  In this paper we introduce and study some geometric objects associated to
Artin monoids. The Deligne complex for an Artin group is a cube complex that
was introduced by the second author and Davis (1995) to study the K(\pi,1)
conjecture for these groups. Using a notion of Artin monoid cosets, we
construct a version of the Deligne complex for Artin monoids. We show that for
any Artin monoid this cube complex is contractible. Furthermore, we study the
embedding of the monoid Deligne complex into the Deligne complex for the
corresponding Artin group. We show that for any Artin group this is a locally
isometric embedding. In the case of FC-type Artin groups this result can be
strengthened to a globally isometric embedding, and it follows that the monoid
Deligne complex is CAT(0) and its image in the Deligne complex is convex. We
also consider the Cayley graph of an Artin group, and investigate properties of
the subgraph spanned by elements of the Artin monoid. Our final results show
that for a finite type Artin group, the monoid Cayley graph embeds
isometrically, but not quasi-convexly, into the group Cayley graph.
