On the variety generated by the monoid of triangular \(2\times 2\) matrices over a two-element field. (Q2907022)

Let \(\mathcal T_n(F)\) be the monoid of all upper triangular \(n\times n\) matrices over a finite field \(F\). It is shown by \textit{M. V. Volkov} and \textit{I. A. Gol'dberg} [Math. Notes 73, No. 4, 474-481; translation from Mat. Zametki 73, No. 4, 502-510 (2003; Zbl 1064.20056)] that the monoid \(\mathcal T_n(F)\) is inherently nonfinitely based if \(|F|>2\) and \(n>3\), but the cases where \(|F|>2\) and \(n=2,3\) and where \(|F|=2\) are left open; it is also shown that the monoid \(\mathcal T_n(F)\) is not inherently nonfinitely based when \(n<4\) or \(|F|=2\).NEWLINENEWLINE In this paper, it is shown that the monoid \(\mathcal T_2(F)\) is finitely based when \(|F|=2\), and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety \(\mathbf T_2(F)\) generated by \(\mathcal T_2(F)\) with \(|F|=2\) are determined. It is shown that \(\mathbf T_2(F)\) has seven maximal subvarieties, each of which is defined within \(\mathbf T_2(F)\) by one identity.