wu :: forums « wu :: forums - Elementary Equivalence » Welcome, Guest. Please Login or Register. May 28th, 2022, 10:43pm RIDDLES SITE WRITE MATH! Home Help Search Members Login Register
 wu :: forums    riddles    putnam exam (pure math) (Moderators: Grimbal, william wu, SMQ, Icarus, Eigenray, towr)    Elementary Equivalence « Previous topic | Next topic »
 Pages: 1 Reply Notify of replies Send Topic Print
 Author Topic: Elementary Equivalence  (Read 492 times)
0.999...
Full Member

Gender:
Posts: 156
 Elementary Equivalence   « on: Aug 6th, 2015, 7:31pm » Quote 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:
Posts: 500
 Re: Elementary Equivalence   « Reply #1 on: Aug 30th, 2015, 3:54pm » Quote 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 Notify of replies Send Topic Print

 Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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