Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

Abstract.

It has been shown by the authors that the mapping$\chi : {\bf V} \longrightarrow {\bf V}\cap{\bf B}$,where B is the pseudovariety of finite bands, is a complete retraction of the lattice L(F) of pseudovarieties of finite semigroups onto the lattice of pseudovarieties of bands. It follows that the classes of the induced congruence $\overline{\chi}$ on L(F), or on the lattice of subpseudovarieties L(W) for any subpseudovariety W of F, are intervals. In this paper we solve the membership problem for the upper limit of the classes of $\overline{\chi}$ restricted to L(W) for various ${\bf W} \in {\bf L(F)}$, including F itself, and provide bases of pseudoidentities for certain cases.