Semilattices with a group of automorphisms

We investigate semilattices expanded by a group F of automorphisms acting as new unary basic operations. We describe up to isomorphism all simple algebras of this kind in case that F is commutative. Finally, we present an example of a simple algebra that does not fit in the previous description, if F is not commutative

Self-rectangulating varieties of type 5

We show that a locally finite variety which omits abelian types is self–rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type–set {5}. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the congruence extension property.