Bernd S. W. Schröder

A Connection between Fixed-Point Theorems and Tiling Problems

The classic Banach Contraction Principle states that any contraction on a complete metric space has a unique fixed point. Rather than requiring that a single operator be a contraction, we consider a minimum involving a set of powers of that operator and derive fixed-point results. Ordinary analytical techniques would be extremely unwieldy, and so we develop a method for attacking this problem by considering a related problem on tiling the integers.

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.

Upper and Lower Bounds

Upper and lower bounds have already been defined in Definitions 2.1.4 and 2.3.1. From their use in Zorn’s Lemma, as well as their occurrences in the proofs of Dilworth’s Chain-Decomposition Theorem 2.5.7 and Proposition 2.6.7 (in both proofs, sets were defined in terms of their upper bounds), the reader can already infer that bounds of sets play an important role in ordered sets. In this chapter we consider various types of bounds and relate them to open problems as well as to each other.

The Copnumber of a Graph is Bounded by [3/2 genus (G)] + 3

We prove that the copnumber of a finite connected graph of genus g is bounded by [3/2g]+3. In particular this means that the copnumber of a toroidal graph is bounded by 4. We also sketch a proof that the copnumber of a graph of genus 2 is bounded by 5.

Algorithms for the fixed point property

This survey exhibits various algorithms to decide the question if a given ordered set P has the fixed point property resp. if P has a fixed point free order-preserving self-map. While a depth-first search algorithm for a fixed point free map is easily written it is also quite inefficient. We discuss a reduction algorithm by Xia which can be used to speed up the search for a fixed point free self-map. The ideas used in creating this algorithm show close connections to two problems: the decision whether an ordered set has a fixed point free automorphism and the decision whether a given r-partite graph has an r-clique. The latter two problems are shown to be NP-complete using the work of Goddard, Lubiw and Williamson. The problem to decide whether a given finite ordered set has a fixed point free order-preserving self-map has recently been shown to be NP-complete, thus showing that the above close connection is not by accident. Retraction theorems leading to dismantling algorithms are another approach to the problem. We present the classical dismantling procedure by Rival and extensions by Fofanova, Li, Milner, Rutkowski and the author. These theorems give a polynomial algorithm to decide if an ordered set has the fixed point property for some nice classes of ordered sets (height 1, width 2), and structural insights for other classes (chain-complete ordered sets with no infinite antichains, sets of (interval) dimension 2). The related issue of uniqueness of cores gives an insight into Birkhoffs problem regarding cancellation of exponents. Walkers relational fixed point property for which the analogous problem has a very satisfying solution also is discussed. Another variation on the retraction theme is the use of algebraic topology in deriving fixed point theorems initiated by Baclawski and Bjorner and continued for example by Constantin and Fournier. After a primer on the basic concepts of (integer) homology we present their retraction/dismantling procedures, which always prove acyclicity of the associated simplicial complex and then the fixed point property via a Lefschetz-type fixed point theorem. Differences and similarities with the above combinatorial procedures are pointed out. The main problem in making these results accessible to entirely combinatorial proofs is the lack of a combinatorial/discrete analogue of the continuous concept of a weak (resp. deformation) retract. We also present a class of ordered sets without the fixed point property, such that all proper retracts have the fixed point property. This class is induced via triangulations of n-spheres. Finally, we include an indication how methods developed for work on the fixed point property can be used in other areas. For example, the arguments by Abian, Brown and Pelczar to prove that in a chain-complete ordered set existence of a point P with P ⩽ f(P) implies existence of a fixed point have recently found application to analysis in the work of Carl, Heikkila, Lakhshmikantham and others.

Lagois connections: a counterpart to Galois connections

Abstract In this paper we define a Lagois connection, which is a generalization of a special type of Galois connection. We begin by introducing two examples of Lagois connections. We then recall the definition of Galois connection and some of its properties; next we define Lagois connection, establish some of its properties, and compare these with properties of Galois connections; and then we (further) develop examples of Lagois connections. Via these examples it is shown that, as is the case of Galois connections, there is a plethora of Lagois connections. Also it is shown that several fundamental situations in computer science and mathematics that cannot be interpreted in terms of Galois connections naturally fit into the theory of Lagois connections.

Isotone relations revisited

We propose a new approach towards proving that the fixed point property for ordered sets is preserved by products. This approach uses a characterization of fixed points in products via isotone relations. First explorations of classes of isotone relations are presented. These first explorations give us hope that this approach could lead to advances on the Product Problem.

Retractability and the fixed point property for products

LetP, Q be ordered sets and leta∈P. IfP \ {a} is a retract ofP and setsP and {x∈P: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.

Examples on Ordered Set Reconstruction

We prove that ordered sets are not reconstructible from the maximal deck and the minimal deck together. The construction also produces classes of more than two pairwise nonisomorphic ordered sets that have the same maximal deck and the same minimal deck.

Reconstruction of the Neighborhood Deck of an Ordered Set

We show that the neighborhood deck of an ordered set can be reconstructed from the deck of one point deleted subsets. As a consequence of the above results we reconstruct some maximal cards and present short new proofs of the reconstructibility of ordered sets of width 2 and of the recognizability of N-free ordered sets. We also reconstruct the maximal cards of N-free ordered sets.