quotient type

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In
type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
, a kind of foundation of mathematics, a quotient type is an
algebraic data type In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite type, i.e., a type formed by combining other types. Two common classes of algebraic types are product types (i.e., t ...
that represents a type whose equality relation has been redefined by a given
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
such that the elements of the type are partitioned into a set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es whose
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
is less than or equal to that of the base type. Just as product types and sum types are analogous to the cartesian product and disjoint sum of abstract algebraic structures, quotient types reflect the concept of set-theoretic
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
s, sets whose elements are surjectively partitioned into equivalence classes by a given equivalence relation on the set. Algebraic structures whose underlying set is a quotient are also termed quotients. Examples of such quotient structures include quotient sets, groups,
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
,
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
and, in topology, quotient spaces. In type theories that lack quotient types, setoids – sets explicitly equipped with an equivalence relation – are often used instead.