# Strong amalgamation property

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Changed: 12,16c12,13
 A,B,C ∈ K and A ≠ Ø there exists a structure D ∈ K and embeddings f ' : B → D, g' : C → D such that f 'of = g'og and Im(f ')∩Im(g') = Ø, where Im(f ') = {f '(x) | x ∈ B}.
 A,B,C ∈ K there exists a structure D ∈ K and embeddings f ' : B → D, g' : C → D such that f 'of = g 'og and f '[B]∩g '[C] = (f ' of)[A] = (g ' og)[A], where for any set X and function h on X, h[X] = {h(x) | x ∈ X}.

Removed: 21,22d17
 I don't understand: if f 'of = g'og, how can it be Im(f ')∩Im(g') = Ø? It seems to me that it should be f' of (\mathbf A)  ⊆ Im(f ')∩Im(g'). But perhaps I missed something. Franco Montagna

### Definition

An amalgam is a tuple (A,f,B,g,C) such that A,B,C are structures of the same signature, and f : A → B, g : A → C are embeddings (injective morphisms).

A class K of structures is said to have the strong amalgamation property, or SAP for short, if for every amalgam with A,B,C ∈ K there exists a structure D ∈ K and embeddings f ' : B → D, g' : C → D such that f 'of = g 'og and f '[B]∩g '[C] = (f ' of)[A] = (g ' og)[A], where for any set X and function h on X, h[X] = {h(x) | x ∈ X}.

### Properties that imply the SAP

Amalgamation property  and  [Epimorphism are surjective]? [Superamalgamation property]?

### Properties implied by the SAP

Amalgamation property

[Generate list of all classes that mention the strong amalgamation property together with it's value in that class (under construction)].

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