wu :: forums
wu :: forums - Elementary Equivalence

Welcome, Guest. Please Login or Register.
Nov 26th, 2021, 3:23pm

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   riddles
   putnam exam (pure math)
(Moderators: towr, Icarus, william wu, Eigenray, Grimbal, SMQ)
   Elementary Equivalence
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: Elementary Equivalence  (Read 475 times)
0.999...
Full Member
***





   


Gender: male
Posts: 156
Elementary Equivalence  
« on: Aug 6th, 2015, 7:31pm »
Quote Quote Modify Modify

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 7th, 2015, 1:55am by 0.999... » IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: Elementary Equivalence  
« Reply #1 on: Aug 30th, 2015, 3:54pm »
Quote Quote Modify Modify

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.
IP Logged

Regards,
Michael Dagg
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright 2000-2004 Yet another Bulletin Board