I came accross the following
Theorem: If $A$ is an $\aleph_0$- categorical structure, then the algebraically closed substructures of $A$ satisfy the strong amalgamation principle. (for definitions look at the end).
My questions are:
(1) If $T$ is an $\aleph_0$- categorical theory and $B$ is a model of $T$, not necessarily countable, do the algebraically closed substructures of $B$ satisfy strong amalgamation?
If $B_0,B_1,B_2$ are substructures of $B$ such that $B_0\subset B_1,B_2$, I am interested in particular in the case that $B_0,B_1$ (not $B_2$) are finitely generated.
(2) The terms disjoint and strong amalgamation, do they refer to the same property?
(3) Does anyone know a reference to the above theorem?
Definitions: If $A$ is a structure and $A_0\subset A$, $A_0$ is $algebraically\;\; closed$ if every finite set $B$ that is definable with parameters from $A_0$ is a subset of $A_0$.
If $A_0\subset A_1,A_2$, the triple $(A_0,A_1,A_2)$ have the $strong\; amalgamation \; property$, if there is a structure $A_3$ and embeddings $f:A_1\rightarrow A_3$ and $g:A_2\rightarrow A_3$ such that $f[x]=g[x]$, for all $x\in A_0$ and $f[A_1]\cap g[A_2]=f[A_0]$.