Monomial ideals and toric rings of Hibi type arising from a finite poset

Ene, Viviana; Herzog, Jürgen; Mohammadi, Fatemeh

doi:10.1016/j.ejc.2010.11.006

Cited by 39 publications

(74 citation statements)

References 17 publications

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“…In 1987, Hibi [Hib87] introduced a class of algebras which nowadays are called Hibi rings. They are defined using finite posets and naturally appear in various algebraic and combinatorial contexts; see for example [HH05], [EHM11], [How05] and [KP]. Hibi rings are toric K-algebras defined over a field K. They are normal Cohen-Macaulay domains and their defining ideal admits a quadratic Gröbner basis.…”

Section: Introductionmentioning

confidence: 99%

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

2019

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Section: Introductionmentioning

confidence: 99%

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

2019

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“…Further, the relation ≤ [a,b] on P is antisymmetric follows from the fact that if p i ≤ a p j , then we have i ≤ j. In Example 4.1 observe that ≤ [1,2] is not a transitive relation on P , and hence P [a,b] need not be a poset. i) Let u be a monomial in S. Then support of u, denoted by supp(u), is defined as…”

Section: Cohen Macaulay Multipartite Graphsmentioning

confidence: 99%

“…Example 4.5. Let P = {p 1 , p 2 , p 3 } and F = {≤ 1 , ≤ 2 , ≤ [1,2] }, where ≤ 1 , ≤ 2 , ≤ [1,2] are given as in Examples 3.2 and 4.1. We associate a following graph on vertices set V = {X a,i : a, i ∈ [3]}:…”

Section: Cohen Macaulay Multipartite Graphsmentioning

confidence: 99%

Certain classes of Cohen–Macaulay multipartite graphs

Kumar

2019

Communications in Algebra

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The Cohen-Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen-Macaulay property of a graph arising from a family of reflexive and antisymmetric relations on a set. We use this result to find classes of multipartite graphs which are Cohen-Macaulay. IntroductionGraphs and simplicial complexes play an important role in combinatorial commutative algebra. In order to see the relationship between commutative algebra and combinatorics, one can associate monomial ideals to graphs or simplicial complexes. Many authors have studied the connection between the algebraic properties of these ideals and the combinatorial properties of the corresponding combinatorial objects, see [4, Chapter 9]. In this article, our main focus is to study the edge ideal of a graph. A graph is called Cohen-Macaulay if the corresponding edge ideal is Cohen-Macaulay.The Cohen-Macaulay property of graphs has been well studied for various classes. Herzog-Hibi ([3]) prove that a bipartite graph is Cohen-Macaulay if and only if it is arising from a poset. For a finite poset and r, s ∈ N, Ene-Herzog-Mohammadi ([2]) associated a monomial ideal generated in degree s, to the set of all multichains of length r in a poset, and proved that this ideal is Cohen-Macaulay. Note that if s = 2, then these ideals are edge ideals of some r-partite graphs. Motivated by these results, we associate a monomial ideal to a family of posets, and find a class of Cohen-Macaulay r-partite graphs.Our main tool in this article is the following well known relationship between the Stanley-Reisner ideal and its Alexander dual: The Stanley-Reisner ideal is Cohen-Macaulay if and only if its Alexander dual has a linear resolution. For more details see [1, Theorem 3].This paper has been organized in the following manner. In Section 2, we introduce the basic notions which are used throughout the article, more details can be found in [4]. In Section 3, we associate a monomial ideal H r (P) to a family P of partial order relations on a finite set. In Lemma 3.3, we prove that monomial ideal H r (P) has a linear resolution. This forces the Alexander dual of H r (P) to be Cohen-Macaulay.Section 4 is devoted to finding the classes of Cohen-Macaulay r-partite graphs. In Theorem 4.4, we see that the Alexander dual of H r (P) is an edge ideal of an r-partite graph associated to a family of reflexive and antisymmetric relations on a given set. Using this we find classes of Cohen-Macaulay graphs which are recorded in Theorems 4.7 and 4.10. Preliminaries2.1. Notation. The following notation is used throughout the article.

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“…That edge ideals of Cohen-Macaulay bipartite graphs have this form, is an astonishing discovery of J.Herzog and T.Hibi [13]. This class of ideals were generalized in [9] and further studied and generalized in [10] were they were called letterplace ideals, see Section 2.…”

Section: Introductionmentioning

confidence: 95%

Rigid ideals by deforming quadratic letterplace ideals

Fløystad

Nematbakhsh

2018

Journal of Algebra

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We compute the deformation space of quadratic letterplace ideals L(2, P ) of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2, P ) defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases J(2, P ) is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.Date: September 21, 2018.

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Monomial ideals and toric rings of Hibi type arising from a finite poset

Cited by 39 publications

References 17 publications

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

Certain classes of Cohen–Macaulay multipartite graphs

Rigid ideals by deforming quadratic letterplace ideals

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