Pub Feb 23, 2026 Up Jun 10, 2026
Definition 1. 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.
Note. An order homomorphism is also known as an order-preserving function.
Definition 2. 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.
Note. An order antihomomorphism is also known as an order-reversing function.
Definition 3. A function is monotonic if it is increasing or decreasing; strictly monotonic if it is strictly increasing or strictly decreasing.
Proposition 1. Let $X$ and $Y$ be two totally ordered sets. If $f : X \to Y$ is a monomorphism, then $f$ is strictly increasing.