Strong Amalgamation Property

An amalgam is a tuple $\langle \mathbf{A},f,\mathbf{B},g,\mathbf{C}\rangle$ such that $\mathbf{A},\mathbf{B},\mathbf{C}$ are structures of the same signature, and $f:\mathbf{A}\to\mathbf{B}$, $g:\mathbf{A}\to\mathbf{C}$ are embeddings (injective morphisms).

A class $\mathcal{K}$ of structures is said to have the strong amalgamation property, or SAP for short, if for every amalgam $\langle \mathbf{A},f,\mathbf{B},g,\mathbf{C}\rangle$ with $\mathbf{A},\mathbf{B},\mathbf{C}\in\mathcal{K}$ and $A\ne\emptyset$ there exists a structure $\mathbf{D}\in\mathcal{K}$ and embeddings $f ':\mathbf{B}\to\mathbf{D}$, $g':\mathbf{C}\to\mathbf{D}$ such that $f '\circ f=g'\circ g$ and $\mbox{Im}(f ')\cap \mbox{Im}(g')=\mbox{Im}(f'\circ f)$, where $\mbox{Im}(f ')=\{f '(x) | x\in B\}$.

Properties that imply the SAP

Properties implied by the SAP