Editing Monotonic function (section)

== In order theory ==
{{anchor|Monotone function (order theory)}}
Order theory deals with arbitrary [[partially ordered set]]s and [[preorder|preordered sets]] as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not [[total order|total]]. Furthermore, the [[strict order|strict]] relations <math><</math> and <math>></math> are of little use in many non-total orders and hence no additional terminology is introduced for them.

Letting <math>\leq</math> denote the partial order relation of any partially ordered set, a ''monotone'' function, also called ''isotone'', or ''{{visible anchor|order-preserving}}'', satisfies the property

<math display="block">x \leq y \implies f(x) \leq f(y)</math>

for all {{mvar|x}} and {{mvar|y}} in its domain. The composite of two monotone mappings is also monotone.

The [[duality (order theory)|dual]] notion is often called ''antitone'', ''anti-monotone'', or ''order-reversing''. Hence, an antitone function {{mvar|f}} satisfies the property

<math display="block">x \leq y \implies f(y) \leq f(x),</math>

for all {{mvar|x}} and {{mvar|y}} in its domain.

A [[constant function]] is both monotone and antitone; conversely, if {{mvar|f}} is both monotone and antitone, and if the domain of {{mvar|f}} is a [[lattice (order)|lattice]], then {{mvar|f}} must be constant.

Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are [[order embedding]]s (functions for which <math>x \leq y</math> [[if and only if]] <math>f(x) \leq f(y))</math> and [[order isomorphism]]s ([[surjective]] order embeddings).