  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 $\tau_s$ coincides with the
centralizer of any non-zero power of the corresponding non-singular generator
$\sigma_s$. Finally, we prove Birman's conjecture, namely, that the
desingularization map $\eta: SA \to \Z [A]$ is injective, for right-angled
singular Artin monoids.

Group Theory