Discrete mathematics

Commutative property

Factorization

Pure mathematics

Mathematics

Multiplicative function

Algorithm

Arithmetic function

Algebra over a field

Morphism

Syntactic monoid

Mathematical analysis

Free monoid

Semiring

Unique factorization domain

Monoid

Semigroup Forum

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Factorization theory in commutative monoids