k-Decomposable Monomial Ideals

Rahmati-Asghar, Rahim; Yassemi, Siamak

doi:10.1142/s1005386715000656

Cited by 10 publications

(12 citation statements)

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“…(ii) If there is a variable x ∈ X and vertex splittable ideals I 1 and I 2 of k[X\{x}] so that I = xI 1 + I 2 , I 2 ⊆ I 1 and G(I) is the disjoint union of G(xI 1 ) and G(I 2 ), then I is a vertex splittable ideal. With the above notations if I = xI 1 + I 2 is a vertex splittable ideal, then xI 1 + I 2 is called a vertex splitting for I and x is called a splitting vertex for I.Recently, the notion of a k-decomposable ideal was introduced in[19] and it was proved that it is the dual concept for k-decomposable simplicial complexes.In the case k = 0, 0-decomposable simplicial complexes are precisely vertex decomposable simplicial complexes. Considering k = 0 in [19, Definition 2.3], one can see that a shedding monomial u in a 0-decomposable ideal I necessarily should satisfy the property I u = (0), where I u = (v ∈ G(I) : [u, v] = 1) and [u, v] is the greatest common divisor of u and v, while in Definition 2.1, it is not the case, i.e., the way that a monomial ideal splits in Definition 2.1 is different from one in [19, Definition 2.3].…”

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confidence: 99%

On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Moradi¹,

Khosh-Ahang²

2016

MATH. SCAND.

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Abstract. In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex ∆ is vertex decomposable if and only if I ∆ ∨ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of R/I ∆ , when ∆ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph G, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when G is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

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confidence: 99%

On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Moradi¹,

Khosh-Ahang²

2016

MATH. SCAND.

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“…The following theorem was proved in [10]. In order to prove Theorem 2.4, one needs to know the construction of the order of linear quotients for a decomposable ideal.…”

Section: Regularity and Projective Dimension Of The Stanley-reisner Rmentioning

confidence: 99%

“…It is known that a d-dimensional simplicial complex is d-decomposable if and only if it is shellable (see [14,Theorem 3.6]). In [10], the concept of a kdecomposable monomial ideal was introduced and it was proved that a simplicial complex ∆ is k-decomposable if and only if I ∆ ∨ is a k-decomposable ideal.…”

Section: Introductionmentioning

confidence: 99%

Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

Moradi

2017

Asian-European J. Math.

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We study the regularity and the projective dimension of the Stanley-Reisner ring of a k-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of kdecomposable monomial ideals which is the dual concept for k-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a chordal clutter H, an upper bound for reg(I(H)) is given in terms of the regularities of edge ideals of some chordal clutters which are minors of H.2010 Mathematics Subject Classification. Primary 13D02, 05E45, 05E40; Secondary 16E05.

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“…Now we study the behavior of expansion functor on the property of having linear quotients. First, we recall some notations and definitions from [12]:…”

Section: The Expansion Of Some Classes Of Monomial Idealsmentioning

confidence: 99%

The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

Rahmati-Asghar

Yassemi

2016

Communications in Algebra

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In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor.The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid P, if toric ideal of P is generated by double swaps then toric ideal of any expansion of P has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.

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k-Decomposable Monomial Ideals

Cited by 10 publications

References 13 publications

On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

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