Michael S. Roddy

Completions of orthomodular lattices II

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.

Fixed points and products

We prove that if the finite ordered setsP andX have the fixed point property then so too doesP×X.

A geometric description of modular lattices

Baer [1] observed that modular lattices of finite length (for example subgroup lattices of abelian groups) can be conceived as subspace lattices of a projective geometry structure on an ordered point set; the set of join irreducibles which in this case are the cyclic subgroups of prime power order. That modular lattices of finite length can be recaptured from the order on the points and, in addition, the incidence of points with ‘lines’, the joins of two points, or the blocks of collinear points has been elaborated by Kurinnoi [18] , Faigle and Herrmann [7], Benson and Conway [2] , and , in the general framework of the ‘core’ of a lattice, by Duquenne [5]. In [7] an axiomatization in terms of point-line incidence has been given. Here, we consider, more generally, modular lattices in which every element is the join of completely join irreducible ‘points’. We prove the isomorphy of an algebraic lattice of this kind and the associated subspace lattice and give a first order characterization of the associated ‘ordered spaces’ in terms of collinearity and order which appears more natural and powerful. The crucial axioms are a ‘triangle axiom’, which includes the degenerate cases, and a strengthened ‘line regularity axiom’, both derived from [7]. As a consequence, using Skolemization, we get that any variety of modular lattices is generated by subspace latices of countable spaces. The central concept, connecting the geometric structure and the lattice structure, is that of a line interval (p + q)/(p + q) where p and q are points and p, q their unique lower covers. Given a line interval, any choice of one incident point per atom of the line interval produces a line of the space.

On the strong fixed point property

An ordered set which has the fixed point property but not the strong fixed point property is presented

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.

Fixed Points and Products: Width 3

If P and X are ordered sets with the fixed point property, does P×X have the fixed point property? In case one of P and X is finite the answer is yes. Here we answer the question affirmatively when P has width at most three.

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.

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.

On an Example of Rutkowski and Schröder

An ultrafilter construction is used to show that a certain infinite product of ordered sets with FPP, while connected, does not itself have FPP.

An orthomodular lattice

An orthomodular lattice is constructed by taking a homomorphic image of the ortholattice obtained from a certain orthogonality relation on an infinite set.