Small inherently nonfinitely based finite semigroups
An algebra \(A\) (or a variety \(\mathcal V\)) is said to be inherently nonfinitely based if it is locally finite, not finitely based and has the property that every locally finite variety \({\mathcal V}'\) for which \(A\in{\mathcal V}'\) (or \({\mathcal V}\subseteq{\mathcal V}'\), respectively) is also not finitely based. It is shown how to construct all ``forbidden divisors'' for the pseudovariety of not inherently nonfinitely based finite semigroups. Several other results concerning finite semigroups that generate an inherently nonfinitely based variety that is minimal amongst those generated by finite semigroups are obtained along the way. For example, aside the variety generated by the well known six element Brandt monoid \(B_2^1\), a variety of this type is necessarily generated by a semigroup with at least 56 elements (all such semigroups with 56 elements are described in the main result).
inherently nonfinitely based varieties of semigroups
scientific article; zbMATH DE number 1747983