Krohn–Rhodes complexity of Brauer type semigroups

  • Karl Auinger

    Universität Wien, Austria


The Krohn–Rhodes complexity of the Brauer type semigroups Bn\mathfrak{B}_n and An\mathfrak{A}_n is computed. In three-quarters of the cases the result is the ‘expected’ one: the complexity coincides with the (essential) J\mathcal{J}-depth of the respective semigroup. The exception (and perhaps the most interesting case) is the annular semigroup A2n\mathfrak{A}_{2n} of even degree in which case the complexity is the J\mathcal{J}-depth minus 11. For the ‘rook’ versions PBnP\mathfrak{B}_n and PAnP\mathfrak{A}_n it is shown that c(PBn)=c(Bn)c(P\mathfrak{B}_n)=c(\mathfrak{B}_n) and c(PA2n1)=c(A2n1)c(P\mathfrak{A}_{2n-1})=c(\mathfrak{A}_{2n-1}) for all n1n\ge 1. The computation of c(PA2n)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