In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined as a positive integer that does not belong to the numerical semigroup. The computation of gaps of numerical semigroups has been actively studied from the 19t...

We give a characterization for irreducible numerical semi-groups. From this characterization we obtain that every irre-ducible numerical semigroup is either a symmetric or pseudo-symmetric numerical semigroup. We study the minimal presentations of an irreducible numerical semigroup. Separately, we deal with the cases of maximal embedding dimension and multiplicity 3 and 4.

In this paper we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.

let $fneq1,3$ be a positive integer‎. ‎we prove that there exists a numerical semigroup $s$ with embedding dimension three such that $f$ is the frobenius number of $s$‎. ‎we also show that‎ ‎the same fact holds for affine semigroups in higher dimensional monoids‎.

In this paper, we study almost symmetric numerical semigroups generated by 4-elements. Rosales and Garćıa-Sánchez [RG2] proved that every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with the same Frobenius number. Using this result, we concretely construct almost symmetric numerical semigroups generated by 4...

In this paper we introduce a particular semigroup transform $\mathcal{A}$ that fixes the invariants involved in Wilf's conjecture, except embedding dimension. It also allows one to arrange set of not ordinary and irreducible numerical semigroups family rooted trees. We study another transform, having similar features, has been introduced by Bras-Amor\'os, make comparison them. behaviour dimensi...