Numerical semigroups of coated odd elements

Given a numerical semigroup ring $$R=k\llbracket S\rrbracket $$ , an ideal E of S and an odd element $$b \in S$$ , the numerical duplication $$S \bowtie ^b E$$ is a numerical semigroup, whose associated ring $$k\llbracket S \bowtie ^b E\rrbracket $$ shares many properties with the Nagata’s idealization and the amalgamated duplication of R along the monomial ideal $$I=(t^e \mid e\in E)$$ . In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen–Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when $$\mathrm{gr}_{\mathfrak {m}}(I)$$ is Cohen–Macaulay and when $$\mathrm{gr}_{\mathfrak {m}}(\omega _R)$$ is a canonical module of $$\mathrm{gr}_{\mathfrak {m}}(R)$$ in terms of numerical semigroup’s properties, where $$\omega _R$$ is a canonical module of R.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of $\N$ that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of $\N$. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups.

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by [Formula: see text] and [Formula: see text] satisfy the Wilf inequality, then so does their gluing which is minimally generated by [Formula: see text]. We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

For a numerical semigroup S, a positive integer a and a nonzero element m of S, we define a new numerical semigroup R(S,a,m) and call it the (a,m)-rotation of S. In this paper we study the Frobenius number and the singularity degree of R(S,a,m). Moreover, we observe that the rotations of ℕ are precisely the modular numerical semigroups.

<title>Abstract</title> Length density δ(n) measures how close the set of factorization lengths of n in a numerical semigroup is to a full interval. We establish explicit lower bounds and convergence rates, proving that δ(n) → 1 as n → ∞ for arbitrary embedding dimension. Our approach combines extremal length estimates with pruning algorithms for factorization trees, providing both structural insight and computational efficiency. A general pruning template is introduced, clarifying the relation between ordered and unordered pruning; we show unordered pruning is sound only in commutative settings or under commutation relations. Extensions to non-commutative and non-cancellative semigroups are analyzed via right ideals and prefix-ACCP conditions. Numerical experiments illustrate stabilization of δ(n) and confirm asymptotic error bounds. These results place pruning as a unifying tool for asymptotics, computation, and structural analysis of factorization invariants in semigroups.

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

A numerical set with Frobenius number f is a subset T of that contains zero and Each numerical set has an associated semigroup which has the same Frobenius number as T. For a fixed Frobenius number f there are numerical sets. It is known that there is a number γ close to 0.484 such that the ratio of these numerical sets that are mapped to is asymptotically γ. We identify a collection of families N(D, f) of numerical semigroups such that for a fixed D the ratio of the numerical sets that are mapped to N(D, f) converges to a positive limit as f goes to infinity. We denote the limit as γD , these constants sum up to 1 meaning that they asymptotically account for almost all numerical sets.

ABSTRACTWe define the density of a numerical semigroup and study the densities of all the maximal embedding dimension numerical semigroups with a fixed Frobenius number, as well as the possible Frobenius number for a fixed density. We also prove that for a given possible density, in the sense of Wilf’s conjecture, one can find a maximal embedding dimension numerical semigroup with that density.

We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Apéry set, as well as bounds on the elements of the Apéry set. We also find bounds on the Frobenius number and genus, and the asymptotic behavior of the Frobenius number and genus. Finally, we find the embedding dimension of all such numerical semigroups.

The Frobenius problem for Thabit numerical semigroups

We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius number is F and which have exactly k second kind gaps.

In this paper we introduce a particular semigroup transform [Formula: see text] that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of non-ordinary and non-irreducible numerical semigroups in a family of rooted trees. In addition, we study another transform, having similar features, that has been introduced by Bras-Amorós, and we make a comparison of them. In particular, we study the behavior of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture.

This survey presents recent results on minimal free resolutions of numerical semigroup rings. We focus on two classes of numerical semigroups where the resolution is explicitly given: Gorenstein semigroups of embedding dimension 4 that are not a complete intersection and semigroups generated by a sequence of integers in arithmetic progression. Finally, we describe how the resolution is constructed when the semigroup is obtained by gluing of two numerical semigroups of smaller embedding dimension. Along the paper, we provide several non-trivial examples to illustrate our results.