hw1570 - A B are finitely generated substructures of M and...

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MATH 570: MATHEMATICAL LOGIC HOMEWORK 1 Due on Wednesday, Sep 3 1. Define appropriate signatures for (a) vector spaces over Q ; (b) metric spaces. 2. (a) Show by an example that in the signature τ group = ( 1 , ) , a substructure of a group need not be a subgroup. (b) Define a signature for groups different from the one above so that a substructure of a group is a subgroup. 3. If h A B is a τ -homomorphism then the image h ( A ) is a universe of a substructure of B . 4. Prove that if τ does not contain any relation symbols, then any bijective τ -homomorphism is a τ -isomorphism. (That’s why this happens with groups and rings, but not with graphs or orderings.) 5. A structure is called rigid if it has no automorphisms 1 other than the identity. Show that the structures N = ( N , 0 ,S, + , ) and Q = ( Q , 0 , 1 , + , ) are rigid. 6. A structure M is called ultrahomogeneous if given any isomorphism between two finitely generated substructures, it extends to an automorphism of the whole structure, i.e.
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Unformatted text preview: A , B are finitely generated substructures of M and h ∶ A → B is an isomorphism, then there is an automorphism h of M with h ± h . Show that ( Q , <) is ultrahomogeneous. The same proof should also work to show that ( R , <) is ultrahomogeneous. 7. For a signature τ , a τ-sentence is a τ-formula that does not have free variables. For τ-structures A , B , we write A ≡ B if for every τ-sentence φ , A ² φ ⇐⇒ B ² φ . (a) Show that there is a τ group-sentence φ such that M ² φ ⇐⇒ M ³ Z ´ 2 Z . (b) More generally, let τ be a finite signature and A be a finite τ-structure. Show that there is a τ-sentence φ such that for any τ-structure B , B ² φ ⇐⇒ B ³ A . In particular, B ≡ A ⇐⇒ B ³ A . 1 isomorphism with itself...
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