Numerical Semigroups and Applications

Assi, Abdallah; García-Sánchez, Pedro A.

doi:10.1007/978-3-030-54943-5

Cited by 52 publications

(149 citation statements)

References 22 publications

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“…2.3. Minimal presentations and Betti elements [1,17]. Let S be a numerical semigroup minimally generated by {n 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

“…The minimal possible choice is to have a spanning tree connecting them all, once the x i have been chosen. Different choices of x i in C i and different spanning trees will yield different minimal presentations, but they all have the same cardinality (see, for instance, [1,Chapter 4]). As a consequence, all minimal presentations have cardinality equal to s∈Betti(S) (nc(∇ s ) − 1).…”

Section: Preliminariesmentioning

confidence: 99%

“…It is not difficult to conclude that the coefficients of P S are in {−1, 0, 1} and that consecutive nonzero coefficients alternate in sign. On noting the formal identity H S (x) = (1 − x) −1 − s∈G(S) x s , we have (1) P S (x) = 1 + (x − 1)…”

Section: Introductionmentioning

confidence: 99%

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Cyclotomic exponent sequences of numerical semigroups

Ciolan¹,

García-Sánchez²,

Herrera-Poyatos³

et al. 2021

Preprint

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We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of S with unique representations in terms of minimal generators, and the Betti elements b ∈ S for which the set {a ∈ Betti(S) : a ≤S b} is totally ordered with respect to ≤S (we write a ≤S b whenever a − b ∈ S, with a, b ∈ S). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection.

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“…2.3. Minimal presentations and Betti elements [1,17]. Let S be a numerical semigroup minimally generated by {n 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Cyclotomic exponent sequences of numerical semigroups

Ciolan¹,

García-Sánchez²,

Herrera-Poyatos³

et al. 2021

Preprint

Self Cite

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“…We illustrate these ideas with (arguably) the most well-studied step set: Consider the two words u = EEN EN and v = N N EEE from F X . Although u = v, we note that α X (u) = α X (v) = 3E + 2N = (3,2). We may picture the walks from O to (3,2) determined by u and v as in Figure 1.…”

Section: Definitions and Basic Examplesmentioning

confidence: 99%

Lattice paths and submonoids of $\mathbb Z^2$

East¹,

Ham²

2018

Preprint

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We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set X ⊆ Z 2 , there are two naturally associated monoids: F X , the monoid of all X-walks/paths; and A X , the monoid of all endpoints of X-walks starting from the origin O. For each A ∈ A X , write π X (A) for the number of X-walks from O to A. Calculating the numbers π X (A) is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schröder numbers, among many other famous sequences and arrays. Our main results give the precise relationships between finiteness properties of the numbers π X (A), geometrical properties of the step set X, algebraic properties of the monoid A X , and combinatorial properties of a certain bi-labelled digraph naturally associated to X. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane, and present a number of algorithms for computing the combinatorial data associated to finite step sets. Several examples are considered throughout to highlight the sometimessubtle nature of the theoretical results.

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“…A numerical semigroup S is a subset of the natural numbers that contains 0, is closed under addition and has a finite complement N \ S. The numbers in N \ S are called gaps and the largest gap is called the Frobenius number F (S). For general references on numerical ssemigroups see [7], [8].…”

Section: Introductionmentioning

confidence: 99%

Frobenius allowable gaps of Generalized Numerical Semigroups

Singhal¹,

Lin²

2021

Preprint

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A generalised numerical semigroup (GNS) is a submonoid S of N d for which the complement N d \ S is finite. The points in the compliment N d \ S are called gaps. A gap F is considered Frobenius allowable if there is some relaxed monomial ordering on N d with respect to which F is the largest gap. We characterise the Frobenius allowable gaps of a GNS. We extend the notions of symmetric and irreducible numerical semigroups into GNS and show that they have analogous properties. A GNS that has only one Frobenius allowable gap is called a Frobenius GNS. We estimate the number of Frobenius GNS with a given Frobenius gap F ∈ N d and show that it is close to √ 3 F for large d. Here F = (F (1) , . . . , F (d) ) = (F (1) + 1) • • • (F (d) + 1).

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Numerical Semigroups and Applications

Cited by 52 publications

References 22 publications

Cyclotomic exponent sequences of numerical semigroups

Cyclotomic exponent sequences of numerical semigroups

Lattice paths and submonoids of $\mathbb Z^2$

Frobenius allowable gaps of Generalized Numerical Semigroups

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