A subsemiring S of $$\mathbb {R}$$
 is called a positive semiring provided that S consists of nonnegative numbers and $$1 \in S$$
 . Here we study factorizations in both the additive monoid $$(S,+)$$
 and the multiplicative monoid $$(S\backslash \{0\}, \cdot )$$
 . In particular, we investigate when, for a positive semiring S, both $$(S,+)$$
 and $$(S\backslash \{0\}, \cdot )$$
 have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP $$\Rightarrow $$
 BFP and FFP $$\Rightarrow $$
 BFP $$\Rightarrow $$
 ACCP $$\Rightarrow $$
 atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.

Bi-atomic classes of positive semirings

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number $\alpha$, the additive monoid $M_\alpha$ of the evaluation semiring $\mathbb{N}_0[\alpha]$ is atomic. The atomic structure of both the additive and the multiplicative monoids of $\mathbb{N}_0[\alpha]$ has been the subject of several recent papers. Here we focus on the monoids $M_\alpha$, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when $\alpha$ is less than 1, the atoms of $M_\alpha$ are as far from being prime as they can possibly be. Then we establish some results about the elasticity of $M_\alpha$, including that when $\alpha$ is rational, the elasticity of $M_\alpha$ is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).

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On the primality and elasticity of algebraic valuations of cyclic free semirings

A subset $S$ of an integral domain $R$ is called a semidomain provided that the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and Zafrullah in 1990, and this area has been systematically investigated since then. In this paper, we study the divisibility and arithmetic of factorizations in the more general context of semidomains. We are specially concerned with the ascent of the most standard divisibility and factorization properties from a semidomain to its semidomain of (Laurent) polynomials. As in the case of integral domains, here we prove that the properties of satisfying the ascending chain condition on principal ideals, having bounded factorizations, and having finite factorizations ascend in the class of semidomains. We also consider the ascent of the property of being atomic and that of having unique factorization (none of them ascends in general). Throughout the paper we provide several examples aiming to shed some light upon the arithmetic of factorizations of semidomains.

On the arithmetic of polynomial semidomains

For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$ be the semiring of all real numbers $f(\alpha)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over the set of nonnegative integers $\mathbb{N}_0$. In this paper, we study various factorization properties of the additive structure of $\mathbb{N}_0[\alpha, \alpha^{-1}]$. We characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ is atomic. Then we characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies the unique factorization property and show that, when this is not the case, $\mathbb{N}_0[\alpha, \alpha^{-1}]$ has infinite elasticity.

Factorizations in evaluation monoids of Laurent semirings

We say that a Puiseux monoid is exponential provided that it is generated by some of the powers of a rational number. Here we study the atomic properties of exponential Puiseux monoids and semiring...

On the atomic structure of exponential Puiseux monoids and semirings

A Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. We say that a Puiseux monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers, which contains $0$, such that $M$ is generated by the set $\{r^s \mid s \in S\}$. If $M$ is multiplicatively closed then we say that $M$ is an exponential Puiseux semiring. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then we proceed to offer a necessary condition and a sufficient condition for an exponential Puiseux monoid to satisfy the ACCP. We conclude by describing the exponential Puiseux monoids that are semirings.

Sofía Albizu-Campos

Papers

On the atomic structure of exponential Puiseux monoids and semirings

On the atomic structure of exponential Puiseux monoids and semirings