Amalgamation property

Remarks

The structure $D$ is called an amalgam of $B$ and $C$ over $A$.

Example: the class of finite graphs has the amalgamation property. Given $f$ and $g$, take the disjoint union of $B$ and $C$, identify $f(a)$ with $g(a)$ for each $a \in A$, and add no new edges. Two vertices of $B$ are adjacent in the result exactly when they are adjacent in $B$, and the same holds for $C$. So the inclusions of $B$ and $C$ are embeddings, and they agree on $A$.

Non-example: the class of graphs with at most $2$ vertices fails the condition. Let $A$ be one vertex $a$, let $B$ be an edge $\set{a, b}$, and let $C$ be two non-adjacent vertices $a, c$. In any amalgam $D$, the images of $b$ and $c$ are distinct, since one is adjacent to the image of $a$ and the other is not. So $D$ has at least $3$ vertices and is not in the class.

When $A$ may be the empty structure, the amalgamation property implies the joint embedding property: any two members of $\mathcal{K}$ embed into a common member.