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| {{ about|[[mathematics]]|substructures of [[buildings]] in [[civil engineering]]|Superstructure#Engineering }}
| | 49 yrs old Osteopath Courtney Hyslop from Port Coquitlam, has pastimes for instance bonsai trees, property developers in [http://www.mchostingplus.com/forums/index.php/269789-new-launch-rental-2013 singapore apartment for sale] and building. Likes to discover unfamiliar towns and locales like Route of Santiago de Compostela. |
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| In [[mathematical logic]], an '''(induced) substructure''' or '''(induced) subalgebra''' is a [[structure (mathematical logic)|structure]] whose domain is a [[subset]] of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure. Some examples of subalgebras are [[subgroup]]s, [[submonoid]]s, [[subring]]s, [[subfield]]s, subalgebras of [[algebra over a field|algebras over a field]], or induced [[Glossary of graph theory#Subgraphs|subgraphs]]. Shifting the point of view, the larger structure is called an '''extension''' or a '''superstructure''' of its substructure.
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| In [[model theory]], the term '''"submodel"''' is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models.
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| In the presence of relations (i.e. for structures such as [[ordered group]]s or [[graph (mathematics)|graph]]s, whose [[signature (logic)|signature]] is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a '''weak substructure''' (or '''weak subalgebra''') are ''at most'' those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. [[Ordered group]]s, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure.
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| == Definition ==
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| Given two [[structure (mathematical logic)|structures]] ''A'' and ''B'' of the same [[signature (logic)|signature]] σ, ''A'' is said to be a '''weak substructure''' of ''B'', or a '''weak subalgebra''' of ''B'', if
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| * the domain of ''A'' is a subset of the domain of ''B'',
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| * ''f <sup>A</sup>'' = ''f <sup>B</sup>'' | ''A<sup>n</sup>'' for every ''n''-ary function symbol ''f'' in σ, and
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| * ''R <sup>A</sup>'' <math>\subseteq</math> ''R <sup>B</sup>'' <math>\cap</math> ''A<sup>n</sup>'' for every ''n''-ary relation symbol ''R'' in σ.
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| ''A'' is said to be a ''' substructure''' of ''B'', or a ''' subalgebra''' of ''B'', if ''A'' is a weak subalgebra of ''B'' and, moreover,
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| * ''R <sup>A</sup>'' = ''R <sup>B</sup>'' <math>\cap</math> ''A<sup>n</sup>'' for every ''n''-ary relation symbol ''R'' in σ.
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| If ''A'' is a substructure of ''B'', then ''B'' is called a '''superstructure''' of ''A'' or, especially if ''A'' is an induced substructure, an '''extension''' of ''A''.
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| == Example ==
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| In the language consisting of the binary functions + and ×, binary relation <, and constants 0 and 1, the structure ('''Q''', +, ×, <, 0, 1) is a substructure of ('''R''', +, ×, <, 0, 1). More generally, the substructures of an [[ordered field]] (or just a [[field (mathematics)|field]]) are precisely its subfields. Similarly, in the language (×, <sup>−1</sup>, 1) of groups, the substructures of a [[Group (mathematics)|group]] are its [[subgroup]]s. In the language (×, 1) of monoids, however, the substructures of a group are its [[submonoid]]s. They need not be groups; and even if they are groups, they need not be subgroups.
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| In the case of [[graph (mathematics)|graph]]s (in the signature consisting of one binary relation), [[Glossary of graph theory#Subgraphs|subgraphs]], and its weak substructures are precisely its subgraphs.
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| == Substructures as subobjects ==
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| For every signature σ, induced substructures of σ-structures are the [[subobject]]s in the [[concrete category]] of σ-structures and [[structure (mathematical logic)#Homomorphisms|strong homomorphisms]] (and also in the [[concrete category]] of σ-structures and σ-[[structure (mathematical logic)#Homomorphisms|embeddings]]). Weak substructures of σ-structures are the [[subobject]]s in the [[concrete category]] of σ-structures and [[structure (mathematical logic)#Homomorphisms|homomorphisms]] in the ordinary sense.
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| == Submodel ==
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| In model theory, given a structure ''M'' which is a model of a theory ''T'', a '''submodel''' of ''M'' in a narrower sense is a substructure of ''M'' which is also a model of ''T''. For example if ''T'' is the theory of abelian groups in the signature (+, 0), then the submodels of the group of integers ('''Z''', +, 0) are the substructures which are also groups. Thus the natural numbers ('''N''', +, 0) form a substructure of ('''Z''', +, 0) which is not a submodel, while the even numbers (2'''Z''', +, 0) form a submodel which is (a group but) not a subgroup.
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| Other examples:
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| # The [[algebraic numbers]] form a submodel of the [[complex numbers]] in the theory of [[algebraically closed field]]s.
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| # The [[rational numbers]] form a submodel of the [[real numbers]] in the theory of [[field (mathematics)|field]]s.
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| # Every [[elementary substructure]] of a model of a theory ''T'' also satisfies ''T''; hence it is a submodel.
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| In the [[category (mathematics)|category]] of models of a theory and [[embedding]]s between them, the submodels of a model are its [[subobject]]s.
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| == See also ==
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| * [[Elementary substructure]]
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| * [[End extension]]
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| * [[Löwenheim-Skolem theorem]]
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| * [[Prime model]]
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| == References ==
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| * {{Citation | last1=Burris | first1=Stanley N. | last2=Sankappanavar | first2=H. P. | title=A Course in Universal Algebra | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1981}}
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| * {{Citation | last1=Diestel | first1=Reinhard | title=Graph Theory | origyear=1997 | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Graduate Texts in Mathematics | isbn=978-3-540-26183-4 | year=2005 | volume=173}}
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| * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=[[Cambridge University Press]] | location=Cambridge | isbn=978-0-521-58713-6 | year=1997}}
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| [[Category:Model theory]]
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| [[Category:Universal algebra]]
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49 yrs old Osteopath Courtney Hyslop from Port Coquitlam, has pastimes for instance bonsai trees, property developers in singapore apartment for sale and building. Likes to discover unfamiliar towns and locales like Route of Santiago de Compostela.