Hereditarily finitely based semigroups of triangular matrices over finite fields. (Q1955762)

A semigroup \(S\) is finitely based if there exists some finite set of its identities from which all other identities of \(S\) can be deduced. For any semigroup \(S\), let \(\text{var\,}S\) denote the variety generated by \(S\). A finitely based semigroup \(S\) satisfies the stronger property of being hereditarily finitely based if all semigroups in the variety \(\text{var\,}S\) are finitely based. Let \(\mathcal T_n(F_q)\) be the semigroup of all \(n\times n\) upper triangular matrices over the finite field \(F_q\) of order \(q\). The authors investigate the problem of hereditarily finitely basedness of semigroups \(\mathcal T_n(F_q)\) for any \(n\) and \(q\) (recall, that \(q=p^k\) for some prime number \(p\) and natural number \(k\)). They prove that the variety generated by \(\mathcal T_n(F_2)\) is hereditarily finitely based if and only if \(n\leq 2\). Moreover, the semigroup \(\mathcal T_n(F_q)\) is not hereditarily finitely based for any \(n\geq 3\) and any \(q\).