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In this paper, using results of Almeida and Weil, we give criteria for the semidirect or two-sided semidirect product of two locally finite pseudovarieties V and W to satisfy an identity u = v. We illustrate these criteria with various semidirect and two-sided semidirect products of pseudovarieties of f-trivial monoids. In particular, let J 1 denote the class of all finite semilattice monoids and let W i be the sequence of pseudovarieties of monoids defined by W 1 = J 1 and W i +1 = J 1 * * W 2 (the two-sided semidirect product of J 1 by W 2 ). Each W k turns out to be perfectly related to the k-move standard Ehrenfeucht-Fraisse game. The union U k>1 W k is then the class A of all finite aperiodic monoids.

On semidirect and two-sided semidirect products of finite $\mathcal {J}$trivial monoids