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.