Twisted Brauer monoids

We investigate the structure of the twisted Brauer monoid B-n(T), comparing and contrasting it with the structure of the (untwisted) Brauer monoid B-n. We characterize Green's relations and pre-orders on B-n(T), describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of B-n(T) (which is not an ideal), as well as the singular ideal of B-n(T) (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid B-n(T) is idempotent generated, a result previously proved by Maltcev and Mazorchuk.