The automorphism group of the universal homogeneous countable semilattice
by
Dietrich Kuske
Institut für Algebra, TU Dresden, D-01062 Dresden, Germany
Coauthors: Manfred Droste (TU Dresden), John K. Truss (University of Leeds)

Using general techinques developed by Fraisse, one can prove the unique existence of a countable semilattice with the following properties: (1) it embeds any countable semilattice (universality), and (2) any isomorphism of finite subsemilattices can be extended to an automorphism of the whole structure (homogeneity). It turns out that this universal homogeneous countable semilattice is join-distributive. Similar universal homogeneous structures are known for other classes, in particular for partial orders and distributive lattices. Glass, McCleary and Rubin showed that the universal homogeneous countable partial order has a simple automorphism group. Using similar proof techniques, we prove that the universal homogeneous countable semilattice has a largest nontrivial normal subgroup. By Droste and Macpherson, the universal homogeneous countable distributive lattice has exactly three normal subgroups. Since the semilattice can be seen as a structure midway between the partial order and the distributive lattice, we conjecture that the normal subgroup we found is not only the largest one, but that there are no further normal subgroups.

http://www.math.tu-dresden.de/~kuske/index.html

Date received: May 29, 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 # caee-82.