Varieties of finitely approximable semigroups

In this paper we discuss the question of when a variety of commutative or nilpotent semigroups consists only of finitely approximate semigroups. Moreover, we give necessary and sufficient conditions in order to have $\operatorname{Var}S=q\operatorname{Var}S$ (here $\operatorname{Var}S$ denotes the variety generated by the semigroup $S$, and $q\operatorname{Var}S$ the quasivariety generated by $S$).
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