Some operators that preserve the locality of a pseudovariety of semigroups

Let V be a pseudovariety of semigroups. We denote by V' the variety of V-recognizable languages. It is well-known that V' is closed under left deterministic product if and only if V=K(m)V, where (m) denotes the Mal'cev product. We show that if V' is closed under left deterministic product, then so is the variety of languages recognized by semigroups belonging to the semidirect product of V with D_n, for every positive integer n. Similar results are obtained for unambiguous product and concatenation product. We apply these preservation properties to show that the operators K(m)_, D(m)_ and LI(m)_ preserve the locality of monoidal pseudovarieties. Other operators of the form Z(m)_ are considered. This is joint work with Ana Paula Escada.