Bases for certain varieties of completely regular semigroups

Abstract

Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $\mathscr{CR}$. The lattice of subvarieties of $\,\mathscr{CR}$ is denoted by $\mathcal{L}(\mathscr{CR})$. For each variety in an $\bigcap$-subsemilattice $\Gamma$ of $\mathcal{L}(\mathscr{CR})$, we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$, we construct $\mathbf{L}$-classes of all varieties in $\Gamma$. Two figures illustrate the theory.