Geodesic
Locally finite M-solid varieties of semigroups
Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 2, pp. 139-148

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An algebra of type τ is said to be locally finite if all its finitely generated subalgebras are finite. A class K of algebras of type τ is called locally finite if all its elements are locally finite. It is well-known (see [2]) that a variety of algebras of the same type τ is locally finite iff all its finitely generated free algebras are finite. A variety V is finitely based if it admits a finite basis of identities, i.e. if there is a finite set σ of identities such that V = ModΣ, the class of all algebras of type τ which satisfy all identities from Σ. Every variety which is generated by a finite algebra is locally finite. But there are finite algebras which are not finitely based. For semigroup varieties, Perkins proved that the variety generated by the five-element Brandt-semigroup
Keywords: locally finite variety, finitely based variety, M-solidvariety 
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Denecke, Klaus; Pibaljommee, Bundit. Locally finite M-solid varieties of semigroups. Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 2, pp. 139-148. http://geodesic.mathdoc.fr/item/DMGAA_2003_23_2_a4/