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        riddles >> putnam exam (pure math) >> Elementary Equivalence
        
(Message started by: 0.999... on Aug 6^th, 2015, 7:31pm)

Title: Elementary Equivalence
Post by 0.999... on Aug 6^th, 2015, 7:31pm

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.

Title: Re: Elementary Equivalence
Post by Michael Dagg on Aug 30^th, 2015, 3:54pm

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.