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:

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

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

Also see

• Increasing Mapping
• Strictly Decreasing Mapping
• Strictly Monotone Mapping

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$