# 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 ${f}_{i}:A\to {B}_{i}$ are elementary embeddings
for $i\in \{1,2\}$ then there is some $C\in S$ and some elementary embeddings
${g}_{i}:{B}_{i}\to C$ 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.

$$\text{xymatrix}\mathrm{\&}A\text{ar}{[dl]}_{{f}_{1}}\text{ar}{[dr]}^{{f}_{2}}\mathrm{\&}{B}_{1}\text{ar}{[dr]}_{{g}_{1}}\mathrm{\&}\mathrm{\&}{B}_{2}\text{ar}{[dl]}^{{g}_{2}}\mathrm{\&}C\mathrm{\&}$$ |

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

Title | amalgamation property |
---|---|

Canonical name | AmalgamationProperty |

Date of creation | 2013-03-22 13:25:01 |

Last modified on | 2013-03-22 13:25:01 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 8 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 03C52 |

Related topic | FreeProductWithAmalgamatedSubgroup |

Related topic | Confluence |

Related topic | JointEmbeddingProperty |

Defines | amalgamation property |