Factorization in Finitely-Presented Monoids

We study arithmetic properties of factorizations of elements into products of generators, in monoids given with explicit presentations. After relating and comparing this perspective to the more usual approach of factoring into products of atoms, as well as other more recent alternatives, we explore how the relations in the presentation of a monoid affect factorization. In the process, we construct a large class of non-commutative fully elastic monoids. We also show that any finitely-presented cancellative normalizing monoid satisfies the Structure Theorem for Unions. Examples are constructed to demonstrate the sharpness of our results, and exhibit unusual factorization behavior.