nLab monotone function

Idea

A function between preordered sets is called monotone if it respects the (pre)ordering.

When preordered sets are regarded as categories (namely: (0,1)-categories) then monotone functions are equivalently the functors between these.

Definition

In components

Let $S$ and $T$ be preordered sets, that is sets equipped with a reflexive and transitive binary relation $\leq$. (By convention, the same symbol is used for both sets, even though technically it is not the same relation.)

Then a function $f$ from $S$ to $T$ is monotone (increasing), isotone, weakly increasing, or order-preserving if it preserves $\leq$:

$$x \leq y \;\Rightarrow\; f(x) \leq f(y)$$

for all $x, y$ in $S$.

A strictly increasing function is a weakly increasing function that is also injective, at least if $S$ and $T$ are partially ordered. Between arbitrary preordered sets, however, it is probably better to accept as strictly increasing any weakly increasing function that is weakly injective in that $x \leq y$ whenever $f(x) = f(y)$; such a function must be injective if $S$ is a partial order (since $y \leq x$ will also follow) but not necessarily in general.

A function $f$ is monotone decreasing, antitone, weakly decreasing, or order-reversing if it reverses $\leq$:

$$x \leq y \;\Rightarrow\; f(y) \leq f(x)$$

for all $x, y$ in $S$.

A strictly decreasing function is a weakly decreasing function that is also (weakly) injective.

Category-theoretic

As a preordered set is the same thing as a category in which any two parallel morphisms are equal, so a monotone function is simply a functor between such categories. An antitone function is a contravariant functor. That ‘monotone’ may be used for both matches that ‘functor’ may be used for both covariant and contravariant functors.

Strictly increasing (and strictly decreasing) functions are particularly important between linearly ordered sets, where they are the most natural kind of morphism. Between partially ordered sets in general (and between preordered sets using the stricter definition), the strictly increasing functions are simply the monomorphisms (if weakly increasing functions are taken as the morphisms). If we use the weaker definition between preordered sets, then the strictly increasing functions correspond to pseudomonic functors, which is an appropriate sort of higher monomorphism; this is one reason for preferring that definition.

The alternative sort of monotone function on a single proset $S$ is rather different; we mention it here largely because of the potential terminological confusion, but it might as well have its own article if we find a nice name for it. As a functor, it is a functor for which every object is an algebra; the condition is part of the requirements of a Moore closure (a monad on $S$).

Related concepts

• enriched monotone

• injective function

• surjective function

• equivariant function

• opposite poset