Definition:Strictly Increasing/Mapping

Definition

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

Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping.

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

$\forall x, y \in S: x \prec_1 y \implies \map \phi x \prec_2 \map \phi y$

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

Also known as

A strictly increasing mapping is also known as a strictly isotone mapping.

Also see

• Definition:Increasing Mapping
• Definition:Strictly Decreasing Mapping
• Definition:Strictly Monotone Mapping

• Results about increasing mappings can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$