DEF-OMOR. Order Morphisms.
Let $(X,\leq)$ and $(Y,\leq’)$ be two partially ordered sets and $f:X\to Y$ be a function. $f$ is an order homomorphism, also called order-preserving / monotone / isotone, if for all $a$, $b\in X$,
\[a\leq b \enspace\Rightarrow\enspace f(a)\leq' f(b).\]If $\leq$ and $\leq’$ are additionally total, then $f$ is also called increasing.
An order isomorphism is a bijective order homomorphism.
An order endomorphism is an order homomorphism from a partially ordered set $(X,\leq)$ to itself.
An order automorphism is an order endomorphism that is an order isomorphism.
Two partially ordered sets $(X,\leq)$ and $(Y,\leq’)$ are isomorphic, if an order isomorphism between them exists.
DEF-OMOR-OR.
Let $(X,\leq)$ and $(Y,\leq’)$ be two partially ordered sets and $f:X\to Y$ be a function. $f$ is called order-reversing / antitone, if for all $a$, $b\in X$,
\[a\leq b \enspace\Rightarrow\enspace f(b)\leq' f(a).\]If $\leq$ and $\leq’$ are additionally total, then $f$ is also called decreasing.