Special Issue "Current Trends on Monomial and Binomial Ideals"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2019).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Takayuki Hibi
Website
Guest Editor
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita 565-0871, Japan
Interests: computational commutative algebra; discrete mathematics; combinatorics
Prof. Tài Huy Hà
Website
Guest Editor
Department of Mathematics, Tulane University, 6823 St. Charles Avenue, New Orleans, LA 70118, USA
Interests: commutative algebra, computational algebra, combinatorics, algebraic geometry

Special Issue Information

Dear Colleagues,

Recently, new trends on monomial ideals and binomial ideals have occurred. Remarkable developments in, for example, finite free resolutions, syzygies, regularity and symbolic powers of monomial ideals, and binomial ideals arising from combinatorial objects including finite graphs, lattice polytopes, and finite partially ordered sets have been brought about by a large number of authors. The present Special Issue summarizes recent achievements in these topics and stimulates further research that invites breakthroughs in the theory of monomial and binomial ideals.

Prof. Takayuki Hibi
Prof. Tài Huy Hà
Guest Editors

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Keywords

  • Finite free resolution
  • Syzygy
  • Regularity
  • Symbolic power
  • Hibi ring
  • Edge ideal
  • Binomial edge ideal
  • Finite graph
  • Lattice polytope
  • Finite partially ordered set

Published Papers (12 papers)

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Research

Open AccessArticle
The Regularity of Edge Rings and Matching Numbers
Mathematics 2020, 8(1), 39; https://doi.org/10.3390/math8010039 - 01 Jan 2020
Abstract
Let K[G] denote the edge ring of a finite connected simple graph G on [d] and mat(G) the matching number of G. It is shown that reg(K[G]) [...] Read more.
Let K [ G ] denote the edge ring of a finite connected simple graph G on [ d ] and mat ( G ) the matching number of G. It is shown that reg ( K [ G ] ) mat ( G ) if G is non-bipartite and K [ G ] is normal, and that reg ( K [ G ] ) mat ( G ) 1 if G is bipartite. Full article
Open AccessArticle
Odd Cycles and Hilbert Functions of Their Toric Rings
Mathematics 2020, 8(1), 22; https://doi.org/10.3390/math8010022 - 20 Dec 2019
Abstract
Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1s5, an O-sequence (h0,h1,,h2s1)Z [...] Read more.
Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 s 5 , an O-sequence ( h 0 , h 1 , , h 2 s 1 ) Z 0 2 s satisfying the properties that (i) h 0 h 1 h s 1 , (ii) h 2 s 1 = h 0 , h 2 s 2 = h 1 and (iii) h 2 s 1 i = h i + ( 1 ) i , 2 i s 1 , can be the h-vector of a Cohen-Macaulay standard G-domain. Full article
Open AccessFeature PaperArticle
Syzygies, Betti Numbers, and Regularity of Cover Ideals of Certain Multipartite Graphs
Mathematics 2019, 7(9), 869; https://doi.org/10.3390/math7090869 - 19 Sep 2019
Cited by 1
Abstract
Let G be a finite simple graph on n vertices. Let JGK[x1,,xn] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo–Mumford regularity of [...] Read more.
Let G be a finite simple graph on n vertices. Let J G K [ x 1 , , x n ] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo–Mumford regularity of J G s for all s 1 for certain classes of graphs G. Full article
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Open AccessArticle
Algebraic Algorithms for Even Circuits in Graphs
Mathematics 2019, 7(9), 859; https://doi.org/10.3390/math7090859 - 17 Sep 2019
Abstract
We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs. Full article
Open AccessFeature PaperArticle
Faces of 2-Dimensional Simplex of Order and Chain Polytopes
Mathematics 2019, 7(9), 851; https://doi.org/10.3390/math7090851 - 14 Sep 2019
Abstract
Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it [...] Read more.
Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges. Full article
Open AccessArticle
Cohen Macaulay Bipartite Graphs and Regular Element on the Powers of Bipartite Edge Ideals
Mathematics 2019, 7(8), 762; https://doi.org/10.3390/math7080762 - 20 Aug 2019
Abstract
In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals I(G), we also discuss methods to recognize regular elements on I(G)s for all s1 in terms [...] Read more.
In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals I ( G ) , we also discuss methods to recognize regular elements on I ( G ) s for all s 1 in terms of the combinatorics of the graph G. Full article
Open AccessArticle
Cohen-Macaulay and (S2) Properties of the Second Power of Squarefree Monomial Ideals
Mathematics 2019, 7(8), 684; https://doi.org/10.3390/math7080684 - 31 Jul 2019
Abstract
We show that Cohen-Macaulay and (S2) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S/I2 satisfies the Serre condition (S2), [...] Read more.
We show that Cohen-Macaulay and (S 2 ) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S / I 2 satisfies the Serre condition (S 2 ), but is not Cohen-Macaulay. Full article
Open AccessArticle
Compatible Algebras with Straightening Laws on Distributive Lattices
Mathematics 2019, 7(8), 671; https://doi.org/10.3390/math7080671 - 27 Jul 2019
Abstract
We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws. Full article
Open AccessFeature PaperArticle
The Regularity of Some Families of Circulant Graphs
Mathematics 2019, 7(7), 657; https://doi.org/10.3390/math7070657 - 22 Jul 2019
Abstract
We compute the Castelnuovo–Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive [...] Read more.
We compute the Castelnuovo–Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity. Full article
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Open AccessFeature PaperArticle
Toric Rings and Ideals of Stable Set Polytopes
Mathematics 2019, 7(7), 613; https://doi.org/10.3390/math7070613 - 10 Jul 2019
Abstract
In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their [...] Read more.
In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases. Full article
Open AccessFeature PaperArticle
On the Stanley Depth of Powers of Monomial Ideals
Mathematics 2019, 7(7), 607; https://doi.org/10.3390/math7070607 - 08 Jul 2019
Cited by 1
Abstract
In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is [...] Read more.
In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals. Full article
Open AccessArticle
Linear Maps in Minimal Free Resolutions of Stanley-Reisner Rings
Mathematics 2019, 7(7), 605; https://doi.org/10.3390/math7070605 - 06 Jul 2019
Abstract
In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ. Indeed, the differentials in the linear part are simply a compilation of restriction maps in [...] Read more.
In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex Δ . Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of Δ . Along the way, we also show that if a monomial ideal has at least one generator of degree 2, then the linear strand of its minimal free resolution can be written using only ± 1 coefficients. Full article
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