equivalenceinstances

Equivalenceinstances is a term used in mathematics and computer science to refer to instances of an equivalence relation on a given domain. An equivalence relation E on a set A satisfies reflexivity (aEa for all a), symmetry (if aEb then bEa), and transitivity (if aEb and bEc then aEc). The collection of all related pairs forms a partition of A into equivalence classes, known as the E-classes. This concept appears in discussions of normalization, data deduplication, and formal methods where a consistent notion of indiscernibility is required.

In type theory and programming, an equivalence instance is a decision procedure or a setoid that provides an equivalence relation on a type, used to compare elements up to the given relation. When an equivalence is decidable, equality tests reduce to deciding whether two elements are equivalent. Equivalence instances can be implemented by a function that returns a boolean or by a relation used by logical frameworks as the basis of normalization or rewriting.

One can construct quotient structures using equivalence instances, forming A modulo E, denoted A/E, whose elements are the equivalence classes. There is a canonical quotient map pi: A -> A/E sending an element to its class. If E is a congruence with respect to algebraic operations, the quotient inherits the structure of A.

Equivalence closures are often used to generate an equivalence instance from a given relation R by taking the smallest equivalence relation containing R, i.e., the reflexive, symmetric, transitive closure of R. In category theory and algebra, equivalence relations that are compatible with operations are called congruences, and they define well-behaved quotient structures.

Examples include the relation of congruence modulo n on integers, logical equivalence on formulas, and bisimilarity in process calculi. Equivalenceinstances thus provide a formal and practical framework for reasoning about indistinguishability and canonical representations in mathematical and computational settings.