Why is division on natural number a poset but division on Z is not a poset? Is is because we get ordered pair such as (-1,1) and (1,-1) which is not antisymmetric?
2026-04-08 07:14:36.1775632476
Why is division on Z not a poset?
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Just to make sure the notation is clear. If you declare $x\le y$ when $x$ divides $y$, then this defines a partial order on $\mathbb N$ but not on $\mathbb Z$. I think that this is what you are saying. And the reason is indeed that on $\mathbb Z$ this relation is not anti-symmetric, as for instance it is the case that $1\le -1$ and $-1\le 1$.