# Algebra Seminar talk

2021-05-21

Martin **Goldstern***Automorphisms and strongly invariant relations*

Abstract:

A bijection f of A strongly respects a finitary relation rho if f is an automorphism of the structure (A,rho). For a set R of relations, we write Aut(R) for the set of bijections strongly respecting all rho in R, and for a set G of permutations of A we write sInv(G) for the set of all relations strongly respected by all g in G.

We investigate characterizations of the Galois connection sInv-Aut.

In particular, for A=omega1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitray intersections, but is not closed under sInv(Aut(-)).

Our structure (A,R) has an omega-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.

This is joint work with Ferdinand Börner and Saharon Shelah. A preprint is available at arXiv