Definition 1

Let E and F be preordered sets (the preorder relation in each being denoted by $\leqslant$). A mapping f of E into F is said to be increasing (or order-preserving) if the relation $x \leqslant y$ implies $f(x) \leqslant f(y)$; it is decreasing (or order-reversing) if the relation $x \leqslant y$ implies $f(x) \geqslant f(y)$. A mapping of E into F is said to be monotone if it is either increasing or decreasing.

An increasing mapping of E into F becomes decreasing (and vice versa) when we replace one of the preorderings on E or on F by the opposite preordering. Every constant function is both increasing and decreasing; the converse of this statement is usually not true.

For example, if a set E is ordered by the equality relation, the identity mapping of E onto itself is both increasing and decreasing, but not constant if E has more than one element (cf. Exercise 7).