The paper explores the concept of the length-finite factorization property in the context of positive monoids and positive semirings. The authors aim to provide a deeper understanding of the non-unique factorization phenomenon in these structures. They identify a large class of positive monoids that satisfy the length-finite factorization property and compare it with the bounded and finite factorization properties. The paper also discusses the preservation of these properties under certain constructions and poses open questions for further research.
The paper defines atomic monoids and discusses their importance in factorization theory. An atomic monoid is one where every non-invertible element can be written as a sum of irreducible elements (atoms).
Length-finite factorization property states that each element in a monoid has only finitely many factorizations of any given length. The authors introduce this property as a complement to the bounded and finite factorization properties.
The paper focuses on positive monoids (additive submonoids of nonnegative real numbers) and positive semirings (positive monoids closed under multiplication). These structures have been actively studied for their atomic properties and factorizations.
The authors compare the length-finite factorization property with the bounded and finite factorization properties, highlighting their relationships and differences.
The paper provides examples of positive monoids and semirings that satisfy or do not satisfy the length-finite factorization property. It also discusses methods to construct positive semirings from positive monoids and examines the preservation of factorization properties under these constructions.
The authors prove that:

1)

The sum of finitely many co-well-ordered sequences is a co-well-ordered sequence;

2)

If \(M\) be an atomic co-well-ordered positive monoid, then for all \(x \in M\) and \(l \in \mathbb{N}\), every irredundant subset of \(\mathrm{Z}_{l}(x)=\{z \in \mathrm{Z}(x)=\{z \in \mathrm{Z}(M)~|~\pi(z) = x + {\mathcal{U}} (M)\} ~|~ |z| = l\}\) is finite; where \(\mathrm{Z}(M)\) is the free commutative monoid on \({\mathcal{A}}(M/{\mathcal{U}}(M))\), \({\mathcal{U}}(M)\) denote the group of invertible elements of \(M\) and the set \(M/{\mathcal{U}}(M) = \{b + {\mathcal{U}}(M) ~|~ b \in M\}\) is a monoid under the natural operation induced by the operation of \(M\); \({\mathcal{A}}(M)\) denote the set of all the atoms of \(M\) and function \(\pi : \mathrm{Z}(M) \rightarrow M/{\mathcal{U}}(M)\) be the only monoid homomorphism such that \(\pi(a) = a\) for all \(a \in {\mathcal{A}}(M/{\mathcal{U}}(M))\);

3)

Every atomic co-well-ordered positive monoid is an length-finite factorization monoid (\(LFFM\)).

In addition, the authors pose two open questions:

1)

For a positive monoid M consisting of algebraic numbers, is \(E(M)=\left<e^m~|~m\in M\right>\) an length-finite factorization semiring (\(LFFS\)) provided that \(M\) is an \(LFFM\)?

2)

For which \(q \in \mathbb{Q}_{>0}\) is the positive semiring \(\mathbb{N}_{0}[q]\) an \(LFFS\)?

he paper makes a significant contribution to the field of factorization theory by introducing and exploring the length-finite factorization property. This property provides a new perspective on the non-unique factorization phenomenon in positive monoids and semirings. The comparison with other well-studied properties, such as the bounded and finite factorization properties, helps to contextualize the importance of the length-finite factorization property. The examples and constructions provided in the paper offer valuable insights and open up new avenues for further research.
Overall, the paper is well-structured and provides a thorough analysis of the length-finite factorization property, making it a valuable resource for researchers in the field of algebra and factorization theory.