On the arithmetic of monoids of ideals

Factorization in Additive Monoids of Evaluation Polynomial Semirings

“…In Section 3, we also investigate the structure of sets of lengths of evaluation polynomial semirings. The structure of the sets of lengths of many classes of atomic monoids has been the subject of a great deal of investigation for almost three decades: see [16,Chapter 4.7] by Geroldinger and Halter-Koch for background on the Structure Theorem for Sets of Lengths, and see the recent paper [19] by Geroldinger and Khadam (and the references therein) for current results in this direction. For any positive rational q, it was proved by Chapman et al [10] that the set of lengths of every nonzero element of N 0 [q] is an arithmetic progression, provided that N 0 [q] is atomic.…”

On algebraic properties of power monoids of numerical monoids

“…For an additive abelian monoid S, say for a numerical monoid, let P fin (S) denote the set of all finite nonempty subsets of S. Together with set addition as operation, P fin (S) is easily seen to be abelian monoid, called the power monoid of S, and {0 S } is its zero-element. While, clearly, being of interest in its own rights, the arithmetic of power monoids is connected with the arithmetic of other monoids, such as the monoid of ideals of polynomial rings ( [26,Proposition 5.13]).…”

“…The conjecture is backed up by a series of results, some of which we gather in the following theorem. For simplicity, we formulate the results for the power monoid of N 0 (see Section 2 for the involved definitions and [17,Theorem 4.11] and [26,Proposition 5.3] for proofs). All results of Theorem 1.2 are simple consequences of Conjecture 1.1, if it holds true.…”