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I was given the following task to prove

$Z_n = Z/R $ with operations of addition and multiplication. Prove that it is impossible to define any non-trivial partial order on $Z_n$, which can be compatible with addition operation on $Z_n$, i.e $a \le b \to a + c > \le b + c$ for all $c \in Z_n$

For those of you who speaks russian this is the translation of the task

$Z_N$ фактор множества с введенными операциями сложения и умножения. Докажите что на $Z_N$, нельзя ввести никакого нетривиального отношения частичного порядка, с которым была бы согласована операция сложения на $Z_N$, то есть такого, что из $a \le b \to a + c > \le b + c$ for all $c \in Z_n$

I do not understand this task. Could you please explain it to me ?

  1. I know what is partial order, but what does non-trivial partial order means ,
Rustem Sadykov
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  • The trivial partial order on a set is the order where $x\leq x$ for every element $x$ in the set and nothing else is related. It is transitive, antisymmetric and reflexive. – Nico Oct 01 '20 at 06:25

1 Answers1

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I ll make an example which you can generalize. Say you ve got a non trivial partial order on $\mathbb Z_{102}$ which is compatible with addition. Say for the sake of argument that $56 \leq 94$. Note that $1 + 101 = 102 = 0$ modulus $102$. Thus, $-1 = 101$ in the group $\mathbb Z_{102}$.

Note that by repeatedly adding $94 - 56$ on both sides and applying the transitivety law you get $94 \leq 94 + n (94 -56)$ for every natural $n$. Plug in $n = 101$ and you get the equation $94 \leq 56$. But then $94 = 56$ by antisymmetrie, which is false.

Nico
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