Bases for pseudovarieties closed under bideterministic product

Abstract

We show that if $\mathsf{V}$ is a semigroup pseudovariety containing the finite semilattices and contained in $\mathsf{DS}$, then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of $\mathsf{J}$-reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that $\mathsf{DH}\cap \mathsf{ECom}$ is local, for any group pseudovariety $\mathsf{H}$.

The only changes from version 1 to version 2 are the following. The year of the scheme of the Mathematics Subject Classification was corrected to 2010. The font size was changed to 11pt