Posets: Division operation on set of real numbers

Question

Is the division operation on set of all real numbers a poset?

According to me, the reflexive and transitive properties are satisfied, but it is anti-symmetric as $\frac {0}{1}=0$, but $\frac{1}{0}=$ $\infty$.

Is my approach correct?

Answer 1

I'm guessing that by divisibility, you mean the following relation, generalized from the usual one on integers.

For $a,b\in\mathbb R$, we say that $a\mid b$ iff there exists an integer $k$ such that $a=kb$.

It turns out that this isn't a partial order: $1\mid-1$ and $-1\mid1$, while $1\neq-1$. However, if we restrict ourselves to $\mathbb R_0^+$, we can prove that this is the case.

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Reflexivity:

For every real $a$, $a=1\cdot a$, so that $a\mid a$.

Antisymmetry:

Suppose $a\mid b$, $b\mid a$. Then, two integers $k_1$, $k_2$ must exist such that $$a=k_1b,\ b=k_2a.$$ In particular, $$a=k_1k_2a.$$ This is only possible if either $k_1=k_2=1$, or $a=0$. In both cases, $a=b$.

Transitivity:

If $a\mid b$ and $b\mid c$, there exist integers $k_1$, $k_2$ such that $$a=k_1b,\ b=k_2c.$$ In particular, $$a=k_1k_2c,$$ so that $a\mid c$.