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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 referred to as an increasing function.

An increasing mapping is also known as isotone or non-decreasing.

In contexts where the ordering in question is more general than in the context of numbers, the term order-preserving mapping is often more appropriate than increasing mapping.

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

Beware that some authors who use the term order-preserving mapping use it to define what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is referred to as an order embedding.

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.