Orderpreserving

Orderpreserving describes a property of a function or mapping between ordered sets in which the input order is reflected in the output order. More precisely, a function f is order-preserving (also called monotone or isotone) if, whenever a ≤ b in the domain, we have f(a) ≤ f(b) in the codomain. This condition may be strict, meaning a < b implies f(a) < f(b), in which case the function is called strictly increasing or strictly order-preserving. The concept can be defined for any partially or totally ordered sets.

In mathematics, order-preserving maps are fundamental in order theory and lattice theory. They are used to compare structures, define morphisms between posets, and study how order relations are preserved under composition. Simple examples include the identity function on any ordered set, and functions like f(x) = x^3 or f(x) = e^x on the real numbers, which preserve the usual order. Not all increasing-looking functions preserve order on all domains; for instance, f(x) = x^2 is not order-preserving on the entire real line, because negative inputs can map to values larger than those from positive inputs.

A notable applied use is order-preserving encryption (OPE), a cryptographic technique that preserves the order of plaintexts in their ciphertexts. OPE enables range queries and certain comparisons on encrypted databases without decrypting data. However, preserving order leaks information about the relative values and can weaken security, so OPE is chosen with careful consideration of leakage profiles and threat models.

Related concepts include monotone and isotone mappings, and their roles in functional analysis, optimization, and computer science.