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Unformatted text preview: Math 5020 Handout 1 (corrected) Elementary Submodels and Extensions Definition Let L be a firstorder language and A , B be Lstructures. A function h :  A  →  B  is a homomorphism if (a) for every constant symbol c ∈ L , h ( c A ) = c B ; (b) for every nary function symbol f ∈ L and all a 1 ,...,a n ∈  A  , f B ( h ( a 1 ) ,...,h ( a n )) = h ( f A ( a 1 ,...,a n )); (c) for every nary relation symbol R ∈ L and all a 1 ,...,a n ∈  A  , if R A ( a 1 ,...,a n ) then R B ( h ( a 1 ) ,...,h ( a n )). h is an embedding if h is a homomorphism and for every nary relation symbol R ∈ L and all a 1 ,...,a n ∈  A  , R A ( a 1 ,...,a n ) ⇐⇒ R B ( h ( a 1 ) ,...,h ( a n )) . (Note that an embedding is always onetoone.) h is an isomorphism if h is an embedding and h is onto. Definition Let L be a firstorder language and A , B be Lstructures. (i) We say that A and B are isomorphic , and denote by A ∼ = B , if there exists an isomorphism from A to B ; (ii) We say that A is a substructure or a submodel of B , and denote by A ⊆ B , if  A  ⊆  B  and the identity map i :  A  →  B  given by i ( a ) = a for all a ∈  A  is an embedding....
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This note was uploaded on 04/11/2011 for the course MATH 5020 taught by Professor Staff during the Spring '11 term at North Texas.
 Spring '11
 staff
 Math, Set Theory

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