Reducts of Tree-like structures

David Bradley-Williams (UCLAN)

ATB G.205,

Given some first order structure M, a reduct is a structure N on the same universe which 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 omega-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.

When M is an omega-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 my 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.

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