Factorization in Finitely-Presented Monoids

Abstract

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.

24 pages. The second version contains improvements to the exposition and additional references

References in corpus (18)

• Factorization theory: From commutative to noncommutative settings
• Sets of lengths in maximal orders in central simple algebras
• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Factorizations in bounded hereditary Noetherian prime rings
• Structural properties of subadditive families with applications to factorization theory
• Factorizations of Elements in Noncommutative Rings: A Survey
• Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules
• Factorization invariants in half-factorial affine semigroups
• On elasticities of locally finitely generated monoids
• Factorization under Local Finiteness Conditions
• Abstract Factorization Theorems with Applications to Idempotent Factorizations
• On noncommutative bounded factorization domains and prime rings
• A characterization of atomicity
• On the finiteness of certain factorization invariants
• Unique factorization property of non-unique factorization domains II
• On transfer homomorphisms in commutative rings with zero-divisors
• On the arithmetic of power monoids
• Semigroups of ideals and isomorphism problems