Krohn–Rhodes complexity of Brauer type semigroups

Abstract

The Krohn–Rhodes complexity of the Brauer type semigroups ${\mathfrak{B}}_n$ and ${\mathfrak{A}}_n$ is computed. In three-quarters of the cases the result is the ‘expected’ one: the complexity coincides with the (essential) $\mathcal{J}$-depth of the respective semigroup. The exception (and perhaps the most interesting case) is the annular semigroup ${\mathfrak{A}}_{2n}$ of even degree in which case the complexity is the $\mathcal{J}$-depth minus $1$. For the ‘rook’ versions $P{\mathfrak{B}}_n$ and $P{\mathfrak{A}}_n$ it is shown that $c(P{\mathfrak{B}}_n)=c({\mathfrak{B}}_n)$ and $c(P{\mathfrak{A}}_{2n-1})=c({\mathfrak{A}}_{2n-1})$ for all $n\ge 1$. The computation of $c(P{\mathfrak{A}}_{2n})$ is left as an open problem.