Identtiset

Identtiset is a theoretical construct used in mathematics and information theory to denote a subset of a given universal set that consists of elements identified as the same under a specified identity relation. Given a set S and an identity (or equivalence) relation ~ on S, the Identtiset of an element x in S is the equivalence class [x] = { y in S | y ~ x }. The collection of all such classes partitions S into disjoint blocks, so every element of S belongs to exactly one Identtiset.

Identtiset is used in contexts such as algebra, database theory, and data deduplication to formalize the idea of grouping objects by identity under a chosen criterion. The term emphasizes that identity is context-dependent rather than absolute, and that different identity relations produce different groupings of the same underlying data.

Example: In a customer relationship database, records are often considered identical if they share the same customer ID. All records with a given ID form an Identtiset, while records with different IDs belong to different Identtiset.

Properties and variants: Each Identtiset is nonempty and the collection of all Identtiset forms a partition of S; elements belong to one and only one Identtiset. If the identity criterion is refined, the Identtiset become smaller and more numerous; if it is broadened, the Identtiset merge and become larger. In mathematics, this corresponds to moving between finer and coarser equivalence relations, and to constructing quotient sets.

See also: equivalence relation, partition, quotient set, multiset, data deduplication.