# Definition:Decreasing/Mapping

## 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.