# Definition:Strictly Decreasing/Mapping

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## Contents

## Definition

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 decreasing** if and only if:

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

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

## Also known as

A **strictly decreasing mapping** is also known as a **strictly order-reversing mapping**.

## Also see

- Results about
**strictly decreasing mappings**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$