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.     

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.     

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.     

A second note on the equational theory of modular ortholattices

We show how to alter the material in [4] to prove that every variety of modular ortholattices is generated by its simple members. Supported by NSERC Operating Grant 0041702.     

A local trace formula

Supported in part by NSERC Operating Grant A3483.     

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2 ^2n/3 order preserving maps (and 2 ² in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.Supported by ONR Contract N00014-85-K-0769.Supported by NSF Grant DMS-9011850.Supported by NSERC Grants 69-3378 and 69-0259.     

The characters of supercuspidal representations as weighted orbital integrals

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on ap-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character. Supported in part by NSERC Operating Grant A3483.     

Affine isoparametric hypersurfaces

Research supported by an NSERC Postdoctoral Fellowship and NSERC Operating Grant OGP0002501     

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

Hirsch induction and torsion units in group rings

Supported by Canadian NSERC Grant A-5300 and Polish Scientific Grant 211399101.     

In- and circumcenters of manifolds of constant width

Supported by a Canadian NSERC Grant.     

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.     

Ordered sets,pagenumbers and planarity

General bounds for the pagenumber of an ordered set are developed. One bound is derived by first showing that the maximum number of edges in the diagram of a planar ordered setP is 2v(P)-2-ht(P). A construction is given to show that the pagenumber of the product ofn chains is no more than 2n-2. Lastly, some open questions are discussed.Supported in part by NSERC of Canada.Supported by an NSERC Summer Scholarship.     

An elementary proof of the local central limit theorem

We give an elementary proof of the local central limit theorem for independent, non-identically distributed, integer valued and vector valued random variables.Support by a NSF Grant.Supported by an NSERC Grant     

The divergence theorem and the Laplacian in Minkowski space

Let (X, B) be a Minkowski space (finite-dimensional Banach space) with unit ball B. Using a Minkowski definition of unit normal to a hypersurface, a Minkowski analogue of Euclidean divergence is defined. We show that the divergence theorem holds. Using the Minkowski divergence, a Minkowski Laplacian is defined. We prove that this Laplacian is a second-order, constant-coefficient, elliptic, differential operator. Furthermore, the symbol of this Laplacian is computed and used to associate a natural Euclidean structure with (X, B).Supported, in part, by NSERC Operating Grant #4066.     

Finite axiomatizations for existentially closed posets and semilattices

In this paper we exhibit axiomatizations for the theories of existentially closed posets and existentially closed semilattices. We do this by considering an infinite axiomatization which characterizes these structures in terms of embeddings of finite substructures, an axiomatization which exists for any locally finite universal class with a finite language and with the joint embedding and amalgamation properties. We then find particular finite subsets of these axioms which suffice to axiomatize both classes. Research supported by an NSERC Postdoctoral Fellowship. Research supported by NSERC Grant No. A7256.     

Forbidden retracts for finite ordered sets of width at most four

The relationship between the fixed point property and forbidden retracts associated with a forgetful functor is formulated. Finite ordered sets of width at most four with fixed point free automorphisms are described. Linear time algorithms for deciding whether a finite ordered set of width two has the fixed point property and whether a finite ordered set of width at most three has a fixed point free automorphism are presented.     

Automorphisms and isotone self-maps of ordered sets with top and bottom

For an ordered setP letP ^P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP ^P . We establish an inequality relating ¦P ^P ¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship.     

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.