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.     

Fixed points of posets and clique graphs

The fixed point property for partial orders has been the object of much attention in the past twenty years. Recently, M. Roddy ([7]) proved this famous conjecture of Rival (see [6]): the class of finite orders with the fixed point property is closed under finite products.In this article, we prove that a finite order has the fixed point property if the sequence of iterated clique graphs of its comparability graph tends to the trivial graph.     

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.     

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.     

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.     

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.     

Tensor products of contexts and complete lattices

Although the categoryCLC of complete lattices and complete homomorphisms does not possess arbitrary coproducts, we show that the tensor product introduced by Wille has the universal property of coproducts for so-called distributing families of morphisms (and only for these). As every family of morphisms into a completely distributive lattice is distributing, this includes the known fact that in the category of completely distributive lattices, arbitrary coproducts exist and coincide with the tensor products. Since the definition of tensor products is based on the notion of contexts and their concept lattices, many results on tensor products extend from complete lattices to contexts. Thus we introduce two kinds of tensor products for arbitrary families of contexts, a partial and a complete one, and establish universal properties of these tensor products.Presented by B. Jonsson.     

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).     

Remote points in products under CH

Assume CH. Let I be any index set, and let Xi, for i∈I, be a completely regular ccc topological space of weight ω₂. If X=_∏i∈IXi is ccc and non-pseudocompact, then X has remote points.     

Tensor products for bounded posets revisited

The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.     

Reflexive domains and fixed points

In a recent paper in this journal, J. Soto-Andrade and F. J. Varela draw attention to the fact that ifR is a retract of a reflexive domain in a suitable category thenR has the fixed point property. They suggest [1], pp. 1 and 18, that conversely every structure with the fixed point property is a retract of a reflexive domain. In this note it is shown that ifR is a retract of a reflexive domain thenR ^R has the fixed point property. This leads to counterexamples to the suggestion of Soto-Andrade and Varela in the categoryPo of partially ordered sets and monotone maps.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

不动点与平衡点

介绍Brouwer不动点定理、Kakutani不动点定理与数理经济学中平衡点和博弈论中Nash平衡点存在性定理的等价性结果.     

On the automorphism conjecture for products of ordered sets

I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S ₁××S _n under the condition that max_i|S_i|=0(n/logn).     

Fixed point property and formal concept analysis

The purpose of this paper is to interpret, with the language of formal concept analysis, the fixed point free and order-preserving self-mappings of ordered sets as formal concepts of a context. With this interpretation one can derive a practicable algorithm for determining if a given finite ordered set has the fixed point property. As a side product it is proved that dismantlability of finite ordered sets can be tested in polynomial time.     

Tensor products of orthoalgebras

We define a tensor product via a universal mapping property on the class oforthoalgebras, which are both partial algebras and orthocomplemented posets. We show how to construct such a tensor product forunital orthoalgebras, and use the Fano plane to show that tensor products do not always exist.     

Cores and retracts

Theretracts (idempotent, isotone self-maps) of an ordered set are naturally ordered as functions. In this note we characterize the possible ways that one retract can cover another one. This gives some insight into the structure of the ordered set of retracts and leads to a natural generalization of the core of an ordered set.Supported by NSERC Operating Grant 41702.     

Sums, products and negations of contexts and complete lattices

In Formal Concept Analysis, one associates with every context its concept lattice , and conversely, with any complete lattice L the standard context L, constituted by the join-irreducible elements as ‘objects’, the meet-irreducible elements as ‘attributes’, and the incidence relation induced by the lattice order. We investigate the effect of the operators and on various (finite or infinite) sum and product constructions. The rules obtained confirm the ‘exponential’ behavior of and the ‘logarithmic’ behavior of with respect to cardinal operations but show a ‘linear’ behavior on ordinal sums. We use these results in order to establish several forms of De Morgan’s law for the lattice-theoretical negation operator, associating with any complete lattice the concept lattice of the complementary standard context. Received February 7, 2001; accepted in final form January 6, 2006.