On lovely pairs of geometric structures

Abstract

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We characterize linear" theories in terms of properties of the corresponding theory of the lovely pair. For o-minimal theories, we use Peterzil-Starchenko's trichotomy theorem to characterize for suciently general points, the local geometry around it in terms of the thorn U rank of its type.

NIP for some pair-like theories

“…So condition (iii) is satisfied. Theorem 3.2 is used in [2] to show that if T has NIP and is a geometric theory and T P is the theory of lovely pairs of models of T , as defined in [4], then T P has NIP. This provides an interesting class of examples to which Theorem 2.1 applies even when condition (iii) is replaced by (iii) .…”

Geometric structures with a dense independent subset

“…It is shown in [3] that when T is geometric, weakly 1-based, and ω-categorical, then the associated theory T P of lovely pairs is also ω-categorical. This is not the case for the associated theory T ind :…”

“…In this section we will only use the definition of lovely pairs. More information on lovely pairs of geometric structures can be found in [3]. Proof.…”

“…If (M, N ) is a lovely pair of models of T , we can study (M, N ) as an L P structure by interpreting P as N . In [3] it is shown that if (M, N ) and (M ′ , N ′ ) are lovely pairs of models of T , then…”

“…Another such expansion corresponds to lovely pairs [6,3]. Let L be the language of T , let M |= T and let H be a new unary predicate that does not belong to L. For M |= T , we say that (M, H(M )) is a lovely pair if H(M ) is an elementary substructure of M , the predicate satisfies the density property (for any infinite Lformula ϕ(x, b) with parameters in M , ϕ(H(M )) = ∅) and M satisfies the extension property over H(M ) (for any infinite L-formula ϕ(x) with parameters in M , (ϕ(M )\ acl(H(M ) b)) = ∅).…”