Let $Λ$ be a numerical semigroup and $I\subset Λ$ be an ideal of $Λ$. The graph $G_I(Λ)$ assigned to an ideal $I$ of $Λ$ is a graph with elements of $(Λ\setminus I)^*$ as vertices and any two vertices $x,y$ are adjacent if and only if $x+y \in I$. In this paper we give a complete characterization (up to isomorphism ) of the graph $G_I(Λ)$ to be planar, where $I$ is an irreducible ideal of $Λ$. This will finally characterize non planar graphs $G_I(Λ)$ corresponding to irreducible ideal $I$.
In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal.
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations …
We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.
A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a Commutative Algebra and an Algebraic Statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize comb…
Good semigroups form a class of submonoids of $\mathbb{N}^d$ containing the value semigroups of curve singularities. In this article, we describe a partition of the complements of good semigroup ideals, having as main application the description of the Apéry sets of good semigroups. This generalizes to any $d \geq 2$ the results of a recent paper of D'Anna, Guerrieri and Micale, which are proved in the case $d=2$ and only for the standard Apéry set with respect to the smallest nonzero element. Several new results describing good semigroups in $\mathbb{N}^d$ are also provided.
We propose the notion of GAS numerical semigroup which generalizes both almost symmetric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonical ideal which generalizes the notion of canonical ideal in the same way almost symmetric numerical semigroups generalize symmetric ones. We prove that a numerical semigroup with maximal ideal $M$ and multiplicity $e$ is GAS if and only if $M-e$ is an almost canonical ideal of $M-M$. This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau, Goto, Kumashiro, and Matsuoka about 2-AGL semigr…
Ulrich ideals in numerical semigroup rings of small multiplicity are studied. If the semigroups are three-generated but not symmetric, the semigroup rings are Golod, since the Betti numbers of the residue class fields of the semigroup rings form an arithmetic progression; therefore, these semigroup rings are G-regular, possessing no Ulrich ideals. Nevertheless, even in the three-generated case, the situation is different, when the semigroups are symmetric. We shall explore this phenomenon, describing an explicit system of generators, that is the normal form of generators, for the Ulrich ideals…
The structure of the defining ideal of the semigroup ring $k[H]$ of a numerical semigroup $H$ over a field $k$ is described, when the pseudo-Frobenius numbers of $H$ are multiples of a fixed integer.
The Poincaré series of a ring associated to a plane curve was defined by Campillo, Delgado, and Gusein-Zade. This series, defined through the value semigroup of the curve, encodes the topological information of the curve. In this paper we extend the definition of Poincaré series to the class of good semigroup ideals, to which value semigroups of curves belong. Using this definition we generalize a result of Pol: under suitable assumptions, given good semigroup ideals E and K, with K canonical, the Poincaré series of K-E is symmetric to the Poincaré series of E.
In this article, we concurrently explore the class of m-bi-ideals as well as the class of m- interior ideals in an ordered semigroup. We call these class of ideals as m-bi-interior ideals. Here, m-simple, m-regular ordered semigroups and their various subclasses get characterized in diverse manners as per their m-bi-interior ideals.
We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine when such a quotient is Cohen--Macaulay. In particular, this gives a Cohen--Macaulayness criterion for lattice ideals. We also prove a result relating the local cohomology with radical monomial ideal support of an affine semigroup ring to the local cohomology with maximal ideal support of the quotient of the affine semigroup ring by the radical monomial ideal. …
We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.
A sumset semigroup is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. In this work, an algorithm for computing the ideals associated with some sumset semigroups is provided. Using these ideals, we study some factorization properties of sumset semigroups and some additive properties of sumsets. This approach links computational commutative algebra with additive number theory.
Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an arbitrary commutative ring with identity. We classify the irreducible elements of this set, which we call m-irreducible, and we classify the elements that admit decompositions into finite intersections of m-irreducible ideals.
Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given.
The aim of this article is to study the ideal class monoid $\mathcal{C}\ell(S)$ of a numerical semigroup $S$ introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $\mathcal{C}\ell(S)$. We observe that $\mathcal{C}\ell(S)$ is isomorphic to the monoid of ideals of $S$ whose smallest element is 0, which helps to relate $\mathcal{C}\ell(S)$ to the Apéry sets and the Kunz coordinates of $S$. We study some combinatorial and algebraic properties of $\mathcal{C}\ell(S)$, including the reduction number of ideals, and the Hasse diagrams of $\mathcal{C}\ell(S)$ with respect to…
For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant measures how far is $H$ from being symmetric, i.e. $K[H]$ from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For $3$-generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of $H$. Thus, in this setting we can classify those whose residue is at most one (the nearly-Gorenstein ones), and we show the eventual peri…
We present stdPairs.spyx, a SageMath library to compute standard pairs of a monomial ideal over a pointed (non-normal) affine semigroup ring. Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from arXiv:2005.10968v2 [math.AC]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.
