Definition:Monotonicity

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a monotone mapping.

Then whether $\phi$ is increasing or decreasing is known as the monotonicity of $\phi$.

That is:
 * if $\phi$ is such that $x \ \preceq_1 \ y \implies \phi \left({x}\right) \ \preceq_2 \ \phi \left({y}\right)$, then the monotonicity of $\phi$ is increasing
 * if $\phi$ is such that $x \ \prec_1 \ y \implies \phi \left({x}\right) \ \prec_2 \ \phi \left({y}\right)$, then the monotonicity of $\phi$ is strictly increasing


 * if $\phi$ is such that $x \ \preceq_1 \ y \implies \phi \left({y}\right) \ \preceq_2 \ \phi \left({x}\right)$, then the monotonicity of $\phi$ is decreasing
 * if $\phi$ is such that $x \ \prec_1 \ y \implies \phi \left({y}\right) \ \prec_2 \ \phi \left({x}\right)$, then the monotonicity of $\phi$ is strictly decreasing.

Also see

 * Definition:Monotone (Order Theory)