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 $mathsf {DH}capmathsf {ECom}$ is local, for any group pseudovariety $mathsf H$.
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