On certain varieties of semigroups

In this paper we generalize the class of completely regular semigroups (unions of groups) to the class of local monoids, that is the class of all semigroups where the local subsemigroups aSa are local submonoids. The sublattice of this variety (L($\mathcal{L}(\mathcal{M}))$) covers another lattice isomorphic to the lattice of all bands $([x^2 = x])$. Every bandvariety $\mathcal{U}$ has as image the variety $\Phi - \mathcal {U}$, which is the class of all semigroups, where $\Phi$ is a $\mathcal{U}$-congruence (a $\Phi$ b $\Leftrightarrow$ aSa = bSb). It is shown how one can find the laws for $\Phi - \mathcal {U}$ for a given bandvariety $\mathcal{U}$. The laws for for $\Phi - \mathcal {B}$ are given and it is shown that $\Phi - \mathcal {RB}$ = $\mathcal{L(G)(L(V)}:= \{S : aSa \in \mathcal{V}\forall a \in S\})$.