Definition:Increasing/Mapping

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

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

Then $\phi$ is increasing iff:
 * $\forall x, y \in S: x \ \preceq_1 y \ \implies \phi \left({x}\right) \ \preceq_2 \ \phi \left({y}\right)$

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 call it a monotone mapping, which concept has a different meaning on ProofWiki, see Definition:Monotone Mapping.

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

 * Strictly Increasing Mapping
 * Decreasing Mapping
 * Monotone Mapping