Correction to: Numerical semigroups generated by primes

Let S be a numerical semigroup. An element is a special gap of S if is also a numerical semigroup. If a is a positive integer, we denote by the set of all numerical semigroups for which a is a special gap. We say that an element of is -irreducible if it cannot be expressed as the intersection of two numerical semigroups of properly containing it. The main aim of this paper is to describe three algorithmic procedures: the first one calculates the elements of the second one determines whether or not a numerical semigroup is -irreducible and the third one computes all the -irreducibles numerical semigroups.

We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroups we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.

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].

To a numerical semigroup S, Eliahou associated a number $${{\mathrm{E}}}(S)$$ and proved that numerical semigroups for which the associated number is non negative satisfy Wilf’s conjecture. The search for counterexamples for the conjecture of Wilf is therefore reduced to semigroups which have an associated negative Eliahou number. Eliahou mentioned 5 numerical semigroups whose Eliahou number is $$-1$$ . The examples were discovered by Fromentin who observed that these are the only ones with negative Eliahou number among the over $$10^{13}$$ numerical semigroups of genus up to 60. We prove here that for any integer n there are infinitely many numerical semigroups S such that $${{\mathrm{E}}}(S)=n$$ , by explicitly giving families of such semigroups. We prove that all the semigroups in these families satisfy Wilf’s conjecture, thus providing not previously known examples of semigroups for which the conjecture holds.

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.

A numerical semigroup is a cofinite subset of ℤ ≥0 containing 0 and closed under addition. Each numerical semigroup S with smallest positive element m corresponds to an integer point in the Kunz cone 𝒞 m ⊆ℝ m-1 , and the face of 𝒞 m containing that integer point determines certain algebraic properties of S. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of 𝒞 m . We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of 𝒞 m , and our results culminate in a method of “walking” the face lattice of the Kunz cone in a manner analogous to that of a Gröbner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned “walks” and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.

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

Recently, the concept of the [Formula: see text]-numerical semigroup with [Formula: see text]-symmetric properties has been introduced. When [Formula: see text], the classical numerical semigroup with symmetric properties is recovered. In this paper, we further study the [Formula: see text]-numerical semigroup with [Formula: see text]-almost symmetric properties. We also give [Formula: see text]-generalized formulas of Watanabe and Johnson, and introduce [Formula: see text]-Arf numerical semigroup and study its properties.

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.

Let S and T be two numerical semigroups. We say that T is an I(S)-semigroup if T∖{0} is an ideal of S. Given k a positive integer, we denote by Δ(k) the symmetric numerical semigroup generated by {2,2k+1}. In this paper we present a formula which calculates the number of I(S)-semigroups with genus g(Δ(k))+h for some nonnegative integer h and which we will denote by i(Δ(k),h). As a consequence, we obtain that the sequence {i(Δ(k),h)}h∈N is never decreasing. Besides, it becomes stationary from a certain term.

<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.

Factorizations in numerical semigroup algebras

A numerical semigroup is said to be perfect if it does not contain any isolated gaps. In this paper we show that if S is any numerical semigroup, then for each integer n≥2 there exists a perfect numerical semigroup T such that S=Tn (that is, S={x∈ℕ:nx∈T})

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.