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 :


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

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 or monotone decreasing.

Also see

 * Definition:Strictly Decreasing Mapping
 * Definition:Increasing Mapping
 * Definition:Monotone Mapping