The Finite Basis Problem for Finite Semigroups: A Survey

Matrix identities involving multiplication and transposition

“…Here we present a 'unary' modification of the approach used in [53]. According to the classification proposed in the survey [54], this approach is referred to as the critical semigroup method. The formulation of the corresponding result involves two simple operators on unary semigroup varieties.…”

THE SEMIGROUPS B2AND B0ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

“…These results are consequences of the existence of a 'series of critical semigroups', in the terminology of Volkov [34]. (See the last part of Section 1 for a general discussion.)…”

“…The article by Volkov [34] gives a useful overview of finite and nonfinite basability for semigroups, inverse semigroups and monoids. In contrast to the situation for groups, whereby every finite group is finitely based, in each case no algorithm is known (at the time of writing) that will decide, given a finite semigroup of the appropriate type, whether it is finitely based.…”

“…Such a series constitutes a sequence of algebras A n with the following properties: A n is ngenerated, and any subalgebra generated by fewer than n elements belongs to V(A). Although the argument that justifies the following application is elementary (see [34,Section 4.2]), we include it both for completeness' sake and because a little thought will reveal that the formal definition may be tweaked somewhat to achieve the same effect, as we need to do in Section 4. RESULT 1.2 If the sequence A n , n ≥ 1, is a series of critical algebras for an algebra A, then any variety of algebras that (a) contains A and (b) contains no A n , is nonfinitely based.…”

On the Variety Generated by the Monoid of Triangular 2×2 Matrices Over A two-Element field

“…However, the solution to this problem is unknown when restricted to most natural classes containing both finitely based and nonfinitely based examples. In particular, the finite basis problem for finite semigroups has been extensively studied since the end of the 1960s, and many important and interesting results about it have been obtained (see [5, 6, 8-10, 12, 14, 16] and references therein); furthermore, there is a very large volume of work concerning this problem within the variety of semigroups [13,14].…”