# Definition:Increasing/Mapping

## Definition

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is increasing if and only if:

$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$

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

## Also known as

An increasing mapping is also known as order-preserving, isotone and non-decreasing.

Some authors refer to this concept as a monotone mapping, but that term has a different meaning on ProofWiki.

## Also defined as

Some sources insist at the point of definition that $\phi$ be an injection for it to be definable as order-preserving, but this is conceptually unnecessary.

## Also see

• Results about increasing mappings can be found here.