# Definition:Decreasing/Mapping

< Definition:Decreasing(Redirected from Definition:Decreasing Mapping)

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## Contents

## 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 **decreasing** if and only if:

- $\forall x, y \in S: x \preceq_1 y \implies \map \phi y \preceq_2 \map \phi x$

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

## Also known as

A **decreasing** mapping is also known as **order-inverting**, **order-reversing**, **antitone** and **non-increasing**.

Some sources refer to it as **monotonic decreasing**.

## Also see

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.2$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$