Monotonicity

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).

As far as the set of all values M(s) does therefore have an upper bound (either within set S, or besides), and as far as every set which is bounded (from above) does have a least upper bound l, the values M(s) are called converging to the upper limit l as the argument s increases.

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.