Self-rectangulating varieties of type 5
  We show that a locally finite variety which omits abelian types is self-regulating if and only if it has a compatible semilattice term operation. Such varieties must have a 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. 
Keith A. Kearnes (University of Arkansas), {\AA}gnes Szendrei (Bolyai   Institute)