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We study the sets of nonnegative solutions of Diophantine inequalities of the form ax mod bpcx with a; b and c positive integers. These sets are numerical semigroups, which we study and characterize. r

A one-to-one correspondence is described between the set S(m) of numerical semigroups with multiplicity m and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in S(m) a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of N m−1 . Finally, this result is particularized to the symmetric case.

Let {a1, . . . , ap} be the minimal generating set of a numerical monoid S. For any s ∈ S, its Delta set is defined by ∆xiai and xi ∈ N for all i}. The Delta set of a numerical monoid S, denoted by ∆(S), is the union of all the sets ∆(s) with s ∈ S. As proved in [5], there exists a bound N such that ∆(S) is the union of the sets ∆(s) with s ∈ S and s < N . In this work, we obtain a sharpened bound and we present an algorithm for the computation of ∆(S) that requires only the factorizations of a1 elements.

We introduce and study the concept of Arf system of generators for an Arf numerical semigroup. This study allows us to arrange the set of all Arf numerical semigroups in a binary tree and enables us to compute the Arf closure of a given numerical semigroup.