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Abstract LetMbe a cancellative and commutative (additive) monoid. The monoidMis atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also,Msatisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids. 

Abstract A subsetSof an integral domainRis 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. 

A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although factorizations and divisibility in atomic domains have been systematically investigated for more than 30 years, the same aspects in the more general context of atomic semidomains have been considered just recently. Here we study subatomicity in the context of semidomains; that is, we study semidomains satisfying divisibility properties weaker than atomicity. We mostly focus on the Furstenberg property, which is due to P. Clark and motivated by the work of H. Furstenberg on the infinitude of primes, and the almost atomic and quasi-atomic properties, introduced by J. G. Boynton and J. Coykendall in the context of divisibility in integral domains. We investigate these three properties in the context of semidomains, paying special attention to whether they ascend from a semidomain to its polynomial and Laurent polynomial extensions.