Abstract
This paper deals with the calculation of power pseudovarieties which consist of locally commutative semigroups. It provides the solution of some equations and inequalities involving the power operator as well as some upper bounds for specific power pseudovarieties. It turns out that all the pseudovarieties under study are contained in the pseudovariety of semigroups associated with the rational languages for which membership may be tested by a finite automaton which counts occurrences of certain factors.
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