How can division on Natural numbers be a poset?

Question

For natural numbers including 0, how can it be a poset while zero cannot divide zero. Doesn't this mean it isn't reflexive?

Answer 1

$a$ divides $b$ simply means: There is a $k$, s.t. $ak=b$.

Of course $0\cdot 0 = 0$, so $0$ divides $0$.

Answer 2

I think some of your confusion might be coming from whether or not zero "counts" as a natural number. The natural numbers ${\mathbb N}$ can be used to mean either: $${\mathbb N}^0:=\{0,1,2,3,\ldots\}$$ or $${\mathbb N}^{\ast}:=\{1,2,3,\ldots\}.$$ Typically, when one refers to the "natural numbers as a poset order by division", they're referring to ${\mathbb N}^{\ast}$. Although you could define a poset on ${\mathbb N}^0$ with $a\;|\;0$ for all $a\in{\mathbb N}^0$, if you really wanted to include zero in there.