equivalencepreserving

Equivalence-preserving describes a property of a function, transformation, or process in which the equivalence relation on the domain is respected by the mapping. In other words, if two elements are considered equivalent under a given relation, their images under the function are also equivalent under a corresponding relation. This guarantees that the mapping does not distinguish elements that are deemed equivalent.

Formally, let ~ be an equivalence relation on a set X, and let f: X -> Z be a function into a set Z equipped with an equivalence relation ≃. The function f is called equivalence-preserving if whenever a ~ b in X, we have f(a) ≃ f(b) in Z. This condition ensures that f is well-defined on the quotient X/~, and often allows the existence of a unique induced function between quotient structures X/~ and Z/≃.

Example: consider the integers with the equivalence relation a ~ b if a ≡ b (mod 3). The function f(n) = n^2 preserves this equivalence because if a ≡ b (mod 3), then a^2 ≡ b^2 (mod 3). Therefore, f is equivalence-preserving with respect to this relation.

Applications of equivalence-preserving mappings appear in algebra, where they enable quotient constructions, and in computer science and formal semantics, where transformations must preserve program equivalence or indistinguishability under abstraction. Related concepts include compatibility of relations with operations and the notion of well-defined induced mappings on quotient structures.