In this paper, we introduce a new depicting of the so-called numerical semigroup tree T . By exploring computationally this improved picture, relying on the type notion of a semigroup, we found that the number of semigroups of genus g and type t is constant when t is close to g while g grows. We also study the unimodality of various sequences as well as the behavior of the leaves in T . We put forward several conjectures that are supported by various computational experiments.

Let A be a nonempty subset of positive integers. In this paper we study the set of numerical semigroups that fulfill: if {x, y} ⊆ N\S and x > y > min(S\{0}), then x − y ̸ ∈ A.

We consider numerical semigroups S3=⟨d1,d2,d3⟩, which are minimally generated by three positive integers. We revisit the Wilf question for S3 and, making use of identities for degrees of syzygies of such semigroups, we provide a short proof of existence of an affirmative answer. Finally, we find the upper and lower bounds for the rescaled genera of numerical semigroups S3.

Counting edges in factorization graphs of numerical semigroup elements

The number of primes not in a numerical semigroup

A numerical semigroup S is coated with odd elements (Coe-semigroup), if {x − 1, x + 1} ⊆ S for all odd elements x in S. In this note, we will study this kind of numerical semigroup. In particular, we are interested in the study of the Frobenius number, gender and embedding dimension of a numerical semigroup of this type.

This paper investigates the interplay between numerical semigroups and enumerative combinatorics through the lens of Young diagrams, rook polynomials, and the Lah numbers. For a given numerical semigroup S, we associate a Young diagram constructed from the gap set of S. We then compute the rook polynomial corresponding to this diagram and analyze its coefficients. It is observed that these coefficients exhibit a strong connection with the Lah numbers, which count ordered partitions. Our approach introduces a new combinatorial interpretation of numerical semigroup gaps and reveals novel structural links between algebraic and enumerative concepts.

Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups of genus $g$, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\]for some constant $C_d > 0$, which can be made explicit. To do this, we extend the notion of multiplicity and depth to generalized numerical semigroups and show our lower bound is sharp for semigroups of depth 2. We also show other bounds on special classes of semigroups by introducing partition labelings, which extend the notion of Kunz words to the general setting.

Let a and b be positive integers such that a<b and [a,b]={x∈N∣a≤x≤b}. In this work, we will show that A([a,b])={S∣S is a numerical semigroup whose Frobenius number belongs to [a,b]} and is a covariety. This fact allows us to present an algorithm which computes all the elements from A([a,b]). We will prove that A([a,b],m)={S∈A([a,b])∣S has multiplicity m} and is a ratio-covariety. As a consequence, we will show an algorithm which calculates all the elements belonging to A([a,b],m). Based on the above results, we will develop an interesting algorithm that calculates all numerical semigroups with a given multiplicity and complexity.

We characterize additive semigroups in { 0 , 1 , 2 , … } of matricial dimension 2 and produce a counterexample to the conjecture that a numerical semigroup whose small elements are lonely has matricial dimension at most 2.

If S is a numerical semigroup, we denote by n ( S ) the cardinality of N ( S ) = { s ∈ S | s < F ( S ) } , F ( S ) = max ( Z / S ) and by g ( S ) the cardinality of N / S . Let q = a b with 1 ≤ b ≤ a , gcd { a , b } = 1 and { k , F } ⊆ N / { 0 } . In this paper we introduce the sets B ( q ) = { S | S is a numerical semigroup and g ( S ) n ( S ) = q } and A ( k ) = { S | S is a numerical semigroup and g ( S ) ≤ k n ( S ) } . The Wilf’s conjecture will be reformulated by using these sets. Also we show two algorithms which compute the elements of the sets A ( k , F ) = { S ∈ A ( k ) | F ( S ) = F } and B ( q , k ) = { S | S is a numerical semigroup, g ( S ) = ak and n ( S ) = bk } .

On quotients of numerical semigroups for almost arithmetic progressions

In this paper we will show that MED$(F,m)=\{S\mid S \mbox{ is a numerical semigroup with maximal embedding dimension, Frobenius number} ~F~ \mbox{and multiplicity}~ m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes MED$(F,m)$ and another one that calculates the elements of MED$(F,m)$ with a given genus. If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in $ MED$(F,m)$, then there exists the smallest element of MED$(F,m)$ containing $X$. This element will be denoted by MED$(F,m)[X]$ and we will say that $X$ one of its MED$(F,m)$-system of generators. We will prove that every element $S$ of MED$(F,m)$ has a unique minimal MED$(F,m)$-system of generators and it will be denoted by MED$(F,m)$msg$(S).$ The cardinality of MED$(F,m)$msg$(S)$, will be called MED$(F,m)$-rank of $S.$ We will also see in this work, how all the elements of MED$(F,m)$ with a fixed MED$(F,m)$-rank are.

The Covariety Of Bracelet Numerical Semigroups With Fixed Frobenius Number

Abstract Maximally embedding dimension (MED) numerical semigroups are a wide and interesting family, with some remarkable algebraic and combinatorial properties. It is well-known that one can construct an MED closure associated to any numerical semigroup. This paper shows two different explicit methods to construct this closure which also shed new light on the very nature of this object.

In this article, we introduce the graphs associated with gap posets of numerical semigroups. We prove that gap poset graphs are either bipartite graphs or graphs with girth 3. Also, we characterize gap poset graphs in terms of connectedness, girth and diameter.

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.

In this article, we classify all symmetric generalized numerical semigroups in N d of embedding dimension 2 d + 1 . Consequently, we show that in the case d > 1 , the property of being symmetric is equivalent to have a unique maximal gap with respect to natural partial order in N d . Moreover, we deduce that when d > 1 , there does not exist any generalized numerical semigroup of embedding dimension 2 d + 1 , which is almost symmetric but not symmetric.

If P is a nonempty finite subset of positive integers, then A(P)={S∣S is a numerical semigroup, S∩P=∅ and max(P) is the Frobenius number of S}. In this work, we prove that A(P) is a covariety; therefore, we can arrange the elements of A(P) in the form of a tree. This fact allows us to present several algorithms, including one that calculates all the elements of A(P), another that obtains its maximal elements (with respect to the set inclusion order) and one more that computes the elements of A(P) that cannot be expressed as an intersection of two elements of A(P), that properly contain it.

Let H = 〈 n 1 , n 2 , … , n e 〉 be a numerical semigroup minimally generated by e elements. We give a new upper bound for the Frobenius number F ( H ) of H with respect to a pair of relatively prime generators ( n i , n j ) of H. Then we will determine the range of F ( H ) when H is stretched.