Definable and Invariant Types in Enrichments of Nip Theories

Abstract

Let T be an NIP L-theory andT be an enrichment. We give a sufficient condition onT for the underlying L-type of any definable (respectively invariant) type over a model ofT to be definable (respectively invariant). These results are then applied to Scanlon's model completion of valued differential fields.Let T be a theory in a language L and consider an expansion T ⊆T in a languageL. In this paper, we wish to study how invariance and definability of types in T relate to invariance and definability of types inT . More precisely, let U ⊧T be a monster model and consider some typep ∈ S(U) which is invariant over some small M ⊧T . Then the reduct p ofp to L is of course invariant under the action of theL-automorphisms of U that fix M (which we will denote asL(M )-invariant), but there is, in general, no reason for it to be L(M )-invariant. Similarly, ifp isL(M )-definable, p might not be L(M )-definable. When T is stable, and ϕ(x; y) is an L-formula, ϕ-types are definable by Boolean combinations of instances of ϕ. It follows that ifp isL(M )-invariant then p is both L(M )invariant and L(M )-definable. Nevertheless, when T is only assumed to be NIP, then this is not always the case. For example one can take T to be the theory of dense linear orders andL = {⩽, P (x)} where P (x) is a new unary predicate naming a convex non-definable subset of the universe. Then there is a definable type inT lying at some extremity of this convex set whose reduct to L = {⩽} is not definable without the predicate. In the first section of this paper, we give a sufficient condition (in the case where T is NIP) to ensure that anyL(M )-invariant (resp. definable) L-type p is also L(M )invariant (resp. definable). The condition is that there exists a model M ofT whose reduct to L is uniformly stably embedded in every elementary extension of itself. In the case where T is o-minimal for example, this happens whenever the ordering on M is complete. The main technical tool developed in this first section is the notion of external separability (Definition (1.2)). Two sets X and Y are said to be externally separable if there exists * Partially supported by ValCoMo (ANR-13-BS01-0006)

Weakly binary expansions of dense meet‐trees

“…However, using unary predicates is the only way to obtain such behaviour in a binary cone‐expansion of $$\mathsf {DMT}$$, as we are about to show. We refer the reader interested in preservation of invariance under reducts to [15].…”

Weakly binary expansions of dense meet-trees

“…However, using unary predicates is the only way to obtain such behaviour in a binary coneexpansion of DMT, as we are about to show. We refer the reader interested to preservation of invariance under reducts to [RS17].…”

Strong Density of Definable Types and Closed Ordered Differential Fields

“…Here we used the characterization of definable types in CODF given in Theorem 3.3. A substitute to Theorem 3.3 can be found in a recent paper of Rideau and Simon [RS17]. Let us recall the precise setting they are working in.…”

“…In [15,Corollary 1.7], the authors provide a sufficient condition onT implying that the underlying L-type of anyL eq (A)-definable type over N is A ∩ R-definable, where R denotes the set of all L-sorts. In the case of T = RCF andT = CODF, we get that such L-type is A ∩ N -definable.…”