Amalgamation property

In the mathematical field of model theory, the amalgamation property is a property of collections of structures that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one.

An "amalgam" can be formally defined as a 5-tuple ("A,f,B,g,C") such that "A,B,C" are structures having the same signature, and "f: A" → "B, g": "A" → "C" are injective morphisms that are referred to as "embeddings".

A class "K" of structures has the amalgamation property if for every amalgam with "A,B,C" ∈ "K" and "A" ≠ Ø, there exist both a structure "D" ∈ "K" and embeddings "f':" "B" → "D, g':" "C" → "D" such that

:$f\text{'}circ\; f\; =\; g\text{'}\; circ\; g.$

trong amalgamation property

A class "K" of structures has the "strong amalgamation property" (SAP) if for every amalgam with "A,B,C" ∈ "K" there exist both a structure "D" ∈ "K" and embeddings "f':" "B" → "D, g': C" → "D" such that

:$f\text{'}\; circ\; f\; =\; g\text{'}\; circ\; g$

::and

:$f\; \text{'}\; [B]\; cap\; g\; \text{'}\; [C]\; =\; (f\; \text{'}\; circ\; f)\; [A]\; =\; (g\; \text{'}\; circ\; g)\; [A]$

::where for any set "X" and function "h" on "X,"

:$h\; lbrack\; X\; brack\; =\; lbrace\; h(x)\; mid\; x\; in\; X\; brace.$

ee also

* Age (model theory)

References

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* Entries on [http://math.chapman.edu/cgi-bin/structures.pl?Amalgamation_property amalgamation property] and [http://math.chapman.edu/cgi-bin/structures.pl?Strong_amalgamation_property strong amalgamation property] in [http://math.chapman.edu/cgi-bin/structures.pl online database of classes of algebraic structures] (Department of Mathematics and Computer Science, Chapman University).