Monotone maps and divisibility

Categories are direct generalisations of transitivity. This is formalized by the hom class congruence $\frac{C}{Hom} = R$ of a category which produces a thin quotient category $R$. In this way, category theory is a natural development of order theory. There is a stronger condition, which is that in the most common categories structure preserving maps produce induced inclusions in corresponding partial orders. A family of common input morphisms then produces a partially ordered family of induced inclusions, whose structure is determined by a comma category.

Example. Let $(\mathbb{Z}_+, |)$ be the divisibility lattice of the positive integers. Then there are six fundamental monotone maps from the divisibility of the positive integers to other partial orders.

identity : $(\mathbb{Z}_+, |) \to (\mathbb{Z}_+, |)$ which maps any positive integer to itself

radical : $(\mathbb{Z}_+, |) \to (\mathbb{Z}_+, |)$ which maps any positive integer to its square-free part

signature : $(\mathbb{Z}_+,|) \to (\mathbb{Y}, \leq)$ which maps any positive integer to its prime signature, which is a monotone map with respect to Young's lattice

$\omega : (\mathbb{Z}_+,|) \to (\mathbb{Z}, \leq) $ which maps any positive integer to its number of distinct prime factors

$\Omega : (\mathbb{Z}_+,|) \to (\mathbb{Z}, \leq) $ which maps any positive integer to its number of prime factors counting multiplicity

one : $(\mathbb{Z}_+, |) \to (\{1\}, |)$ which maps any positive integer to the terminal preorder

Each of these monotone maps produces an induced inclusion of the divisibility lattice in an induced preorder. This follows from the fact that $x \subseteq y \Rightarrow f(x) \subseteq f(y)$ is an inclusion of the relation $\subseteq$ in $\subseteq_f$. Induced preorders are therefore larger then input preorders. The constant map produces a complete relation $(\mathbb{Z}_+)^2$ and the identity map produces the divisibility lattice $(\mathbb{Z}_+,|)$. All the induced preorders together form a suborder of the lattice of preorders:

Let $C$ be a category. Then the morphisms of $C$ are preordered by either the input action preorder or the output action preorder. Corresponding to their respective preorders are the input action and output action categories, whose morphic preorders are the corresponding action preorders and whose morphisms are the respective input or output actions that relate one morphism to another. The connected components of these categories are then the comma categories $X/C$ and $C/X$ of each object of the category. In particular, we can form a comma category $(\mathbb{Z}_+,|)/Ord$ of all monotone maps starting from $(\mathbb{Z}_+, |)$.

Proposition. output actions in the category of partial orders and monotone maps are increasing with respect to induced preorders.

Proof. Let $f : (X, \subseteq) \to (Y, \subseteq)$, $g: (Y, \subseteq) \to (Z,\subseteq)$ and $g \circ f : (X, \subseteq) \to (Z,\subseteq)$ be monotone maps. Then let $x,y \in X$ and suppose that $f(x) \subseteq f(y)$. By the monotonicity of $g$ in $Y$ we have that $f(x) \subseteq f(y)$ implies $g(f(x)) \subseteq g(f(y))$. It follows that $\subseteq_f$ is a subrelation of $\subseteq_{g \circ f}$. $\square$

We have that morphisms in the category $Ord$ produce induced inclusions in the lattice of preorders. This does not, however, account for intermediate induced inclusions like those in the divisibility example. In order to account for these intermediate inclusions, we can now use the comma category $(\mathbb{Z}_+,|)/Ord$. Output actions are increasing with respect to induced relations, so for intemerdiate inclusions we can find corresponding monotone maps that map smaller morphisms with respect to the output action preordering to larger ones.

In the example described above, we can map the prime signature to $\omega$ by the size of the signature and $\Omega$ by the sum. Likewise, we can map the radical to $\omega$ by the number of distinct prime factors. Any monotone map can be converted to the map to the trivial preorder, by mapping everything to the same output value. As a result, the ordering on induced relations corresponds to the output action ordering of the comma category.