On the finite basis problem for the monoids of triangular boolean matrices

Abstract

Let $${{\fancyscript{T}\fancyscript{B}_n}}$$ denote the submonoid of all upper triangular boolean n × n matrices. It was shown by Volkov and Goldberg that $${{\fancyscript{T}\fancyscript{B}_n}}$$ is nonfinitely based if n > 3, but the cases when n = 2, 3 remained open. In this paper, it is shown that the monoid $${{\fancyscript{T}\fancyscript{B}_2}}$$ is finitely based, and a finite identity basis for the monoid $${{\fancyscript{T}\fancyscript{B}_2}}$$ is given. Moreover, it is shown that $${{\fancyscript{T}\fancyscript{B}_3}}$$ is inherently nonfinitely based. Hence, $${{\fancyscript{T}\fancyscript{B}_n}}$$ is finitely based if and only if n ≤ 2.