Definition:Monotone (Order Theory)
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Definition
Ordered Sets
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 monotone if and only if it is either increasing or decreasing.
Note that this definition also holds if $S = T$.
Real Functions
This definition continues to hold when $S = T = \R$.
Thus, let $f$ be a real function.
Then $f$ is monotone if and only if it is either increasing or decreasing.
Sequences
Let $\struct {S, \preceq}$ be a totally ordered set.
A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is monotone if and only if it is either increasing or decreasing.
Notes
This can also be called monotonic.