On the lattice of completely regular monoid varieties
For any subvariety \(V\) of the variety \(CR\) of completely regular semigroups, let \(MV\) be the variety of completely regular monoids consisting of the monoids whose completely regular semigroup reducts belong to \(V\), and \(\langle MV\rangle\) the subvariety of \(CR\) generated by the completely regular semigroup reducts of the members of \(MV\). Then \(V\to\langle MV\rangle\) is a complete endomorphism of the lattice \(L(CR)\) of subvarieties of \(CR\), and the image of this endomorphism is in a natural way isomorphic to the lattice \(L(MCR)\) of subvarieties of \(MCR\). In [\textit{L. Polák}, Semigroup Forum 36, 253-284 (1987; Zbl 0638.20032)] a representation of the interval \([SL,CR]\) has been given: each subvariety of \(CR\) containing the variety \(SL\) of semilattices is represented by an order preserving mapping between two fixed posets. The above mentioned observation, together with a translation of Polák's machinery to the new circumstances, allows to establish a corresponding representation theorem for the interval \([MSL,MCR]\) of \(L(MCR)\). The description of the lattice of band monoids given by S. Wismath follows as an easy consequence.
scientific article; zbMATH DE number 167675
scientific article