Differences

This shows you the differences between two versions of the page.

strong_amalgamation_property [2010/08/20 20:32] (current)
jipsen created
Line 1: Line 1:
+=====Strong 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{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 ===
+[[Amalgamation property]] and [[Epimorphisms are surjective]]
+
+[[Superamalgamation property]]
+
+=== Properties implied by the SAP ===
+[[Amalgamation property]]
+