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I cannot figure out the solution to this exercise in Marker. Can someone help me?

$(Z \oplus Z, +, 0) \not\equiv (Z, +, 0)$

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It's hard to tell without the text what the question is. Perhaps the problem is to show these models are not first-order equivalent in the language of + with identity 0, i.e. to find a first-order sentence satisfied in one but not the other model? –  hardmath Mar 17 '11 at 17:38

1 Answer 1

up vote 10 down vote accepted

EDIT: The sentence $\exists z \forall y \exists x (x+x=y \vee x+x+z=y)$ is true in $\mathbb{Z}$ but not in $\mathbb{Z} \oplus \mathbb{Z}$.

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Elementary equivalence is not the same thing as isomorphism, so this is not sufficient. (Two structures need not even be the same cardinality to be elementarily equivalent.) It is not clear that the assertion that a monoid is generated by a single element is expressible in a first-order sentence in the language with $0$ and $+$. –  Apollo Mar 17 '11 at 17:54
    
I had forgotten Marker uses $\equiv$ for elementary equivalence. Editing. –  Chris Eagle Mar 17 '11 at 18:02
    
No complaints with that one. –  Apollo Mar 17 '11 at 18:12
    
Thank you very much for your fast response. –  Sumac Mar 17 '11 at 18:17

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