scholarly journals A Criterion for a Monomial Ideal to have a Linear Resolution in Characteristic 2

10.37236/4082 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
E. Connon ◽  
Sara Faridi
Keyword(s):  
Monomial Ideal ◽  
Linear Resolution ◽  
Characteristic 2 ◽  
Combinatorial Condition ◽  
Necessary And Sufficient

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

scholarly journals Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals GORENSTEIN AND Sr PATH IDEALS OF CYCLES

2014 ◽  
Vol 57 (1) ◽  
pp. 7-15 ◽  
Author(s):  
DARIUSH KIANI ◽  
SARA SAEEDI MADANI ◽  
NAOKI TERAI
Keyword(s):  
Directed Graph ◽  
Monomial Ideal ◽  
N Cycle ◽  
Symbolic Powers ◽  
Linear Resolution ◽  
Directed Paths

AbstractLet R = k[x1,…,xn], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or $\lceil \frac{n}{n-t-1}\rceil\geq r+3$. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n − 2.

scholarly journals n-TH Root Selections in Fields

2019 ◽  
Vol 33 (1) ◽  
pp. 106-120
Author(s):  
Paweł Gładki
Keyword(s):  
Multiplicative Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Field Extensions ◽  
Partial Inverse ◽  
Characteristic 2 ◽  
Necessary And Sufficient

AbstractIn this work we generalize the results of [9] to the higher level case: we define n-th root selections in fields of characteristic ≠ 2, that is subgroups of the multiplicative group of a field whose existence is equivalent to the existence of a partial inverse of the x ↦ xn function, provide necessary and sufficient conditions for such a subgroup to exist, study their existence under field extensions, and give some structural results describing the behaviour of maximal n-th root selection fields.

scholarly journals ON UNIFORM DECISION PROBLEMS AND ABSTRACT PROPERTIES OF SMALL OVERLAP MONOIDS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 217-233
Author(s):  
MARK KAMBITES
Keyword(s):  
Isomorphism Problem ◽  
Decision Problems ◽  
Asymptotic Estimates ◽  
Case Complexity ◽  
Abstract Structure ◽  
Combinatorial Condition ◽  
Necessary And Sufficient ◽  
Small Overlap ◽  
The Way

We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2) presentation; by counting canonical presentations we obtain asymptotic estimates for the number of non-isomorphic monoids admitting a-generator, k-relation presentations of a given length. We demonstrate an algorithm to transform an arbitrary presentation for a C(m) monoid (m at least 2) into a canonical C(m) presentation, and a solution to the isomorphism problem for C(2) presentations. We also find a simple combinatorial condition on a C(4) presentation which is necessary and sufficient for the monoid presented to be left cancellative. We apply this to obtain algorithms to decide if a given C(4) monoid is left cancellative, right cancellative or cancellative, and to show that cancellativity properties are asymptotically visible in the sense of generic-case complexity.

scholarly journals Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

2013 ◽  
Vol 120 (7) ◽  
pp. 1714-1731 ◽  
Author(s):  
E. Connon ◽  
S. Faridi
Keyword(s):  
Monomial Ideal ◽  
Necessary Condition ◽  
Linear Resolution

scholarly journals Stability of characters and filters for weighted semilattices

2020 ◽  
Author(s):  
Yemon Choi ◽  
Mahya Ghandehari ◽  
Hung Le Pham
Keyword(s):  
Weight Function ◽  
Large Class ◽  
Stability Property ◽  
Closed Set ◽  
Link Type ◽  
Set Systems ◽  
Combinatorial Condition ◽  
Necessary And Sufficient

AbstractWe continue the study of the AMNM property for weighted semilattices that was initiated in Choi (J Aust Math Soc 95(1):36–67, 2013. 10.1017/S1446788713000189). We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this “filter stability” property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in loc. cit., and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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