Fixed points in partially ordered sets

Bacławski, Kenneth; Björner, Anders

doi:10.1016/0001-8708(79)90045-8

Cited by 98 publications

(67 citation statements)

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“…-{0}) is the simplicial complex of the barycentric subdivision of Q. See for instance [BB,Example 1.3]. Thus x(^(Q ~ (0})) = 1 where x is the Euler characteristic.…”

Section: Ry^r2 If and Only If S(ry) Es(r2)mentioning

confidence: 99%

A Partial Order on the Regions of R n Dissected by Hyperplanes

Edelman¹

1984

Transactions of the American Mathematical Society

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Abstract. We study a partial order on the regions of R" dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introduction. Let % -{77,, H2,.. .,Hk] be a set of hyperplanes in R". Then the components of R" -UHe.xH form a set 9t of open n-cells we will call regions. Traditionally 9t has been studied in terms of enumeration, for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in %. For a thorough discussion of this problem see Zaslavsky's monograph [Zl] and the references therein. See also [OS and Te].In this paper we will study 9? from a different perspective. Our concern will be the way in which one can move from one region to another by "walking through" one hyperplane at a time. To analyze this situation we define a partial order on 9t relative to a base region where the cover relations correspond to " walking through" a hyperplane and moving further away from the base.The paper is organized as follows: In §1 we define the partial order on 3? and begin to study its structure. This culminates in Theorem 1.11 where an explicit computation of its Mobius function is performed. In §11 we examine the special case where all the hyperplanes go through the origin. In certain cases we are able to compute the homotopy type of these posets. The next three sections are devoted to applications. §111 applies our work to a partial order on a zonotope. §IV deals with the weak Bruhat order of a Weyl group as a special case of our posets. As a corollary we show that the homotopy type of the weak Bruhat order is that of a sphere.Finally, §V applies our results to a partial order on the acyclic orientations of a graph. Some of the relationships used in § §III, IV and V appear in an enumerative context in the excellent article of Greene and Zaslavsky [GZ].Most of our poset notation is standard and can be found in [A]. When more than one partial order is being used we will sometimes use subscripts to indicate where the relation is, i.e., x ^Py means that x is less than or equal toy in the poset P. By A(.P) we mean the simplicial complex of chains (totally ordered sets) in P. We will think of A(7") as its geometric realization. By P* we mean the order dual of P, i.e.,

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“…-{0}) is the simplicial complex of the barycentric subdivision of Q. See for instance [BB,Example 1.3]. Thus x(^(Q ~ (0})) = 1 where x is the Euler characteristic.…”

Section: Ry^r2 If and Only If S(ry) Es(r2)mentioning

confidence: 99%

A Partial Order on the Regions of R n Dissected by Hyperplanes

Edelman¹

1984

Transactions of the American Mathematical Society

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“…Rival, in [Riv76], has introduced a notion of dismantlability (by irreducibles) for finite posets, and this has been extended to arbitrary posets in [BB79]. We do not recall the definitions here, but rather record a characterization, which follows from earlier results in [Sto66].…”

Section: Selection Mapsmentioning

confidence: 99%

Homotopies and the Universal Fixed Point Property

Szymik

2014

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A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.2010 MSC: 54H25, 55M20, 47H10.Every continuous self-map of a non-empty, compact, convex subspace of a topological vector space has at least one fixed point. This theorem of Brouwer-Schauder-Tychonoff is a milestone of fixed point theory, and the applications of it are numerous. Whenever the existence of fixed points is guaranteed for every self-map, the space is called a fixed point space. This property is shared by many other spaces besides the ones already covered by that famous result. But, what if the self-map varies continuously? Is there also a continuous choice of fixed points? This is the question that we address here. It is surprising that there is no previous literature on this topic in the general topological setting, despite the property being clearly desirable: in practice, it is genuinely less useful to know the existence of a solution to a problem, if this solution does not depend continuously on the initial conditions. The Banach fixed point theorem provides a class of examples where we do know that the fixed points depend continuously on the initial conditions (Example 2.1). However, the result evidently applies to metric spaces and contractions only. 1 arXiv:1210.6496v3 [math.GN]

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“…Let us mention that the Hopf-Lefschetz fixed point theorem for order-preserving maps on a finite poset was developed and proved by Baclawski-Björner [1] (see also Baclawski [2]). …”

Section: Global Asymptotic Stability Theoremsmentioning

confidence: 99%

“…The answer to Conjecture 6.2 is also affirmative if n = 4, and this case n = 4 can be proved by the reduction method as in the proof of (a) of Theorem 3.2. However, this approach is out of the question for n ≥ 5 because the number of switching functions f i x 1 x n of n variables grows astronomically as n increases. Thus the case n ≥ 5 of Conjecture 6.2 remains open.…”

Section: A Conjecturementioning

confidence: 99%

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Solution of the Boolean Markus–Yamabe Problem

Shih

Ho²

1999

Advances in Applied Mathematics

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The main point in the design of content addressable memory would be under what conditions the state possessing the total information can attract all other states in the phase of the system. The problem can be formulated as a global asymptotic stability problem of Boolean dynamical systems. In this article we give a complete answer to this global asymptotic stability problem. The conditions employed involve the Hamming distance on the phase space 0 1 n as well as the spectral condition on the Jacobian Boolean matrix of F 0 1 n → 0 1 n evaluated at each point of 0 1 n . This article furnishes a complete solution of the Boolean Markus-Yamabe problem.

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Fixed points in partially ordered sets

Cited by 98 publications

References 9 publications

A Partial Order on the Regions of R n Dissected by Hyperplanes

A Partial Order on the Regions of R n Dissected by Hyperplanes

Homotopies and the Universal Fixed Point Property

Solution of the Boolean Markus–Yamabe Problem

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