monotonicityany

Monotonicityany is a term used to describe a property of a function with respect to its input arguments. In this usage, a function is said to have monotonicityany when it is monotone in each argument separately, with all other arguments held fixed. The idea emphasizes that increasing any single input does not decrease the output, regardless of the values of the remaining inputs.

Formally, let f: X1 × ... × Xn → Y where each Xi is a poset (with the order ≤i). f is monotone in the i-th argument if xi ≤i xi' implies f(x1,...,xi,...,xn) ≤ f(x1,...,xi',...,xn). A function is monotone in every argument, i.e., has monotonicityany, if it is monotone in each i = 1,...,n. If this holds, f is often described as being componentwise monotone or monotone in all coordinates.

Examples include f(a,b) = a + b on integers and f(a,b) = max(a,b), both of which are monotone in each argument. A function that is not monotone in one argument, such as f(a,b) = a - b, generally does not have monotonicityany. Some functions may be monotone in certain arguments only under domain restrictions.

Variants and clarifications: monotonicityany is related to the standard notion of monotone (isotone) and to componentwise monotonicity. The product order on X1 × ... × Xn supports this interpretation: if f is monotone in each argument, it is monotone with respect to the product order.

Applications and context: this concept appears in fixed-point theory, lattice theory, and programming language semantics, where monotone functions preserve order and enable fixed-point constructions. In databases and optimization, monotone properties support reliable evaluation and convergence guarantees.

See also: isotone function, monotone function, lattice, componentwise monotonicity, fixed point theorem.