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

## Congruence lattice

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

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