On the finite basis problem for the monoids of triangular Boolean matrices. (Q634761)

Let \(\mathcal{TB}_n\) denote the submonoid of all upper triangular Boolean \(n\times n\) matrices. It was shown by Volkov and Goldberg that \(\mathcal{TB}_n\) is inherently nonfinitely based for \(n\geq 4\), but the cases when \(n=2\), \(3\) remained open [see \textit{M. V. Volkov} and \textit{I. A. Goldberg}, ``The finite basis problems for monoids of unitriangular Boolean matrices'', RIMS Kokyuroku 1366, 205-214 (2004)]. In this paper it is shown that the monoid \(\mathcal{TB}_2\) is finitely based, and a finite identity basis for the monoid \(\mathcal{TB}_2\) is given. Moreover, it is shown that the monoid \(\mathcal{TB}_3\) is inherently nonfinitely based. Hence, the monoid \(\mathcal{TB}_n\) is finitely based if and only if \(n\leq 2\).