The definition of monotonicity is valid in the cases of partially ordered sets as well as preordered sets along with real numbers. We avoid the terms decreasing, increasing in this case as they give a conventional representation in a pictorial form which is not applicable to the not total orders.
An isotone is a monotone function, also called order preserving. The antitone is the dual notion, also known as order reversing or anti monotone. This implies that an order reversing function will satisfy the following property
$a\ \leq\ b$
$\Rightarrow$ $g\ (a)\ \geq\ g\ (b)$
for every $a$, $b$ lying in the domain of the function. It is important to note that the composition of two monotone mappings is again a monotone.
When we have a constant function, it lies in both categories, as it is both antitone and monotone. This condition is a vice versa situation.