Definition:Decreasing/Mapping

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 decreasing :


 * $\forall x, y \in S: x \preceq_1 y \implies \phi \left({y}\right) \preceq_2 \phi \left({x}\right)$

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.

Also see

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