Examples of Powers of Ordered Sets with the Fixed Point Property

This note presents examples of non-dismantlable ordered sets P with the fixed point property such that P ^A with A an arbitrary antichain and P ^C with C being a four crown have the fixed point property.     

On a fixed point theorem for partially ordered sets

An elementary combinatorial proof is presented of the following fixed point theorem: Let P be a finite partially ordered set with a cut-set X. If every subset of X has either a meet or a join, then P has the fixed point property. This theorem is strengthened to include a certain class of infinite partially ordered sets, as well.     

Antichain cutsets

A subset A of an ordered set P is a cutset if each maximal chain of P meets A; if, in addition, A is an antichain call it an antichain cutset. Our principal result is a characterization, by means of a forbidden configuration, of those finite ordered sets, which can be expressed as the union of antichain cutsets.     

Retractability and the fixed point property for products

LetP, Q be ordered sets and letaP. IfP \ {a} is a retract ofP and setsP and {xP:x>p} (or its dual) have the fixed point property then, for each chain complete setP,P×Q has the fixed point property if and only if (P\{a})×Q has this property. This establishes the fixed point property for some products of ordered sets which are beyond the reach of all known product theorems.The work of the first of authors was supported in part by the K.B.N. Grant No. 2 2037 92 03.     

A note on maximal antichains in ordered sets

We examine the question of when two consecutive levels in a product of -chains form an ordered set such that for any antichain, there is a maximal antichain disjoint from it. We characterize the pairs of consecutive levels in the product of t2 -chains that have this property. We also show that there is no upper bound on the heights of ordered sets having this property.The graph of an ordered set is the graph whose points are the elements of the ordered set, and whose edges are the ordered set's 2-element maximal antichains. We construct a class of ordered sets of all widths at least three such that the graph of each ordered set is a path, and we construct an ordered set of infinite width having a connected graph.Research supported by NSERC undergraduate student summer research fellowship, and by NSERC operating grant 69-3378Research supported by ONR contract N00014-85-K-0769     

Fixed point theorems for ordered sets P with P ∖ {a,c} as retract

We prove fixed point theorems for ordered sets P that have a retract with two points less than P and show how they can be used to prove the fixed point property for various well-known and various new ordered sets.     

A fixed point theorem with applications to truncated lattices

We prove a fixed point theorem related to the set P2 of [17]. The result gives access to nontrivial infinite ordered sets with the fixed point property. We also show how the result can be used to provide an elementary proof of part of Baclawski and Björner’s results on truncated lattices.Dedicated to the memory of Ivan RivalReceived December 1, 2002; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

The fixed point property in ordered sets of width two

We consider the fixed point property (FPP) in an ordered set of width two (every antichain contains at most two elements). The necessary condition of the FPP and a number of equivalent conditions to the FPP in such sets is established. The product theorem is proved, as well.     

Comparability invariance of the fixed point property

We show that the fixed point property is comparability invariant for finite ordered sets; that is, if P and Q are finite ordered sets with isomorphic comparability graphs, then P has the fixed point property if and only if Q does. In the process we give a characterization of comparability invariants which can also be used to give shorter proofs of some known results.     

Fixed points of products and the strong fixed point property

This problem motivates the present work: If ordered sets X and Y both have the fixed point property for order preserving maps has their product as well? Here we present a related condition — the so-called strong fixed point property — which arises from naive attempts to solve the problem. We are concerned with determining the nature and extent of this property. Several questions are raised concerning its relation to the fixed point property and other conditions such as dismantlability and contractibility.     

On canonical antichains

An antichain A of a well-founded quasi-order Q is canonical if for every ideal F of Q, F has an infinite antichain if and only if F∩A is infinite. In this paper we characterize the obstructions to having a canonical antichain. As an application we show that, under the induced subgraph relation, the class of finite graphs does not have a canonical antichain. In contrast, this class does have a canonical antichain with respect to the subgraph relation.     

A Note on Blockers in Posets

The blocker A* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized.(2) The blocker A* of a symmetric antichain in the partition lattice is characterized.(3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.AMS Subject Classification: 05C35, 05D05, 06A07.     

Gliding hump properties of matrix domains

Gliding hump properties play an important role in numerous topics in analysis, for instance, they are used as a substitute for the uniform boundedness principle. Since examples of sequence spaces having certain gliding hump properties are rare, the main aim of this paper is to present classes of infinite matrices A such that the matrix domain E _A has a certain gliding hump property whenever a given sequence space E has this property.     

There are no infinite order polynomially complete lattices, after all

If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power has an antichain of size . Hence there are no infinite opc lattices. However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices. Received November 2, 1998; accepted in final form March 19, 1999.     

On retractable sets and the fixed point property

Call a subsetA of an ordered setP retractable tob P iff the map that mapsA tob and leaves all other points fixed is a retraction. We prove fixed point theorems for sets that contain a retractable set and also use this tool to study the fixed point property for products. The results in this paper show that three classical approaches to the fixed point property: irreducible points, cutsets and lexicographic sums can be viewed as special cases of the situation described above.Presented by I. Rival.This research was funded in part by ONR grant nr. N00014-89-J-1824.     

Some progress on the Aharoni–Korman conjecture

Aharoni and Korman (Order 9 (1992) 245) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. Related to both Aharoni's extension of the König duality theorem and Dilworth's theorem, this is an important conjecture in the theory of infinite orders. It is verified for ordered sets of the form C×P, where C is a chain and P is finite, and for ordered sets with no infinite antichains and no infinite intervals.     

A combinatorial proof of a fixed point property

A class of finite simplicial complexes, called pseudo cones, is developed that has a number of useful combinatorial properties. A partially ordered set is a pseudo cone if its order complex is a pseudo cone. Pseudo cones can be constructed from other pseudo cones in a number of ways. Pseudo cone ordered sets include finite dismantlable ordered sets and finite truncated noncomplemented lattices. The main result of the paper is a combinatorial proof of the fixed simplex property for finite pseudo cones in which a combinatorial structure is constructed that relates fixed simplices to one another. This gives combinatorial proofs of some well known non-constructive results in the fixed point theory of finite partially ordered sets.     

Compactness and subsets of ordered sets that meet all maximal chains

LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP ₁ andP ₂ refer to two particular ordered sets considered below. It is shown that the space of maximal chains (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP ₁ or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP ₁ orP ₂ and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

A generalization of Hanani's theorem on partial order

The theorem of Hanani referred to in the title of this note states that, if a partially ordered set has cardinal not exceeding 1/2n(n+3) (n=1,2,...), then it can be expressed as the union ofn sets each of which is either a chain or an antichain. A corresponding result is now obtained for a set equipped with an arbitrary number of partial order relations.