As a generalization of the facet ideal of a forest, we define monomial ideal of forest type and show that monomial ideals of forest type are pretty clean. As a consequence, we show that if I is a monomial ideal of forest type in the polynomial ring S, then Stanley's decomposition conjecture holds for S/I. The other main result of this article shows that a clutter is totally balanced if and only if it has the free vertex property, and which is also equivalent to say that its edge ideal is a monomial ideal of forest type or is generated by an M sequence. 相似文献
We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals. 相似文献
The notion of apartitionable simplicial complex is extended to that of asignable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone
fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable,
strictly contains the class of convex polytopes. A general sufficient condition, termedtotal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial
members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational
complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation,
the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial
fans can be computed in polynomial time, extending the analogous statement for convex polytopes.
The research of S. Onn was supported by the Alexander von Humboldt Stifnung, by the Fund for the Promotion of Research at
the Technion, and by Technion VPR fund 192–198. 相似文献
We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field and for an arbitrary intersection of monomial prime ideals (Pi)i∈[s] of S such that each Pi is not contained in the sum of the other (Pj)j≠i. 相似文献
Let Δ be a one-dimensional simplicial complex. Let be the Stanley–Reisner ideal of Δ. We prove that for all and all intermediate ideals J generated by and some minimal generators of , we have 相似文献
We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two. 相似文献
In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a left-module on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring.Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes.More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families.Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work. 相似文献
We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs.
We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.
of simplicial complexes G, we introduce the notion of a
-C-space. In the definition of a C-space, open disjoint families vi refine coverings ui. The nerves of these families are zero-dimensional complexes. In our definition, the nerve of a family vi must embed in the complex Gi of the class
. We give a complete characterization of bicompact
Kalai has conjectured that a simplicial complex can be partitioned into Boolean algebras at least as roughly, as a shifting-preserving collapse sequence of its algebraically shifted complex. In particular, then, a simplicial complex could (conjecturally) be partitioned into Boolean intervals whose sizes are indexed by its iterated Betti numbers, a generalization of ordinary homology Betti numbers. This would imply a long-standing conjecture made (separately) by Garsia and Stanley concerning partitions of Cohen-Macaulay complexes into Boolean intervals. We prove a relaxation of Kalai’s conjecture, showing that a simplicial complex can be partitioned into recursively defined spanning trees of Boolean intervals indexed by its iterated Betti numbers. 相似文献
We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular,
on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential
d and the exterior product ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called
A∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)n = 0 of the second degree, n = 2.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 3–37, July, 2008. 相似文献
The purpose of this note is to point out a careless error in the algebraic criterion of shellability of a pure simplicial complex Δ given in [1Anwar, I., Raza, Z. (2015). Quasi-linear quotients and shellability of pure simplicial complexes. Commun. Algebra 43:4698–4704.[Taylor &; Francis Online], [Web of Science ®], [Google Scholar]]. 相似文献