# 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 **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.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings - 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.2$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$