## Amalgamation Property

An \emph{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 \emph{amalgamation property} 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 $A\neq\emptyset$ 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 $f '\circ f=g'\circ g$.