# Algebra Seminar talk

2019-03-29

Norbert **Sauer***Copies and equimorphic structures*

Abstract:

A copy of a relational structure A is a subset of its domain which induces a structure isomorphic to A.

The domain of the hypergraph of copies of A is the domain of A and the edges of the hypergraph of copies of A are the copies of A. In the case of countable structures properties of this hyper graph will be discussed and related to the conjectures of Bonato-Tardif and Thomassé concerning the number of twins of trees and countable structures. Two structures are twins if they are equimorphic but not isomorphic.

Theorems for the number of twins of scattered trees and ordered structures and aleph-0 categorical structures will be discussed. (Note that the special case of structures with only the equality relation is covered by the Cantor-Schröder-Bernstein theorem.)