In mathematics, monotonicity is a characteristic of certain functions between partially ordered sets.
A map M between a set T and a set S (both of which are partially ordered sets) has equal monotonicity if, for any two members s and t of T, if s < t, then M(s) ≤ M(t); or the map M has opposite monotonicity if, for any two members s and t of T, if s < t, then M(s) ≥ M(t).
A constant map has both equal monotonicity and opposite monotonicity; conversely, if M has both equal monotonicity and oppositive monotonicity, and if T is totally ordered, then M must be constant.
The notion of monotonicity allows one to express the principal instances of convergence (to a limit):
Given that a commensurate difference relation is defined between the members of S; that is, such that for any four (not necessarily distinct) members g, h, j, and k of S, either g - h ≤ j - k, or g - h ≥ j - k, and given that M from T to S is a map of equal monotonicity, then the values M(s) are called converging (to an upper limit), as the argument s increases, if either:
- the set T has a last and largest member (which M maps explicitly to the corresponding limit value l in set S); or
- for each member m of T, there exists a member n > m such that for any two further members x > y with y > n, M(n) - M(m) ≥ M(x) - M(y).
Similarly one may consider convergence of the values M(s) to a lower limit, as the argument s decreases; as well as convergence involving maps of opposite monotonicity.
The article on monotonic functions considers applications of monotonicity to real-valued functions.