# Quotient algebra

In mathematics, a **quotient algebra**, (where algebra is used in the sense of universal algebra), also called a **factor algebra**, is obtained by partitioning the elements of an algebra into equivalence classes given by a congruence relation, that is an equivalence relation that is additionally *compatible* with all the operations of the algebra, in the formal sense described below.

## Compatible relation

Let *A* be a set (of the elements of an algebra ), and let *E* be an equivalence relation on the set *A*. The relation *E* is said to be *compatible* with (or have the *substitution property* with respect to) an *n*-ary operation *f* if for all whenever implies . An equivalence relation compatible with all the operations of an algebra is called a congruence.

## Congruence lattice

For every algebra on the set *A*, the identity relation on A, and are trivial congruences. An algebra with no other congruences is called *simple*.

Let be the set of congruences on the algebra . Because congruences are closed under intersection, we can define a meet operation: by simply taking the intersection of the congruences .

On the other hand, congruences are not closed under union. However, we can define the closure of any binary relation *E*, with respect to a fixed algebra , such that it is a congruence, in the following way: . Note that the (congruence) closure of a binary relation depends on the operations in , not just on the carrier set. Now define as .

For every algebra , with the two operations defined above forms a lattice, called the *congruence lattice* of .

## Quotient algebras and homomorphisms

A set *A* can be partitioned in equivalence classes given by an equivalence relation *E*, and usually called a quotient set, and denoted *A*/*E*. For an algebra , it is straightforward to define the operations induced on *A*/*E* if *E* is a congruence. Specifically, for any operation of arity in (where the superscript simply denotes that it's an operation in ) define as , where denotes the equivalence class of *a* modulo *E*.

For an algebra , given a congruence *E* on , the algebra is called the *quotient algebra* (or *factor algebra*) of modulo *E*. There is a natural homomorphism from to mapping every element to its equivalence class. In fact, every homomorphism *h* determines a congruence relation; the kernel of the homomorphism, .

Given an algebra , a homomorphism *h* thus defines two algebras homomorphic to , the image h() and The two are isomorphic, a result known as the *homomorphic image theorem*. Formally, let be a surjective homomorphism. Then, there exists a unique isomorphism *g* from onto such that *g* composed with the natural homomorphism induced by equals *h*.

## See also

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}