Automorphisms and strongly invariant relations
Ferdinand Börner, Martin Goldstern, Saharon Shelah · Algebra Universalis · 2023
Abstract We investigate characterizations of the Galois connection $${{\,\textrm{Aut}\,}}$$ Aut - $${{\,\textrm{sInv}\,}}$$ sInv between sets of finitary relations on a base set A and their automorphisms. In particular, for $$A=\omega _1$$ A = ω 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 $${\textrm{sInv Aut}}$$ sInv Aut . Our structure (A, R) has an $$\omega $$ ω -categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.