The global of a pseudovariety of semigroups is the pseudovariety of semigroupoids it generates where its members are viewed as one-vertex semigroupoids. When the global of the pseudovariety is characterized by properties of the local semigroups of its semigroupoids, the pseudovariety is said to be local. We show that Mal'cev operators LI(malcev)_, K(malcev)_ and D(malcev)_ preserve the locality of monoidal pseudovarieties. In the process, we deal with the localization operator L_ and the semidirect product operator _* D establishing some interplay between them.
This is a joint work with Alfredo Costa.