ \alpha \beta_h= \alpha \gamma_h$. We also get a new condition equivalent to $ m$-permutability.