LAUSR

Let $$T_n(\mathbb {F})$$Tn(F) and $$UT_n(\mathbb {F})$$UTn(F) be the semigroups of all upper triangular $$n\times n$$n×n matrices and all upper triangular $$n\times n$$n×n matrices with 0s and/or 1s on the main diagonal over a field $$\mathbb {F}$$F with $$\mathsf {char}(\mathbb {F})=0$$char(F)=0, respectively. In this paper, we address the finite basis problem for $$T_2(\mathbb {F})$$T2(F) and $$UT_2(\mathbb {F})$$UT2(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 $$T_2(\mathbb {F})$$T2(F) and $$UT_2(\mathbb {F})$$UT2(F) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety $$\mathsf {var} UT_2(\mathbb {F})$$varUT2(F) generated by $$UT_2(\mathbb {F})$$UT2(F) and the involution monoid variety $$\mathsf {var} T_2(\mathbb {F})$$varT2(F) generated by $$T_2(\mathbb {F})$$T2(F). Moreover, $$\mathsf {var} UT_2(\mathbb {F})$$varUT2(F) cannot be defined within $$\mathsf {var} T_2(\mathbb {F})$$varT2(F) by any finite set of identities.