Monotonicity

In mathematics, monotonicity is a characteristic of certain maps between totally ordered sets:
a map M: Σ → S, between set Σ and set S (both of which are totally ordered sets)
has equal monotonicity if for any two members σ and τ of Σ holds (σ < τ) ⇒ M( σ ) ≤ M( τ ); or map M has opposite monotonicity if for any two members σ and τ of Σ holds (σ < τ) ⇒ M( σ ) ≥ M( τ ).

Only maps which have both equal monotonicity and opposite monotonicity are 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, i.e.
such that for any four (not necessarily distinct) members g, h, j, and k of S holds g - h ≤ j - k, or g - h ≥ j - k,
and given that M: Σ → S is a map of equal monotonicity,

the values M( σ ) are called converging (to an upper limit), as the argument σ increases, if and only if:

either the set Σ has a last and largest member (which M maps explicitly to the corresponding limit value l in set S); or

for each member μ of Σ there exists a member ν > μ such that for any two further members χ > φ with φ > ν holds

M( ν ) - M( μ ) ≥ M( χ ) - M( φ ).

As far as the set of all values M( σ ) 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( σ ) are called converging to the upper limit l as the argument σ increases.

Similarly one may consider convergence of the values M( σ ) to a lower limit, as the argument σ decreases; as well as convergence involving maps of opposite monotonicity.

The article on monotonic functions considers applications of monotonicity to real-valued functions.