Multifunctions and the fixed point property for products of ordered sets

Sufficient conditions for the fixed point property for products of two partially ordered sets are proved. These conditions are formulated in terms of multifunctions (functions with non-empty sets as values).     

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.     

The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order defined on P so that ab holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is -maximal, (ii) X is a -antichain, and (iii) X is -dominating; we call such a subset a -good subset of P. A -good subset is a retract of P and any two -good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a -good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the core of P (a -good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.Research partially supported by grants from the National Natural Science Foundation of China and The Natural Science Foundation of Shaanxi province.Research supported by NSERC grant #69-0982.     

On the strong fixed point property

An ordered set which has the fixed point property but not the strong fixed point property is presentedSupported by NSERC Operating Grant A7884.Supported by NSERC Operating Grant 41702.     

The strong fixed point property for lexicographic sums

The following theorem is proved: If Q=L{P _t tT} is a finite lexicographic sum of posets such that T and all P _t have the strong fixed point property then Q has the strong fixed point property. Moreover we show the strong fixed point property for two more classes of posets.     

The strong fixed point property for small sets

It is shown that every partially ordered set with the fixed point property and with ten or fewer elements actually has the strong fixed point property.Supported by NSERC Operating Grant A7884.Supported by NSERC Operating Grant 41702.     

The fixed point property for small sets

There exist exactly eleven (up to isomorphism and duality) ordered sets of size 10 with the fixed point property and containing no irreducible elements.The great part of the work presented here has been done when the author was visiting Ivan Rival at the University of Ottawa, Department of Computer Science.     

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.     

On the number of nondismantlable posets with the fixed point property

We give a lower bound for the number of orders on a finite set that are not dismantlable and have the fixed point property. To do so we also derive an upper bound for the number of orders on a finite set that are dismantlable.The author was partially funded by the Office of Naval Research under Contract N00014-88-K-0455.     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?     

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.     

Fixed point property for 11-element sets

We introduce retractable points and show how this notion provides the key for a classification of all sets with 11 elements that have the fixed point property.     

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.     

Generalizations of Tarski's fixed point theorem for order varieties of complete meet semilattices

A poset P is -conditionally complete ( a cardinal) if every set X P all of whose subsets of cardinality < have an upper bound has a least upper bound. For we characterize the subposets of a -complete poset which can occur as the set of fixed points of some montonic function on P. This yields a generalization of Tarski's fixed point theorem. We also show that for every the class of -conditionally complete posets forms an order variety and we exhibit a simple generating poset for each such class.Research supported in part by NSERC while the author was visiting Professor Ivo Rosenberg at the Université de Montreal.Research supported in part by NSF-grant DMS-8703743.     

Subsets of the square with the continuous and order preserving fixed point property

This study characterizes the convex sets whose complements in the unit square exhibit the fixed point property for mappings which are jointly continuous and order preserving. Hence, one can readily construct simple sets with this fixed point property, but which neither have the fixed point property individually for continuous mappings nor for order preserving mappings. This is the first characterization of any non-trivial set with this property.     

Brush spaces and the fixed point property

We introduce the notions of a brush space and a weak brush space. Each of these spaces has a compact connected core with attached connected fibers and may be either compact or non-compact. Many spaces, both in the Hausdorff non-metrizable setting and in the metric setting, have realizations as (weak) brush spaces. We show that these spaces have the fixed point property if and only if subspaces with core and finitely many fibers have the fixed point property. This result generalizes the fixed point result for generalized Alexandroff/Urysohn Squares in Hagopian and Marsh (2010) [4]. We also look at some familiar examples, with and without the fixed point property, from Bing (1969) [1], Connell (1959) [3], Knill (1967) [7] and note the brush space structures related to these examples.     

Strengthened fixed point property and products in ordered sets

Strengthened fixed point property for ordered sets is formulated. It is weaker than the strong fixed point property due to Duffus and Sauer and stronger than the product property meaning that A × Y has the fixed point property whenever A has the former and Y has the latter. In particular, doubly chain complete ordered sets with no infinite antichain have the strengthened fixed point property whenever they have the fixed point property, which yields a transparent proof of the well-known theorem saying that doubly chain complete ordered sets with no infinite antichain have the product property whenever they have the fixed point property. The new proof does not require the axiom of choice. Presented at the Summer School on General Algebra and Ordered Sets, Malá Morávka, 4–10 September 2005.     

Some fixed point results without monotone property in partially ordered metric-like spaces

The purpose of this paper is to obtain the fixed point results for F-type contractions which satisfies a weaker condition than the monotonicity of self-mapping of a partially ordered metric-like space. A fixed point result for F-expansive mapping is also proved. Therefore, several well known results are generalized. Some examples are included which illustrate the results.     

Fixed points and products

We prove that if the finite ordered setsP andX have the fixed point property then so too doesP×X.Supported by NSERC Operating Grant 41702.     

Dimension two,fixed points and dismantlable ordered sets

When does the fixed point property of a finite ordered set imply its dismantlability by irreducible elements? For instance, if it has width two. Although every finite ordered set is dismantlable by retractible (not necessarily irreducible) elements, surprisingly, a finite, dimension two ordered set, need not be dismantlable by irreducible elements. If, however, a finite ordered set with the fixed point property is N-free and of dimension two, then it is dismantlable by irreducibles. A curious consequence is that every finite, dimension two ordered set has a complete endomorphism spectrum.