# can I find a compatible ordering in $\Bbb{Z}_m$?

Can we define a compatible ordering in $\Bbb{Z}_m$?

The ordering is compatible with the operation if:

$x,y,z \in G$ with $x\leq y$ then $x\cdot z\leq y\cdot z$

I have the belief that we can not find this ordering, but how can I prove it?

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Hint: Consider whether $x<e$ or $x>e$ for arbitrary $x$; where $e$ is the identity element.
In case you ask about $(\mathbb{Z}/m,+)$: Try to prove that ordered groups are trivial or infinite.
It appears that OP is interested in the multiplicative monoid of $\Bbb{Z}_{m}$, where $m$ looks arbitrary. –  Andreas Caranti Mar 23 '13 at 18:21