The monoid of $$2 \times 2$$2×2 triangular boolean matrices under skew transposition is non-finitely based

Let $$\mathscr {T\!B}_n$$ be the involution semigroup of all upper triangular boolean $$n\times n$$ matrices under the ordinary matrix multiplication and the skew transposition. It is shown by Auinger et al. that the involution semigroup $$\mathscr {T\!B}_n$$ is non-finitely based if $$n > 2$$, but the case when $$n=2$$ still remains open. In this paper, we give a sufficient condition under which an involution semigroup is non-finitely based. As an application, we show that the involution semigroup $$\mathscr {T\!B}_2$$ is non-finitely based. Hence $$\mathscr {T\!B}_n$$ is non-finitely based for all $$n \ge 2$$.