Quotient algebra

In mathematics, a quotient algebra, (where algebra means algebraic structure 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.

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.

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