The 116 reducts of (ℚ, <,a)

Abstract

This article aims to classify those reducts of expansions of (ℚ, <) by unary predicates which eliminate quantifiers, and in particular to show that, up to interdefinability, there are only finitely many for a given language. Equivalently, we wish to classify the closed subgroups of Sym(ℚ) containing the group of all automorphisms of (ℚ, <) fixing setwise certain subsets. This goal is achieved for expansions by convex predicates, yielding expansions by constants as a special case, and for the expansion by a dense, co-dense predicate. Partial results are obtained in the general setting of several dense predicates.