Pseudo-symmetric Numerical Semigroups Research Articles

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

In this article we solve the following problem: “The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing.” We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation.

On the Planarity of Graphs Associated with Symmetric and Pseudo Symmetric Numerical Semigroups

Let S(m,e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph GS an S(m,e)-graph if there exists a numerical semigroup S∈S(m,e) with V(GS)={x:x∈g(S)} and E(GS)={xy⇔x+y∈S}, where g(S) denotes the gap set of S. The aim of this article is to discuss the planarity of S(m,e)-graphs for some cases where S is an irreducible numerical semigroup.

Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

In this article, we study minimal graded free resolutions of Cohen-Macaulay tangent cones of some monomial curves associated to 4-generated pseudo symmetric numerical semigroups. We explicitly give the matrices in these minimal free resolutions.

The examination of the quotient of numerical semigroup with RF-matrices

In this paper, we study quotients of a numerical semigroups with RF (Row-Factorization) matrices. We prove a formula for the Frobenious number of quotients of some families of numerical semigroups. Moreover, we examine half of the numerical semigroups, pseudo-symmetric numerical semigroups.

Gömme Boyutu Üç Olan Bazı Pseudo-Simetrik Sayısal Yarıgruplarının Delta Kümeleri Üzerine

Let S be a numerical semigroup. The catenary degree of an element s in S is a non-negative integer used to measure the distance between factorizations of s. The catenary degree of the numerical semigroup S is obtained at the maximum catenary degree of its elements. The maximum catenary degree of S is attained via Betti elements of S with complex properties. The Betti elements of S can be obtained from all minimal presentations of S. A presentation for S is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of S is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in S.
 In this study, we will mainly express the given above invariants of a special pseudo-symmetric numerical semigroup family in terms of its generators.

SYMMETRIC AND PSEUDO-SYMMETRIC NUMERICAL SEMIGROUPS VIA YOUNG DIAGRAMS AND THEIR SEMIGROUP RINGS

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into an over semigroup and its dual. It is also given exactly for what kind of numerical semigroup $S$, the semigroup ring $\Bbbk[\![S]\!]$ has at least one Gorenstein subring and has at least one Kunz subring.

Gluing semigroups and strongly indispensable free resolutions

We study strong indispensability of minimal free resolutions of semigroup rings focusing on the operation of gluing used in the literature to take examples with a special property and produce new ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine simple gluings of [Formula: see text]-generated non-symmetric, [Formula: see text]-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many new complete intersection semigroups of any embedding dimensions, having strongly indispensable minimal free resolutions.

Almost symmetric numerical semigroups

We study almost symmetric numerical semigroups and semigroup rings. We describe a characteristic property of the minimal free resolution of the semigroup ring of an almost symmetric numerical semigroup. For almost symmetric semigroups generated by $4$ elements we will give a structure theorem by using the \lq\lq row-factorization matrices", introduced by Moscariello. As a result, we give a simpler proof of Komeda's structure theorem of pseudo-symmetric numerical semigroups generated by $4$ elements. Row-factorization matrices are also used to study shifted families of numerical semigroups.

STAR OPERATIONS ON KUNZ DOMAINS

We study star operations on Kunz domains, a class of analyti- cally irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mi- mouni and Park. We also nd an estimate for the number of star operations in a particular case, and a precise counting in a sub-case.

BETTI NUMBERS FOR CERTAIN COHEN–MACAULAY TANGENT CONES

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

Parametrizing numerical semigroups with multiplicity up to 5

In this work, we give parametrizations in terms of the Kunz coordinates of numerical semigroups with multiplicity up to [Formula: see text]. We also obtain parametrizations of MED semigroups, symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text]. These parametrizations also lead to formulas for the number of numerical semigroups, the number of MED semigroups and the number of symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text] and given conductor.

On pseudo symmetric monomial curves

ABSTRACTWe study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen–Macaulay.

Genus of numerical semigroups generated by three elements

Let H=〈a,b,c〉 be a numerical semigroup generated by three elements and let R=k[H] be its semigroup ring over a field k. We assume that H is not symmetric and assume that the defining ideal of R is defined by maximal minors of the matrix (XαYβZγYβ′Zγ′Xα′). Then we will show that the genus of H is determined by the Frobenius number F(H) and αβγ or α′β′γ′. In particular, we show that H is pseudo-symmetric if and only if αβγ=1 or α′β′γ′=1.Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups H=〈a,b,c〉 with given Frobenius number.

One half of a pseudo-symmetric numerical semigroup

We prove that a numerical semigroup is irreducible if and only if it is one half of a pseudo-symmetric numerical semigroup. As a consequence we find that every numerical semigroup can be obtained by dividing a pseudo-symmetric numerical semigroup by 4.

Contractions of a numerical semigroup

Given a numerical semigroup S, a positive integer a and m ∈ S \{ 0}, we introduce the set C(S,a,m )= {x ∈ N | aw(xmod m) x} ,w here{w(0),w(1),...,w(m − 1)} is the Ap´ ery set of m in S , which is a numerical semigroup and that we call (a,m)- contraction of S. We study the Frobenius number and the singularity degree of C(S,a,m). We also characterize the contractions C(S,a,m) that are symmetric and pseudo-symmetric numerical semigroups. Finally we see that the contractions of N are solutions of modular Diophantine inequalities.

Pseudo-symmetric numerical semigroups with three generators

This paper gives a solution to the Diophantine Frobenius problem for pseudo-symmetric numerical semigroups with embedding dimension three. We also describe a presentation for this kind of semigroup and a formula to determine whether or not a numerical semigroup with embedding dimension three is pseudo-symmetric.

Pseudo-symmetric modular Diophantine inequalities

In this paper we study and characterize those Diophantine inequalities ax mod b x whose set of solutions is a pseudo-symmetric numerical semigroup.

Acute Semigroups, the Order Bound on the Minimum Distance, and the Feng–Rao Improvements

We introduce a new class of numerical semigroups, which we call the class of acute semigroups and we prove that they generalize symmetric and pseudosymmetric numerical semigroups, Arf numerical semigroups, and the semigroups generated by an interval. For a numerical semigroup /spl Lambda/={/spl lambda//sub 0/ i})=/spl nu//sub i+1/ for all i/spl ges/m. We prove that the only numerical semigroups for which the sequence (/spl nu//sub i/) is always nondecreasing are ordinary numerical semigroups. Furthermore, we show that a semigroup can be uniquely determined by its sequence (/spl nu//sub i/).

The oversemigroups of a numerical semigroup

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.