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.

中文翻译：

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数值半群中的乘法因式分解

数值半群是一个非空加性闭子集 [公式：见文本] 与 [公式：见文本]。数值半群的算术，即加性因式分解性质已经得到了很好的研究。另一方面，它们的乘法性质几乎没有受到关注。如果 [Formula: see text] 或 [Formula: see text]，则 [Formula: see text] 中的乘法因式分解（作为素数的乘积）是唯一的。但是，如果有 [Formula: see text] 和 [Formula: see text]，则 [Formula: see text] 中的乘法因式分解不再是唯一的。本文的目的是介绍这种以前未研究过的数值半群结构。具体来说，我们对不可约元素进行了分类，并描述了非唯一乘法分解如何在数值半群中进行。