Automorphisms and strongly invariant relations

Abstract

We investigate characterizations of the Galois connection {{,textrm{Aut},}}-{{,textrm{sInv},}} between sets of finitary relations on a base set A and their automorphisms. In particular, for A=omega _1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under {textrm{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.