Equational theories of the Boolean matrix monoid BRn with involutions

Abstract

Let $B{R}_n$ be the monoid of all $n\times n$ Boolean matrices with 1s on the main diagonal. It is known that $B{R}_n$ can be identified with the monoid of all reflexive binary relations on an n-element set under composition. The monoid $B{R}_n$ admits two natural unary operations: the transposition ^T and the skew transposition ^D, which makes $B{R}_n$ an involutory monoid. Let $B{U}_n$ (resp. $C{B}_n$) be the submonoid of $B{R}_n$ consisting of all $n\times n$ upper triangular Boolean matrices with 1s on the main diagonal (resp. all convex Boolean matrices) and $C{U}_n=C{B}_n\cap B{U}_n$ the monoid of all convex upper triangular Boolean matrices. Denote by $D{C}_n$ the double Catalan monoid which is a submonoid of $C{B}_n$.

In this paper, we explore equational theories of the involutory monoids $(B{R}_n,{}^T)$ and $(B{R}_n,{}^D)$. Firstly, we have completely solved the finite basis problems for $(B{R}_n,{}^T)$ and $(B{R}_n,{}^D)$. It is shown that the involutory monoid $(B{R}_n,{}^T)$ is finitely based if and only if $n⩽4$, and $(B{R}_n,{}^D)$ is finitely based if and only if $n⩽2$. Secondly, we study the equational equivalence problem for involutory submonoids of $(B{R}_n,{}^T)$ and $(B{R}_n,{}^D)$, respectively. It is proved that the involutory monoid $(B{R}_n,{}^T)$ satisfies the same identities as $(C{B}_n,{}^T)$ for each positive integer n, but does not satisfy the same identities as $(D{C}_n,{}^T)$ for each $n⩾2$. It is also proved that the involutory monoids $(C{U}_n,{}^D)$, $(D{C}_n,{}^D)$, $(C{B}_n,{}^D)$ and $(B{U}_n,{}^D)$ satisfy the same identities for each positive integer n, but do not satisfy the same identities as $(B{R}_n,{}^D)$ for each $n⩾3$.