On a semigroup variety of György Pollák

Let P be the variety of semigroups defined by the identity xyzx ≈ x². By a result of György Pollák, every subvariety of P is finitely based. The present article is concerned with subvarieties of P and the lattice they constitute, where the main result is a characterization of finitely generated subvarieties of P. It is shown that a subvariety of P is finitely generated if and only if it contains finitely many subvarieties, and the identities defining these varieties are described. Specifically, it is decidable when a finite set of identities defines a finitely generated subvariety of P. It follows that the finitely generated subvarieties of P constitute an incomplete lattice while the non-finitely generated subvarieties of P constitute an interval. It is also shown that given any pair of finitely generated subvarieties of P, a finite semigroup that generates their meet is computable.