Definition:Decreasing

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 $S = T$.

This definition continues to hold when $S = T = \R$; thus:

Let $f$ be a real function.

Then $f$ is decreasing if and only if:

$x \le y \implies \map f y \le \map f x$.

Let $\struct {S, \preceq}$ be a totally ordered set.

Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is decreasing if and only if:

$\forall j, k \in A: j < k \implies a_k \preceq a_j$