Multiplicative factorization in numerical semigroups

Abstract

A numerical semigroup is a nonempty additively closed subset [Formula: see text] with [Formula: see text]. The arithmetic, that is the additive factorization properties, of numerical semigroups has been well studied. Their multiplicative properties, on the other hand, have received little, if any, attention. If [Formula: see text] or [Formula: see text], then multiplicative factorization (as products of primes) in [Formula: see text] is unique. However, if there is [Formula: see text] with [Formula: see text], then multiplicative factorization in [Formula: see text] is no longer unique. The purpose of this paper is to introduce this previously unstudied structure of numerical semigroups. Specifically, we classify the irreducible elements and provide a description of how non-unique multiplicative factorization can be in numerical semigroups. In addition, we show that multiplicative numerical semigroups belong to the class of [Formula: see text]-monoids, but yet are not Krull.