Let $(X,\lhd)$ and $(Y,\lhd’)$ be two ordered sets and $f:X\to Y$ be a function. $f$ is an order homomorphism, if for all $a$, $b\in X$,
\[a\lhd b \enspace\rimp\enspace f(a)\lhd' f(b).\]
If $\lhd$ and $\lhd’$ are total, $f$ is also called increasing.
If $\lhd$ and $\lhd’$ are total and strict, $f$ is also called strictly increasing.
- Synonym of order homomorphism: order-preserving
Let $(X,\lhd)$ and $(Y,\lhd’)$ be two ordered sets and $f:X\to Y$ be a function. $f$ is an order antihomomorphism, if for all $a$, $b\in X$,
\[a\lhd b \enspace\rimp\enspace f(b)\lhd' f(a).\]
If $\lhd$ and $\lhd’$ are total, $f$ is also called decreasing.
If $\lhd$ and $\lhd’$ are total and strict, $f$ is also called strictly decreasing.
- Synonym of order antihomomorphism: order-reversing
A function is monotonic if it is increasing or decreasing.
A function is strictly monotonic if it is strictly increasing or strictly decreasing.
Let $X$ and $Y$ be two totally ordered sets. If $f:X\to Y$ is a monomorphism, then $f$ is strictly increasing.