Krohn-Rhodes complexity of Brauer type semigroups. (Q1941297)

The Krohn-Rhodes complexity of a finite semigroup \(S\) is the minimal number of factors of the form \(A_i\wr G_i\), where \(A_i\) is aperiodic and \(G_i\) is a group, such that \(S\) divides an iterated wreath product composed of these factors. A ``Brauer type semigroup'' is, informally speaking, a ``natural'' subsemigroup of the Jones-Martin semigroup of all partition diagrams. In the paper under review the author determines the Krohn-Rhodes complexity for many Brauer type semigroups, including the classical Brauer semigroup, the partial version of the Brauer semigroup and the annular semigroup. In most cases the results is ``as expected'', that is the Krohn-Rhodes complexity is given by the essential \(\mathcal J\)-depth of the respective semigroup. The exception is the annular semigroup of even degree in which case the complexity is the \(\mathcal J\)-depth minus one.