The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ × 2 matrices

Abstract

Let $$T_n(\mathbb {F})$$ and $$UT_n(\mathbb {F})$$ be the semigroups of all upper triangular $$n\times n$$ matrices and all upper triangular $$n\times n$$ matrices with 0s and/or 1s on the main diagonal over a field $$\mathbb {F}$$ with $$\mathsf {char}(\mathbb {F})=0$$ , respectively. In this paper, we address the finite basis problem for $$T_2(\mathbb {F})$$ and $$UT_2(\mathbb {F})$$ as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both $$T_2(\mathbb {F})$$ and $$UT_2(\mathbb {F})$$ are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety $$\mathsf {var} UT_2(\mathbb {F})$$ generated by $$UT_2(\mathbb {F})$$ and the involution monoid variety $$\mathsf {var} T_2(\mathbb {F})$$ generated by $$T_2(\mathbb {F})$$ . Moreover, $$\mathsf {var} UT_2(\mathbb {F})$$ cannot be defined within $$\mathsf {var} T_2(\mathbb {F})$$ by any finite set of identities.