Let \(\mathbb{N}_0\) denote the set of nonnegative integers, and \(\mathbb{N}\) the set of positive integers. Then \((\mathbb{N}_0, +)\) is a monoid, and so is \((\mathbb{N},\times)\). These two monoids have a quite different structure. The first is generated by \(\{1\}\), while the second is generated by the set of all prime positive integers.
A numerical semigroup is a subset \(S\) of \(\mathbb{N}_0\) closed under addition and with finite complement in \(\mathbb{N}_0\). The set \(S_1=(S\setminus\{0\})\cup\{1\}\) becomes a submonoid of \(\mathbb{N}\) under multiplication, and its structure and factorization properties differ from those of \(S\).
The authors start by giving a description of the atoms (irreducibles, elements that cannot be expressed as a product of two non-unit elements) of \(S_1\), and determine which are strong atoms (an atom \(a\) is strong if the only atom that divides \(a^n\) for every positive integer \(n\) is \(a\)). Let \(\mathbb{P}\) be the set of positive integer primes. Then, the atoms of \(S_1\) are either elements of \(S\cap\mathbb{P}\), or elements of the form \(x p\), with \(x\in \mathbb{N}\setminus S_1\) and \(p\in \mathbb{P}\cap S\), or of the form \(a=q_1^{m_1}\dots q_t^{m_t}\), with \(a\in S\) and \(\{q_1,\ldots, q_t\}=\mathbb{P}\setminus S\). No atom of \(S_1\) (if \(S_1\) is not \(\mathbb{N}\)) is a prime element, and only the atoms in \(\mathbb{P}\cap S\) are strong atoms.
Once the atoms of \(S_1\) are clearly described and determined, the authors study some factorization invariants and the structure of \(S_1\). It turns out that \(S_1\) is half-factorial (for any element, all its factorizations have the same number of atoms) if and only if \(S_1\) is factorial (factorizations are unique up to arrangement of factors), and this happens precisely when \(S\) is either the whole set of nonnegative integers or \(\mathbb{N}_0\setminus\{1\}=\langle 2,3\rangle\). Moreover, it is shown that in any factorization of an element in \(S_1\), the combined number of atoms of the form \(p\) or \(x p\) is constant. Sets of lengths for several kinds of elements in \(S_1\) are explicitly described, showing that the system of sets of lengths of \(S_1\) contains arbitrarily large intervals. Later, lower and upper bounds for the elasticity of \(S_1\) are given.
The last section is devoted to the algebraic structure of \(S_1\), which is never completely integrally closed (and thus it cannot be a Krull monoid) unless \(S_1\) is the set of positive integers. The authors show that \(S_1\) is always a simple \(C\)-monoid, and they give an explicit formula for its exponent.
The paper contains many good and illustrative examples, and several open questions. Among these, the first one derived in a sequence of interesting discussions and computational experiments with the first author.