multipliceres

Multipliceres is a term used in some theoretical discussions to denote a class of combinatorial invariants that encode the multiplicative factorization structure of elements within a multiplicative monoid, such as the positive integers under multiplication. The concept builds on the idea that many questions about factorization can be studied by examining the multiset of irreducible factors and their multiplicities.

Definition and construction: For a multiplicative monoid M and an element x in M, a multiplicere of x is defined as the multiset of irreducible factors of x, counted with their multiplicities relative to a chosen factorization basis. In the integers, where the irreducibles are primes, the multiplicere of n corresponds to the multiset consisting of its prime factors with the appropriate multiplicities. In other monoids with non-unique factorization, the multiplicere captures the pattern of factors across possible factorizations.

Properties: The mapping x -> multiplicere(x) is compatible with multiplication in the sense that the multiplicere of a product reflects the combination of the multipliceres of its factors. In unique factorization domains this reduces to the familiar prime exponent multisets, while in other structures the multiplicere records non-uniqueness of factorization.

Examples: In the integers, multiplicere(18) equals the multiset {2, 3, 3}. If an element has multiple distinct factorizations in a given domain, its multiplicere may reflect different valid multisets that represent the same element.

Relation to broader theory: Multipliceres relate to factorization invariants and to the study of elasticity, catenary degree, and factorization types within factorization theory. They provide a compact descriptor for comparing how elements factor multiplicatively across different algebraic settings.

See also: prime factorization, factorization theory, multiplicative monoid, elasticity, catenary degree.