Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $\mathrm{𝒞ℛ}$. The lattice of subvarieties of $\mathrm{𝒞ℛ}$ is denoted by $ℒ\left(\mathrm{𝒞ℛ}\right)$. For each variety in an $\bigcap$-subsemilattice $\Gamma$ of $ℒ\left(\mathrm{𝒞ℛ}\right)$, 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 $𝐋$-classes of all varieties in $\Gamma$. Two figures illustrate...

In this paper we consider different relations on the set P(V) of all proper hypersubstitutions with respect to a given variety V and their properties. Using these relations we introduce the cardinalities of the corresponding quotient sets as degrees and determine the properties of solid varieties having given degrees. Finally, for all varieties of bands we determine their degrees.