Congruences,equational theories and lattice representations

We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday     

A strange pigeon-hole principle

Using Ramsey theory, we establish the following pigeon-hole type principle: From a large number of random variables (functions, vectors, etc.) one can always select two, X and Y, such that P(X < Y) 1/2. We apply the principle for a poset problem.     

On well-ordered mono-unary algebras

Given a linearly ordered set (A, _R ) and an R-monotone function f: AA, we give a necessary and sufficient condition on A, f, _R , involving generating sets and forbidden subalgebras, for _R to be a well-ordering.Partially supported by Hungarian National Foundation for Scientific Research Grant nr. 1813.     

Comparison theorems for weak splittings of bounded operators

Summary Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wonicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502     

Sphere orders

Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P _i } of ordered sets such that eachP _i is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769.     

The number of linear extensions of bipartite graphs

The number of linear extensions among the orientations of a bipartite graph is maximum just if the orientation itself is bipartite, the natural one.     

On the width of an orientation of a tree

There are 2 ^n-1 ways in which a tree on n vertices can be oriented. Each of these can be regarded as the (Hasse) diagram of a partially ordered set. The maximal and minimal widths of these posets are determined. The maximal width depends on the bipartition of the tree as a bipartite graph and it can be determined in time O(n). The minimal width is one of [/2] or [/2]+1, where is the number of leaves of the tree. An algorithm of execution time O(n + ² log ) to construct the minimal width orientation is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

A short proof that N 3 is not a circle containment order

A partially ordered set P is called a circle containment order provided one can assign to each xP a circle C _x so that . We show that the infinite three-dimensional poset N ³ is not a circle containment order and note that it is still unknown whether or not [n]³ is such an order for arbitrarily large n.     

Quantitative characterization of chaotic instabilities in semiconductors

A method for quantitative characterization of chaotic dynamical systems is discussed. An electronic instrument for determining the number of independent variablesk ^*, involved in the motion, is described. It allows one to obtain these in real time from a single observable. The suggested technique has been applied to quantification of strange attractors underlying chaotic instabilities in semi-insulating GaAsCr, and n-Ge, irradiated with high energy electrons. In n-Ge, for instance, the measured numbersk ^* range from 2 to 4 depending on control parameters. These measurements reveal the highly deterministic nature of the observed chaotic oscillations. The physical mechanisms responsible for the current instabilities and chaotic behaviour are discussed.