On Singular Artin Monoids

In this paper we study some combinatorial aspects of the singular Artin monoids. Firstly, we show that a singular Artin monoid SA can be presented as a semidirect product of a graph monoid with its associated Artin group A. Such a decomposition implies that a singular Artin monoid embeds in a group. Secondly, we give a solution to the word problem for the FC type singular Artin monoids. Afterwards, we show that FC type singular Artin monoids have the FRZ property. Briefly speaking, this property says that the centralizer in SA of any non-zero power of a standard singular generator τs coincides with the centralizer of any non-zero power of the corresponding non-singular generator σs. Finally, we prove Birman’s conjecture, namely, that the desingularization map η : SA → Z[A] is injective, for right-angled singular Artin monoids. AMS Subject Classification: Primary 20F36.