Krohn–Rhodes complexity of Brauer type semigroups
Karl Auinger · Portugaliae Mathematica · 2013
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.