A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigroups in terms of Gröbner bases. Specifically, a numerical semigroup is universally free if and only if all initial ideals of its corresponding toric ideal are complete intersections. Furthermore, we establish several equalities among the toric bases o…
Let R be a two-dimensional regular local ring with maximal ideal \mathfrak m, and let \wp be a simple complete \mathfrak m-primary ideal which is residually rational. Let R_0:= R\subsetneqq ...\subsetneqq R_r be the quadratic sequence associated to \wp, let Γ_\wp be the value-semigroup associated to \wp, and let ((e_j(\wp))_{0\leq j\leq r} be the multiplicity sequence of \wp. We associate to \wp a sequence of natural integers, the formal characteristic sequence of \wp, and we show that the value-semigroup, the multiplicity sequence and the formal characteristic sequence are equivalent data. Fu…
The aim of this work is to reduce the complexity of the available algorithms for computing the generator sets of a semigroup ideal by using the Hermite normal form. In order to achieve it we introduce the concept of decomposable semigroup. If a semigroup is decomposable, the computation of its ideal is equivalent to compute the ideals of each semigroup in the decomposition, thus obtaining a reduction of the complexity of the algorithms. Furthermore, since these computations are mutually independent, they can be carried out in parallel. The concept of decomposable variety is introduced and a co…
This paper considers numerical semigroups $S$ that have a non-principal relative ideal $I$ such that $μ_S(I)μ_S(S-I)=μ_S(I+(S-I)) $. We show the existence of an infinite family of such which $I+(S-I)=S\backslash\{0\}$. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.
We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.
The t-class semigroup of an integral domain is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of a Noetherian domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup.
Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property that there are atoms $u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$ (thus, $\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\mathscr U_k (H)$ are almost arithmeti…
An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals terminates. It was asserted by Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by Grams in the seventies with an explicit construction of a neat example. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams' first construction.…
In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup $S$ and a semigroup ideal $E\subseteq S$, produces a new numerical semigroup, denoted by $S\Join^b\E$ (where $b$ is any odd integer belonging to $S$), such that $S=(S\Join^b\E)/2$. In particular, we characterize the ideals $E$ such that $S\Join^bE$ is almost symmetric and we determine its type.
The t-class semigroup of an integral domain R, denoted S_t(R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain, then S_t(R) is a Clifford semigroup. This paper aims to describe the idempotents of S_t(R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prufer domains of finite character.
We generalize the Bernstein-Sato polynomials of Budur, Mustata and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein-Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. In order to prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.
We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…
Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree $i$ has dimension exactly dim(R/I) - i. This Cohen-Macaulay characterization reduces to the Eagon-Reiner theorem by Alexander duality when R is a polynomial ring. The proof exploits a graded ring-theoretic generalization of the Zeeman spectral sequence, thereby also providing a combinatorial topological version for polyhedral cel…
Let $\mathcal{S}$ be a commutative semigroup, and let $T$ be a sequence of terms from the semigroup $\mathcal{S}$. We call $T$ an (additively) {\sl irreducible} sequence provided that no sum of its some terms vanishes. Given any element $a$ of $\mathcal{S}$, let ${\rm D}_a(\mathcal{S})$ be the largest length of the irreducible sequence such that the sum of all terms from the sequence is equal to $a$. In case that any ascending chain of principal ideals starting from the ideal $(a)$ terminates in $\mathcal{S}$, we found the sufficient and necessary conditions of ${\rm D}_a(\mathcal{S})$ being f…
Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal and $I$ is a radical monomial ideal (i.e., $R/I$ is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of $R/I$ is a topological property of the "geometric realization" of the cell complex associated with $I$. Moreover, we can give a squarefree modules/constructible sheaves version of this result.
We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtai…
In this note we observe that the Frobenius number and therefore the conductor of a numerical semigroup can be obtained from the maximal socle degree of the quotient of the corresponding semigroup algebra by the ideal generated by the biggest generator of the semigroup.
The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_ν}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semi…
This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call cores of $S$ includes the semigroup rings $k[H]$ of numerical semigroups $H$, but much larger than the class of numerical semigroup rings. For $R=k[H]$ and $M \in \operatorname{Max}R$, our result eventually shows that $μ_{R}(M) \in \{1,2,μ(H)\}$ where $μ_{R}(M)$ (resp. $μ(H)$) stands for the minimal number of generators of $M$ (resp. $H$), which covers in t…
The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prufer v-multiplication domain (PVMD) is of Krull type (in the sense of Griffin) if and onl…
We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.
The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t-multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Prufer domains and Bezout domains of finite character.
We study almost symmetric semigroups generated by odd integers. If the embedding dimension is four, we characterize when a symmetric semigroup that is not complete intersection or a pseudo-symmetric semigroup is generated by odd integers. Moreover, we give a way to construct all the almost symmetric semigroups with embedding dimension four and type three generated by odd elements. In this case we also prove that all the pseudo-Frobenius numbers are multiple of one of them and this gives many consequences on the semigroup and its defining ideal.
The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
