proof isomorphism implies elementary equivalence

**nsarocket17** · October 26th 2009, 01:11 PM

I need to show that if A and B are isomorphic, then A and B are elementarily equivalent with a proof by induction on formulas.
A and B are first-order structures for a first-order language, L.

A and B are isomorphic if there is a bijection h: domain of A -> domain of B such that
h(c^A)=c^B
h(f^A(a1,...,an))=f^B(a1),...,f^B(an)
(a1,...,an) contained in P^A iff (h(a1),...,h(an)) contained in P^B.

I also am given that
Th(A)={psi l A l= psi}
A is logically equivalent to B iff Th(A) =Th(B)
*where Th(A) means the theory of A

**tonio** · October 26th 2009, 02:59 PM

Originally Posted by nsarocket17

I need to show that if A and B are isomorphic, then A and B are elementarily equivalent with a proof by induction on formulas.

What are you talking about, anyway?? Groups, rings, algebras, vector spaces, over what field or ring, what equivalent, what formulas...what?!

Tonio

**Taluivren** · October 27th 2009, 03:35 AM

Hi nsarocket17,

Here are the hints:

First prove this lemma:
Let $L$ be a language and $\mathcal{A}=\langle A,...\rangle$ and $\mathcal{B}=\langle B,...\rangle$ be $L$ -structures and let $h:A \rightarrow B$ be a homomorphism. Then for every term $t=t[v_0,...,v_{n-1}]$ and for all elements $a_0,...,a_{n-1}$ of $A$ , the following holds: $h(\bar{t}^{\mathcal{A}}[a_0,...,a_{n-1}])=\bar{t}^{\mathcal{B}}[h(a_0),...,h(a_{n-1})]$ .

Proof is very easy, by induction on $t$ .

What you're trying to prove is then immediate consequence of the following proposition:
Let $L$ be a language and $\mathcal{A}=\langle A,...\rangle$ and $\mathcal{B}=\langle B,...\rangle$ be $L$ -structures, let $h:A \rightarrow B$ be an isomorphism, let $F=F[v_0,...,v_{n-1}]$ be a formula of $L$ and let $a_0,...,a_{n-1}$ be elements of $A$ . Then $F$ is satisfied in $\mathcal{A}$ by the sequence $(a_0,...,a_{n-1})$ if and only if $F$ is satisfied in $\mathcal{B}$ by sequence $(h(a_0),...,h(a_{n-1}))$ .

Proof of the proposition is again very easy, by induction on $F$ . If $F$ is atomic you just use that isomorphisms preserve relations and the lemma above. Induction steps for logical connectives and for quantification are more or less straightforward verifications.

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