Location: BH 1214

Reducts of Tree-like structures

**Abstract**

Given some first order structure **M**, a reduct of **M**
is a structure **N** on the same universe such that **N**
is definable in **M**. Informally, a reduct of a structure is
obtained by forgetting about some of the structure. Given this
notion, the general problem is to take a structure **M** and
classify all of its reducts up to interdefinability. This has
been demonstrated to be feasible in a number of cases where
**M** is ω-categorical and homogeneous in a finite
(often binary) language. All such classifications have been found
to confirm the conjecture of Simon Thomas, that every countably
infinite structure which is (ultra)homogeneous in a finite
relational language has only finitely many reducts up to
interdefinability. Don't worry I will define all these concepts
when they are introduced.

When **M** is an ω-categorical structure on a countably
infinite set **X**, the problem of classifiying the reducts of
**M** is the same as classifying the closed subgroups of
Sym(**X**) which contain Aut(**M**); considering the
natural Polish topology on Sym(**X**). So then we can consider
using what we know about infinite permutation groups to classify
reducts.

In this talk I will describe the classification of the reducts of a family of tree-like structures called semilinear orderings; these are partial orderings which are not linear, but for every element the elements below it are linearly ordered. My theorem concerns those countably infinite semilinear orderings which were called 2-homogenous, and described up to isomorphism, by Manfred Droste. Their automorphism groups are examples of Jordan groups, so I will explain what that means and how my proof makes use of many results from the study of infinite, primitive Jordan groups.