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 :


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

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