THE FINITE BASIS PROBLEM FOR INVOLUTION SEMIGROUPS OF TRIANGULAR MATRICES

Abstract

Let $T_{n}(\mathbb{F})$ be the semigroup of all upper triangular $n\times n$ matrices over a field $\mathbb{F}$ . Let $UT_{n}(\mathbb{F})$ and $UT_{n}^{\pm 1}(\mathbb{F})$ be subsemigroups of $T_{n}(\mathbb{F})$ , respectively, having $0$ s and/or $1$ s on the main diagonal and $0$ s and/or $\pm 1$ s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that $UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$ and $T_{2}(\mathbb{F})$ as involution semigroups under the skew transposition are nonfinitely based for any field $\mathbb{F}$ .