handout01

# handout01 - Math 5020 Handout 1(corrected Elementary...

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Unformatted text preview: Math 5020 Handout 1 (corrected) Elementary Submodels and Extensions Definition Let L be a first-order language and A , B be L-structures. A function h : | A | → | B | is a homomorphism if (a) for every constant symbol c ∈ L , h ( c A ) = c B ; (b) for every n-ary 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 n-ary 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 n-ary 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 one-to-one.) h is an isomorphism if h is an embedding and h is onto. Definition Let L be a first-order language and A , B be L-structures. (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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handout01 - Math 5020 Handout 1(corrected Elementary...

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