Definition:Strictly Increasing/Mapping

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

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

Then $\phi$ is strictly increasing if:

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

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

  • Results about increasing mappings can be found here.