# Definition:Monotone (Order Theory)

Jump to navigation
Jump to search

## Definition

### Ordered Sets

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping.

Then $\phi$ is **monotone** if and only if it is either increasing or decreasing.

Note that this definition also holds if $S = T$.

### Real Functions

This definition continues to hold when $S = T = \R$.

Thus, let $f$ be a real function.

Then $f$ is **monotone** if and only if it is either increasing or decreasing.

### Sequences

Let $\struct {S, \preceq}$ be a totally ordered set.

A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is **monotone** if and only if it is either increasing or decreasing.

## Notes

This can also be called **monotonic**.