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The Krohn–Rhodes complexity of the Brauer type semigroups and is computed. In three-quarters of the cases the result is the ‘expected’ one: the complexity coincides with the (essential) -depth of the respective semigroup. The exception (and perhaps the most interesting case) is the annular semigroup of even degree in which case the complexity is the -depth minus . For the ‘rook’ versions and it is shown that and for all . The computation of is left as an open problem.