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 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.
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): $\S 7$