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Topic: Elementary Equivalence (Read 565 times) |
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0.999...
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Elementary Equivalence
« on: Aug 6th, 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.
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| « Last Edit: Aug 7th, 2015, 1:55am by 0.999... » |
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Michael Dagg
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Re: Elementary Equivalence
« Reply #1 on: Aug 30th, 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 first-order
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
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