Definition 7.1. Let M ⊆ A and p(x) ∈ S(A). We say that p is an heir of p ↾ M or that p inherits from M if for every φ(x, y) ∈ L(M) if φ(x, a) ∈ p for some tuple a ∈ A, then φ(x, m) ∈ p for some tuple m ∈ M. We say that p is a coheir of p ↾ M or that p coinherits from M if p is finitely satisfiable in M. The same definitions apply to global types, i.e., to the case A = ℭ. These definitions also make sense for types in infinitely many variables.

Remark 7.2. tp(a/Mb) inherits from M if and only if tp(b/Ma) coinherits from M.

Proof. It is just a matter of writing down the definitions.

Lemma 7.3.

1. 1. If p(x) ∈ S(M), then p inherits and coinherits from M.

2. 2. If M ⊆ A and p(x) ∈ S(A) coinherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q coinherits from M.

3. 3. If M ⊆ A and p(x) ∈ S(A) inherits from M, then for every B ⊇ A there is some q(x) ∈ S(B) such that p ⊆ q and q inherits from M.