Weakly Strict Regular Semigroups

Weakly strict regular semigroups WS represent a generalization of strict regular semigroups. For the intersection of WS with some usual e-varieties of regular semigroups, such as left regular orthodox semigroups and similar e-varieties V, we provide WS ⋂ V with a basis of identities and forbidden semigroups. The latter, by exclusion, characterize the given e-variety. We do this also for WS. Coupled with some results of Hall, Churchill and Trotter, we characterize in this way also the e-varieties of completely regular semigroups as well as regular semigroups which are locally completely regular. The e-varieties studied are depicted in a diagram.