Let $(X,\preceq)$ and $(Y,\preceq’)$ be two preordered sets and $f:X\to Y$ be a function. $f$ is an order homomorphism, also called order-preserving / monotone, if for all $a$, $b\in X$,
\[a\preceq b \enspace\rimp\enspace f(a)\preceq' f(b).\]If $\preceq$ and $\preceq’$ are total, $f$ is called increasing.
Let $(X,\preceq)$ and $(Y,\preceq’)$ be two preordered sets and $f:X\to Y$ be a function. $f$ is called order-reversing / antitone, if for all $a$, $b\in X$,
\[a\preceq b \enspace\rimp\enspace f(b)\preceq' f(a).\]If $\preceq$ and $\preceq’$ are total, $f$ is also called decreasing.
A homomorphism between strict partial orders is defined analogously. In addition:
If $f:X\to Y$ is a homomorphism between two strictly totally ordered sets $(X,\prec)$ and $(Y,\prec’)$, i.e. for all $a$, $b\in X$,
\[a\prec b \enspace\rimp\enspace f(a)\prec' f(b),\]then $f$ is also called strictly increasing. Similarly, if $f$ is order-reversing, i.e. for all $a$, $b\in X$,
\[a\prec b \enspace\rimp\enspace f(b)\prec' f(a),\]then $f$ is also called strictly decreasing.
Let $(X,\preceq)$ and $(Y,\preceq’)$ be two totally ordered sets. If $f:X\to Y$ is a monomorphism, then $f$ is strictly increasing.