Rewriting Systems and Embedding of Monoids in Groups

In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system ℜ that satisfies the condition that each rule in ℜ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from ℜ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris that Artin monoids embed in their groups.