Identities of semigroups of triangular matrices over finite fields.
Ever since \textit{P. Perkins} [J. Algebra 11, 298-314 (1969; Zbl 0186.03401)] gave the first example of a finite semigroup which did not have a finite basis for its laws, the question of precisely which semigroups are finitely based has been of great interest. A stronger property than being nonfinitely based is being inherently nonfinitely based. A locally finite variety is `inherently nonfinitely based' if every locally finite variety containing it is nonfinitely based. Since the variety generated by any finite algebra is locally finite, it makes sense to refer to a finite semigroup as being inherently nonfinitely based. This paper proves the interesting result that the semigroup \(T_n(K)\) of all upper triangular \(n\times n\) matrices over a finite field \(K\) is inherently nonfinitely based if and only if \(|K|>2\) and \(n>3\). Note that the varieties generated by these semigroups do not contain Perkin's semigroup. Sufficiency is established by using a criterion of \textit{M. V. Sapir} [Math. USSR, Sb. 61, No. 1, 155-166 (1988); translation from Mat. Sb., Nov. Ser. 133(175), No. 2(6), 154-166 (1987; Zbl 0634.20027)] while necessity is proved by showing that \(T_n(K)\) can be embedded in a locally finite, finitely based variety when \(|K|=2\) or \(n\leq 3\).
scientific article; zbMATH DE number 2115592