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.