Square-free divisor complexes of certain numerical semigroup elements

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 a submonoid of $${\mathbb {N}}$$ with finite complement in $${\mathbb {N}}$$ . A generalized numerical semigroup is a submonoid of $${\mathbb {N}}^{d}$$ with finite complement in $${\mathbb {N}}^{d}$$ . In the context of numerical semigroups, Wilf’s conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf’s conjecture, to the class of $${\mathcal {C}}$$ -semigroups, was proposed by Garcia-Garcia, Marin-Aragon, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf’s conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by Garcia-Garcia, Marin-Aragon, and Vigneron-Tenorio.

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to frac{3f}{4}. We denote the number of numerical semigroups of Frobenius number f by N(f). While N(f) is not monotonic we prove that N(f)<N(f+2) for every f.

The computation of the clique number of a graph is a fundamental problem in graph theory, which has many applications in computational chemistry, bioinformatics, computer, and social networking. A subset S of non-negative integers N 0 is called a numerical semigroup if it is a submonoid of N 0 and has a finite complement in N 0 . The graph associated with numerical semigroup S is denoted by G [ S ] and is defined by the vertex set { x : x ∈ g ( S ) } and the edge set { xy ⇔ x + y ∈ S } . In this article, we compute the clique number and the minimum degree of those graphs, which can be associated with symmetric numerical semigroups of embedding dimension 2. Moreover, on this basis, we give some bounds for the atom-bond sum-connective index of graphs G [ S ] in terms of the harmonic index, the first Zagreb index, the sum-connectivity index, the maximum degree, the minimum degree, the chromatic number, and the clique number.

A subset Δ of non-negative integers N 0 is called a numerical semigroup if it is a submonoid of N 0 and has a finite complement in N 0 . A graph G Δ is called a Δ ( α , β ) -graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that V ( G Δ ) = { v i : i ∈ N 0 ∖ Δ } and E ( G Δ ) = { v i v j ⇔ i + j ∈ Δ } . In this article, we compute the Δ ( α , β ) -graphs for irreducible and Arf numerical semigroups having a metric dimension of 2. It is proved that if Δ be an irreducible and arf numerical semigroup then there are exactly 2 and 8 non-isomorphic Δ ( α , β ) -graphs respectively, whose metric dimension is 2.

A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to g? We outline Zhai's proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.

A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N. An important example of numerical semigroup is given by the Weierstrass semigroup at one point of a curve. In the theory of algebraic geometry codes, Weierstrass semigroups are crucial for defining bounds on the minimum distance as well as for defining improvements on the dimension of codes. We present these applications and some theoretical problems related to classification, characterization and counting of numerical semigroups.

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.

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

A numerical semigroup S is an additive subsemigroup of the nonnegative integers, containing zero, with finite complement. Its multiplicity m is its smallest nonzero element. The Apéry set of S is the set of elements . Fixing a numerical semigroup, we ask how many elements of its Apéry set have nonunique factorization and define several new invariants.

A generalized numerical semigroup is a submonoid of ℕ d with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups defined starting from numerical semigroups. In particular, we prove that the class of the so called T -stripe generalized numerical semigroups satisfies a generalization of Wilf’s conjecture. Some partial results for the generalized Wilf’s conjecture, together with characterizations for the properties of quasi-irreducibility and quasi-symmetry, are obtained for the so called T - graded semigroups and for the generalized numerical semigroups S ⊆ℕ d such that elements of ℕ d \ S belong to the coordinate axes.

Given a ring R and a semigroup S the semigroup ring R[S] inherits the properties of S and R. If no restrictions are posed on the semigroup S and on the ring R the class of semigroup rings is very large. There is a wide literature on this subject. Gilmer's book [15] is the classical reference in the commutative case, i.e. when both the ring R and the semigroup S are commutative. The book contains a deep study of the conditions under which the semigroup ring R[S] has given ring theoretic properties. The following short survey considers a very particular class of semigroup rings: the ring of coefficient R is supposed to be a field and, for the main part of the results, the semigroup S is supposed to be a numerical semigroup, i.e. an additive submonoid of N, with finite complement in N. Also within this particular class of semigroup algebras over a field, the paper deals only with some themes and it is far from being complete. Results and proofs are ranged through two different sources. On one side there is the classical ring theory, a rich, historically settled and well known theory. On the other side there are more elementary but simetimes quicker arguments which come from studying a less rich structure as that of semigroups. In doing so I hope to be not too far from R. Gilmer's open research attitude and from his tolerant view. Here, as in many other fields of human life, there is not a "Unique Thought", but several different points of view and techniques may coexist and be reciprocally useful.

A numerical set T is a subset of N0 that contains 0 and has finite complement. The atom monoid of T is the set of x∈N0 such that x+T⊆T. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup S and show that numerical sets with atom monoid S are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when S has small type.

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of [Formula: see text] with finite complement in [Formula: see text]. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in [Formula: see text] we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in [Formula: see text] we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). Its implementation allowed us to compute (for various dimensions) the number of numerical semigroups of higher genus than has previously been computed.

Let $S$ be a numerical semigroup, and $\operatorname{msg}(S)$ its minimal system of generators. Then $\mathrm{m}(S)=\min(\operatorname{msg}(S))$, $\mathrm{M}(S)=\max(\operatorname{msg}(S))$, $\mathrm{e}(S)$, which is the cardinality of $\operatorname{msg}(S)$, and $\mathscr{S}(S)=\frac{\mathrm{M}(S)}{\mathrm{m}(S)}$, are called the multiplicity, comultiplicity, embedding dimension, and elasticity of $S$, respectively. Let $m$ and $M$ be positive integers, and let $q$ be a rational number greater than $1$. In this paper, we will study the following sets: \begin{itemize} \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{m}(S)=m \mbox{ and } \mathrm{M}(S)=M\}$, \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{m}(S)=m \mbox{ and } \mathscr{S}(S)\leq q\}$, and \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{e}(S)=3 \mbox{ and } \mathscr{S}(S)=q\}$. \end{itemize}