Monotone Function

Definition

A monotone function (or order-preserving function) is a function between partially ordered sets that preserves the order relation.

Given partially ordered sets $(P,{\le }_P)$ and $(Q,{\le }_Q)$, a function $f:P\to Q$ is monotone if:

$\forall x,y\in P.x{\le }_{P}y⟹f(x){\le }_{Q}f(y)$

Variants

• Monotone increasing (or isotone): The standard definition above
• Monotone decreasing (or antitone): Reverses order, i.e., $x{\le }_{P}y⟹f(x){\ge }_{Q}f(y)$
• Strictly monotone: $x{<}_{P}y⟹f(x){<}_{Q}f(y)$

Properties

• The composition of monotone functions is monotone
• The identity function on any poset is monotone
• Monotone functions form the morphisms in Poset (Category)
• Every monotone function induces a functor between posets viewed as categories

Examples

• The function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^3$ is strictly monotone increasing
• The function $g:\mathbb{R}\to \mathbb{R}$ defined by $g(x)=-x$ is strictly monotone decreasing
• Inclusion maps between ordered sets are monotone
• The floor and ceiling functions are monotone

Relation to Continuity

In the context of topological spaces with order topologies, continuous monotone functions have special properties and play an important role in order theory and topology.

Related Concepts

• Partial Order: The structure preserved by monotone functions
• Poset (Category): The category where morphisms are monotone functions
• Functor: Monotone functions as functors between thin categories
• Continuous Map: A related but distinct notion in topology
• Galois Connection: A pair of monotone functions with special properties