Monotonicity in Order Theory
Order theory deals with arbitrary partially ordered sets and preordered sets in addition to 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. Furthermore, the strict relations < and > are of little use in many non-total orders and hence no additional terminology is introduced for them.
A monotone function is also called isotone, or order-preserving. The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property
- x ≤ y implies f(x) ≥ f(y),
for all x and y in its domain. It is easy to see that the composite of two monotone mappings is also monotone.
A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then 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 embeddings (functions for which x ≤ y if and only if f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).
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