Heirs and externally definable subsets of polynomially bounded structures

Let M ≺ N be L-structures and let p ∈ Sn(M) be an n-type. An ex-tension q ∈ Sn(N) of p is called an heir of p if the following holds: For every L-formula ϕ(x1,..., xn, ȳ) with parameters in M, IF there is some β ̄ ∈ N y ̄ with ϕ(x ̄ , β̄) ∈ q, THEN there is some b ̄ ∈ M y ̄ with ϕ(x ̄ , b̄) ∈ p. Definition (externally definable) Let M be an L-structure. A subset S of Md is called externally definable if there is some M ′ M and a subset S ′ of (M ′)d, definable in M ′ with parameters from M ′ such that S = S ′ ∩Md. 2. An example Let R be an o-minimal expansion of the real field and let M be the set of germs of R-definable functions R − → R at +∞. Then M R and the set V = {a ∈ M | there is r ∈ R with |a | < r} ⊆ M of “finite elements ” of M is externally definable: