Amalgamation property

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'circ f = g' 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' circ f = g' circ g

::and

:f ' [B] cap g ' [C] = (f ' circ f) [A] = (g ' 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

*
* 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).


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