# Krohn–Rhodes complexity of Brauer type semigroups

### Karl Auinger

Universität Wien, Austria

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## 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.

## Cite this article

Karl Auinger, Krohn–Rhodes complexity of Brauer type semigroups. Port. Math. 69 (2012), no. 4, pp. 341–360

DOI 10.4171/PM/1921