# nLab amalgamation property

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Idea

A class of structures in model theory has the amalgamation property if for any three structures $A,B,C$ and embeddings $f_B: A\hookrightarrow B$, $f_C: A\hookrightarrow C$, there exist embeddings $g_B:B\hookrightarrow D$ and $g_C: C\hookrightarrow D$ such that $g_B\circ f_B = g_C\circ f_C$.

One of the simplest cases is when the free amalgam of structures $B\oplus_A C$ exists.

## References

The amalgamation method for generating strongly minimal theories is introduced in

• Ehud Hrushovski, A new strongly minimal set. Ann. Pure Appl. Logic,

62(2):147–166, 1993. Stability in model theory, III (Trento, 1991).

Recent references include

• Ehud Hrushovski, Groupoids, imaginaries and internal covers, arxiv/math.LO/0603413
• Uri Andrews, Amalgamation constructions and recursive model theory, thesis, pdf
• John T. Baldwin, Alexei Kolesnikov, Saharon Shelah, The amalgamation spectrum, J. Symbolic Logic 74:3 (2009) 914-928, MR2548468, doi
• J D Brody, Model theory of graphs, thesis, pdf

Last revised on August 9, 2016 at 15:46:54. See the history of this page for a list of all contributions to it.