Semidirect products of pseudovarieties from the universal algebraist's point of view

Several problems in finite semigroup theory and its applications ask for effective algorithms to decide whether a given finite semigroup belongs to the semidirect product V ∗ W of pseudo-varieties V and W for which such algorithms are known. Much work has been done with the special case V ∗ D, where D is the class of all finite semigroups S such that se=e for e,s ∈ S with e2=e. A new approach is proposed to treat these problems. First, note that similar problems may be phrased for suitable varieties. Then, translate results back to pseudovarieties. This method is illustrated with simple proofs of Simon's theorem SI ∗ D = LSI and Therien and Weiss's characterization of Com ∗ D (where SI and Com denote the pseudovarieties of all finite semi- lattices and commutative semigroups, respectively) as well as the solution of some equations in X of the form X ∗ V = W.

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