The set of solutions of a proportionally modular Diophantine inequality

Abstract

The set of solutions of the inequality ax mod b cx is a numerical semigroup. We present in this paper a tool for finding the set of minimal generators of this set, and thus the set of solutions to such an inequality. This tool will also enable us to give characterizations of those numerical semigroups that are the set of integer solutions of inequalities of this form. Finally, we give a deeper study of the embedding dimension three case.

Symmetries on almost symmetric numerical semigroups

“…Theorem 4.4. [RGU2] Let H be a numerical semigroup with e(H) = n. Then H is proportionally modular if and only if for some rearrangement of its generators {a 1 , a 2 , ..., a n } the following conditions hold:…”

Ideals of proportionally modular numerical semigroups

“…, p} such that a 1 a 2 · · · a h a h+1 · · · a p . The following result follows from Theorem 31 and Corollary 18 of [10].…”

“…Another result that will be used several times along this paper and that appears in [10] as Remark 24 is the following.…”

“…The aim of this section is to introduce some results related to proportionally modular numerical semigroups that we can find in [9] and [10], and that will be useful along this paper.…”

“…(2) If we analyze the proof of Theorem 31 of [10], we observe that if S is a proportionally modular numerical semigroup in the same conditions of Lemma 3, then…”