Equivalence class
have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set
These equivalence classes are constructed so that elements
meaning, that every element of the set belongs to exactly one equivalence class.
The set of the equivalence classes is sometimes called the quotient set or the quotient space of
has some structure (such as a group operation or a topology) and the equivalence relation
is compatible with this structure, the quotient set often inherits a similar structure from its parent set.
satisfying the three properties:[1] The equivalence class of an element
For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets.
which maps each element to its equivalence class, is called the canonical surjection, or the canonical projection.
When such an element is chosen, it is called a representative of the class.
The choice of a representative in each class defines an injection from
Since its composition with the canonical surjection is the identity of
such an injection is called a section, when using the terminology of category theory.
For example, in modular arithmetic, for every integer m greater than 1, the congruence modulo m is an equivalence relation on the integers, for which two integers a and b are equivalent—in this case, one says congruent—if m divides
Each class contains a unique non-negative integer smaller than
The use of representatives for representing classes allows avoiding to consider explicitly classes as sets.
In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class.
In the preceding example, this function is denoted
and produces the remainder of the Euclidean division of a by m. Every element
the following statements are equivalent: An undirected graph may be associated to any symmetric relation on a set
This occurs, for example, in the character theory of finite groups.
Such a function is a morphism of sets equipped with an equivalence relation.
In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.
In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map.
However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.
The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.
A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.
Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set
either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on
Both the sense of a structure preserved by an equivalence relation, and the study of invariants under group actions, lead to the definition of invariants of equivalence relations given above.
