# amalgamation property

A class of $L$-structures $S$ has the amalgamation property if and only if whenever $A,B_{1},B_{2}\in S$ and $f_{i}:A\rightarrow B_{i}$ are elementary embeddings for $i\in\{1,2\}$ then there is some $C\in S$ and some elementary embeddings $g_{i}:B_{i}\rightarrow C$ for $i\in\{1,2\}$ so that $g_{1}(f_{1}(x))=g_{2}(f_{2}(x))$ for all $x\in A$. That is, the following diagram commutes.

 $\xymatrix{&{A}\ar[dl]_{f_{1}}\ar[dr]^{f_{2}}&\\ {B_{1}}\ar[dr]_{g_{1}}&&{B_{2}}\ar[dl]^{g_{2}}\\ &{C}&}$

Compare this with the free product with amalgamated subgroup for groups and the definition of pushout there.

Title amalgamation property AmalgamationProperty 2013-03-22 13:25:01 2013-03-22 13:25:01 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 03C52 FreeProductWithAmalgamatedSubgroup Confluence JointEmbeddingProperty amalgamation property