Finite basis problem for involution monoids of unitriangular boolean matrices

Abstract

Let $$(BU_n, \,^*\,)$$ be the involution monoid of all Boolean upper triangular $$n\times n$$ matrices with 1s on the main diagonal under the ordinary matrix multiplication and the skew transposition. The involution monoid $$(BU_2, \,^*\,)$$ is easily seen to be finitely based. In this paper, we shown that $$(BU_n, \,^*\,)$$ is non-finitely based for each $$n \ge 3$$, which answers an open question posed by Auinger et al. Therefore involution monoid $$(BU_n, \,^*\,)$$ is finitely based if and only if $$n = 2$$.