Inherently nonfinitely based finite semigroups

A locally finite variety is called inherently nonfinitely based if it is not contained in any finitely based locally finite variety. A finite universal algebra is called inherently nonfinitely based if it generates an inherently nonfinitely based variety. In this paper a description of inherently nonfinitely based finite semigroups is given; it is proved that the set of such semigroups is recursive and that the property of a finite semigroup to be inherently nonfinitely based is mainly determined by the structure of its subgroups. It is also shown that there exists a unique minimal inherently nonfinitely based variety of semigroups consisting not only of groups. It is not known whether there exists an inherently nonfinitely based variety of groups.
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