Definition:Strictly Increasing/Mapping

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping.

Then $\phi$ is strictly increasing if:


 * $\forall x, y \in S: x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$

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