Ohkuma Structures
by
Charles Holland
Bowling Green State University

We define an Ohkuma structure as a relational structure whose automorphism group acts uniquely transitively on the points of the structure. More than 40 years ago, Ohkuma studied totally ordered sets (chains) whose automorphism groups act uniquely transitively, that is, for each x, y in the chain there is exactly one automorphism of the chain taking x to y. The obvious example is the chain of integers. But Ohkuma constructed others and proved that every such is isomorphic to a subset (indeed, a subgroup) of the reals. Ohkuma chains have turned out to be important in the study of lattice-ordered groups.

We investigate Ohkuma betweenness chains, Ohkuma circles, Ohkuma betweenness circles, and Ohkuma graphs. All of these are related in some way to Ohkuma chains, but there are interesting differences. We will completely characterize Ohkuma cirlces and almost characterize Ohkuma betweenness chains and and Ohkuma betweenness circles.

As for Ohkuma graphs, the investigation is still in the exploratory stages. Here is a challenge for the readers of this abstract: It is easy to see that every graph with no more than 2 vertices is Ohkuma. Now, find another one.

Date received: February 6, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caed-05.