Abstract A natural operation on numerical semigroups is taking a quotient by positive integer. If $\mathcal {S}$ of semigroup with k generators, we call -quotient. We give necessary condition for given to be -quotient and present, each $k \ge 3$ , the first known family that cannot written as also examine probability randomly selected generators

Abstract Let $$A \subset \mathbb Z$$ A ⊂ Z be a finite subset. We denote by $${{\,\mathrm{\mathcal {B}}\,}}(A)$$ B ( ) the set of all integers $$n \ge 2$$ n ≥</mm...

In this paper we study and characterize those Diophantine inequalities axmod b ≤ x whose set of solutions is a symmetric numerical semigroup. Given two integers a and b with b = 0 we write a mod b to denote the remainder of the division of a by b. Following the notation used in [8], a modular Diophantine inequality is an expression of the form ax mod b ≤ x. The set S(a, b) of integer solutions ...

Given a numerical semigroup S, let M(S) = S \{0} and (lM(S)− lM(S)) = {x ∈ N0 : x + lM(S) ⊆ lM(S)}. Define associated numerical semigroups B(S) := (M(S)−M(S)) and L(S) := ∪l=1(lM(S)− lM(S)). Set B0(S) = S, and for i ≥ 1, define Bi(S) := B(Bi−1(S)). Similarly, set L0(S) = S, and for i ≥ 1, define Li(S) := L(Li−1(S)). These constructions define two finite ascending chains of numerical semigroups ...

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the r Feng-Rao number is obtained.

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabilizing ascending chain whose first step consists...

We recall L-shapes, which are minimal distance diagrams, related to weighted 2-Cayley digraphs, and we give the number and the relation between minimal distance diagrams related to the same digraph. On the other hand, we consider some classes of numerical semigroups useful in the study of curve singularity. Then, we associate L-shapes to each numerical 3-semigroup and we describe some main inva...

We introduce a new sequence τ associated to a numerical semigroup similar to the ν sequence used to define the order bound on the minimum distance and to describe the Feng–Rao improved codes. The new sequence allows a nice description of the optimal one-point codes correcting generic errors and to compare them with standard codes and with the Feng–Rao improved codes. The relation between the τ ...