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| [[Image:Amalgamation property.svg|thumb|200px|alt=Amalgamation Property commutative diagram|A [[commutative diagram]] of the amalgamation property.]]
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| In the mathematical field of [[model theory]], the '''amalgamation property''' is a property of collections of [[structure (mathematical logic)|structures]] that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one.
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| This property plays a crucial role in [[Fraïssé's theorem]] which characterises classes of finite structures which arise as
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| [[Age_(model_theory)|age]]s of countable homogeneous structures.
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| The diagram of the amalgamation property appears in many areas of mathematical logic. Examples include in [[modal logic]] as an incestual accessibility relation, and in [[lambda calculus]] as a manner of reduction having the [[Church–Rosser property]].
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| ==Definition==
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| 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 (logic) |signature]], and ''f: A'' → ''B, g'': ''A'' → ''C'' are injective morphisms that are referred to as ''embeddings''.
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| 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
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| :<math>f'\circ f = g' \circ g. \, </math>
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| ==Examples==
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| * The class of sets where the embeddings are injective functions and the amalgam is simply the union of the two sets.
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| * The class of [[free group]]s where the embeddings are injective homomorphisms and the amalgam is the [[quotient group]] <math>B*C/A</math>, where * is the [[free product]].
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| * The class of finite [[linear order]]ings.
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| A similar but different notion to the amalgamation property is the [[joint embedding property]]. To see the difference, first consider the class ''K'' (or simply the set) containing three models with linear orders, ''L''<sub>1</sub> of size one, ''L''<sub>2</sub> of size two, and ''L''<sub>3</sub> of size three. This class ''K'' has the joint embedding property because all three models can be embedded into ''L''<sub>3</sub>. However, ''K'' does not have the amalgamation property. The counterexample for this starts with ''L''<sub>1</sub> containing a single element ''e'' and extends in two different ways to ''L''<sub>3</sub>, one in which ''e'' is the smallest and the other in which ''e'' is the largest. Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of ''e''.
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| Now consider the class of [[algebraically closed field]]s. This class has the amalgamation property since any two field extensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embedded into a common field when the [[Characteristic (algebra)|characteristic]] of the fields differ.
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| ==Strong amalgamation property==
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| 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<nowiki>':</nowiki>'' ''B'' → ''D, g': C'' → ''D'' such that
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| :<math>f' \circ f = g' \circ g \, </math>
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| ::and
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| :<math>f '[B] \cap g '[C] = (f ' \circ f)[A] = (g ' \circ g)[A] \, </math>
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| ::where for any set ''X'' and function ''h'' on ''X,''
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| :<math>h \lbrack X \rbrack = \lbrace h(x) \mid x \in X \rbrace. \, </math>
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| ==See also==
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| * [[Span (category theory)]]
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| * [[Pushout (category theory)]]
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| * [[Joint embedding property]]
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| * [[Fraïssé%27s theorem]]
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| ==References==
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| * {{ cite book | last=Hodges | first=Wilfrid | authorlink=Wilfrid Hodges | publisher=[[Cambridge University Press]] | title=A shorter model theory | year=1997 | isbn=0-521-58713-1 }}
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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).
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| [[Category:Model theory]]
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