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Topic: Elementary Equivalence (Read 492 times) 

0.999...
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Elementary Equivalence
« on: Aug 6^{th}, 2015, 7:31pm » 
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Show that the abelian group of integers Z is not elementarily equivalent to the abelian group Z+Z (direct sum). That is, find (show that there exists) a sentence involving the symbols +,*,0 (and =) that is true for one but not for the other.

« Last Edit: Aug 7^{th}, 2015, 1:55am by 0.999... » 
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Michael Dagg
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Re: Elementary Equivalence
« Reply #1 on: Aug 30^{th}, 2015, 3:54pm » 
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I thought someone might have taken this by now since it has interesting analogies in other areas and those ideas are very similar. Elementary equivalent means that any firstorder sentence (logical  meaning in group theory language and using forall and there exists) that is true for one of the groups is true for the other. Elementary equivalence is weaker than isomorphism  in fact, strictly weaker but you are at liberty to think of it as an equivalence relation as it certainly is. In particular, the group Z is cyclic with generators +1. So, you can contrive a sentence asserting this fact involving forall and there exists (not necessarily involving its generators). This will certainly be true in Z but not in Z+Z since Z+Z is not cyclic.


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Regards, Michael Dagg



