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The divisibility poset of positive integers
Definition
Let . The divisibility order on positive integers is
where divisibility is that of Divisibility in : when for some integer . This is a partial order (Partial order and partially ordered set): it is reflexive and transitive by Divisibility is reflexive and transitive on , and is linear: if and then for all integers ; also implies , and , and it is antisymmetric because with gives by If and then and ; hence the set of divisors of a nonzero integer is bounded above by , while gives , so antisymmetry of the integer order (The integers form a totally ordered ring) gives .
For , the interval is
The least element is . There is no greatest element because every positive integer divides a larger positive multiple of itself.
Depends on
- Divisibility in $\mathbb{Z}$: $d \mid a$ when $a = dq$ for some integer $q$
- 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 $d \mid -a$
- Partial order and partially ordered set
- The integers form a totally ordered ring
- 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 $|a|$
Used by
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Sources
- R. Stanley, Enumerative Combinatorics, Volume 1, §§3.8.4–3.8.5 (standard reference, not scraped)