# 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 ${\displaystyle {\mathcal {A}}}$), 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 ${\displaystyle a_{1},a_{2},\ldots ,a_{n},b_{1},b_{2},\ldots ,b_{n}\in A}$ whenever ${\displaystyle (a_{1},b_{1})\in E,(a_{2},b_{2})\in E,\ldots ,(a_{n},b_{n})\in E}$ implies ${\displaystyle (f(a_{1},a_{2},\ldots ,a_{n}),f(b_{1},b_{2},\ldots ,b_{n}))\in E}$. An equivalence relation compatible with all the operations of an algebra is called a congruence.

## Congruence lattice

For every algebra ${\displaystyle {\mathcal {A}}}$ on the set A, the identity relation on A, and ${\displaystyle A\times A}$ are trivial congruences. An algebra with no other congruences is called simple.

Let ${\displaystyle \mathrm {Con} ({\mathcal {A}})}$ be the set of congruences on the algebra ${\displaystyle {\mathcal {A}}}$. Because congruences are closed under intersection, we can define a meet operation: ${\displaystyle \wedge :\mathrm {Con} ({\mathcal {A}})\times \mathrm {Con} ({\mathcal {A}})\to \mathrm {Con} ({\mathcal {A}})}$ by simply taking the intersection of the congruences ${\displaystyle E_{1}\wedge E_{2}=E_{1}\cap E_{2}}$.

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 ${\displaystyle {\mathcal {A}}}$, such that it is a congruence, in the following way: ${\displaystyle \langle E\rangle _{\mathcal {A}}=\bigcap \{F\in \mathrm {Con} ({\mathcal {A}})|E\subseteq F\}}$. Note that the (congruence) closure of a binary relation depends on the operations in ${\displaystyle {\mathcal {A}}}$, not just on the carrier set. Now define ${\displaystyle \vee :\mathrm {Con} ({\mathcal {A}})\times \mathrm {Con} ({\mathcal {A}})\to \mathrm {Con} ({\mathcal {A}})}$ as ${\displaystyle E_{1}\vee E_{2}=\langle E_{1}\cup E_{2}\rangle _{\mathcal {A}}}$.

For every algebra ${\displaystyle {\mathcal {A}}}$, ${\displaystyle ({\mathcal {A}},\wedge ,\vee )}$ with the two operations defined above forms a lattice, called the congruence lattice of ${\displaystyle {\mathcal {A}}}$.

## 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 ${\displaystyle {\mathcal {A}}}$, it is straightforward to define the operations induced on A/E if E is a congruence. Specifically, for any operation ${\displaystyle f_{i}^{\mathcal {A}}}$ of arity ${\displaystyle n_{i}}$ in ${\displaystyle {\mathcal {A}}}$ (where the superscript simply denotes that it's an operation in ${\displaystyle {\mathcal {A}}}$) define ${\displaystyle f_{i}^{{\mathcal {A}}/E}:(A/E)^{n_{i}}\to A/E}$ as ${\displaystyle f_{i}^{{\mathcal {A}}/E}([a_{1}]_{E},\ldots ,[a_{n_{i}}]_{E})=[f_{i}^{\mathcal {A}}(a_{1},\ldots ,a_{n_{i}})]_{E}}$, where ${\displaystyle [a]_{E}}$ denotes the equivalence class of a modulo E.

For an algebra ${\displaystyle {\mathcal {A}}=(A,(f_{i}^{\mathcal {A}})_{i\in I})}$, given a congruence E on ${\displaystyle {\mathcal {A}}}$, the algebra ${\displaystyle {\mathcal {A}}/E=(A/E,(f_{i}^{{\mathcal {A}}/E})_{i\in I})}$ is called the quotient algebra (or factor algebra) of ${\displaystyle {\mathcal {A}}}$ modulo E. There is a natural homomorphism from ${\displaystyle {\mathcal {A}}}$ to ${\displaystyle {\mathcal {A}}/E}$ mapping every element to its equivalence class. In fact, every homomorphism h determines a congruence relation; the kernel of the homomorphism, ${\displaystyle \mathop {\mathrm {ker} } \,h=\{(a,a')\in A\times A|h(a)=h(a')\}}$.

Given an algebra ${\displaystyle {\mathcal {A}}}$, a homomorphism h thus defines two algebras homomorphic to ${\displaystyle {\mathcal {A}}}$, the image h(${\displaystyle {\mathcal {A}}}$) and ${\displaystyle {\mathcal {A}}/\mathop {\mathrm {ker} } \,h}$ The two are isomorphic, a result known as the homomorphic image theorem. Formally, let ${\displaystyle h:{\mathcal {A}}\to {\mathcal {B}}}$ be a surjective homomorphism. Then, there exists a unique isomorphism g from ${\displaystyle {\mathcal {A}}/\mathop {\mathrm {ker} } \,h}$ onto ${\displaystyle {\mathcal {B}}}$ such that g composed with the natural homomorphism induced by ${\displaystyle \mathop {\mathrm {ker} } \,h}$ equals h.