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strong_amalgamation_property [2010/08/20 20:32] (current)
jipsen created
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 +=====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]]
 +