The divisibility poset of positive integers

Let $\mathbb Z_{>0}:=\{n\in\mathbb Z:n>0\}$. The divisibility order on positive integers is

$a\le_{\mid} b\quad\Longleftrightarrow\quad a\mid b,$

where divisibility is that of Divisibility in $\mathbb{Z}$: $d \mid a$ when $a = dq$ for some integer $q$. This is a partial order (Partial order and partially ordered set): it is reflexive and transitive by Divisibility is reflexive and transitive on $\mathbb{Z}$, and is linear: if $d \mid a$ and $d \mid b$ then $d \mid ax + by$ for all integers $x, y$; also $d \mid a$ implies $d \mid ac$, $-d \mid a$ and $d \mid -a$, and it is antisymmetric because $a\mid b$ with $b\ne0$ gives $a=|a|\le|b|=b$ by If $d \mid a$ and $a \ne 0$ then $d \ne 0$ and $|d| \le |a|$; hence the set of divisors of a nonzero integer is bounded above by $|a|$, while $b\mid a$ gives $b\le a$, so antisymmetry of the integer order (The integers form a totally ordered ring) gives $a=b$.

For $a\mid b$, the interval is

$[a,b]_{\mid}=\{d\in\mathbb Z_{>0}:a\mid d\text{ and }d\mid b\}.$

The least element is $1$. There is no greatest element because every positive integer divides a larger positive multiple of itself.