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# 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 contained there.

Defines:

amalgamation property

Related:

FreeProductWithAmalgamatedSubgroup, Confluence, JointEmbeddingProperty

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03C52*no label found*

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## Corrections

Is this the same as pushout? by Dr_Absentius ✓

and and by yark ✓

would be nice by CWoo ✓

Linking by rm50 ✓

and and by yark ✓

would be nice by CWoo ✓

Linking by rm50 ✓