Divides is Partial Ordering on Positive Integers

Theorem

Divides is a partial ordering of $\Z_{>0}$.

Proof

Checking in turn each of the critera for an ordering:

Divides is Reflexive

$\forall n \in \Z: n \backslash n$ from Every Integer Divides Itself‎.

$\blacksquare$

Divides is Transitive

$\forall x, y, z \in \Z: x \backslash y \land y \backslash z \implies x \backslash z$

	\(\displaystyle \)	\(\displaystyle x \backslash y\)	\(\implies\)	\(\displaystyle \exists q_1 \in \Z: q_1 x = y\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle y \backslash z\)	\(\implies\)	\(\displaystyle \exists q_2 \in \Z: q_2 y = z\)	\(\displaystyle \)
	\(\displaystyle \implies\)	\(\displaystyle q_2 \left({q_1 x}\right)\)	\(=\)	\(\displaystyle z\)	\(\displaystyle \)
	\(\displaystyle \implies\)	\(\displaystyle \left({q_2 q_1 x}\right)\)	\(=\)	\(\displaystyle z\)	\(\displaystyle \)
	\(\displaystyle \implies\)	\(\displaystyle q \in \Z: q x\)	\(=\)	\(\displaystyle z\)	\(\displaystyle \)
	\(\displaystyle \implies\)	\(\displaystyle x \backslash z\)	\(\)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Divides is Antisymmetric

We have $\forall a, b \in \Z: a \backslash b \land b \backslash a \implies \left\vert{a}\right\vert = \left\vert{b}\right\vert$ which follows from Integer Absolute Value Greater than Divisors:

	\(\displaystyle \)	\(\displaystyle a \backslash b\)	\(\implies\)	\(\displaystyle \left\vert{a}\right\vert \le \left\vert{b}\right\vert\)	\(\displaystyle \)	Integer Absolute Value Greater than Divisors
	\(\displaystyle \)	\(\displaystyle b \backslash a\)	\(\implies\)	\(\displaystyle \left\vert{b}\right\vert \le \left\vert{a}\right\vert\)	\(\displaystyle \)	Integer Absolute Value Greater than Divisors
	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left\vert{a}\right\vert = \left\vert{b}\right\vert\)	\(\displaystyle \)

If we restrict ourselves to the domain of positive integers, we can see:

$\forall a, b \in \Z_{>0}: a \backslash b \land b \backslash a \implies a = b$

Hence the result.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 14$: Example $14.4$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.1$: Example $4$
• T.S. Blyth: Set Theory and Abstract Algebra (1975):$\S 7$: Example $7.2$