Quotient Algebra - Quotient Algebras and Homomorphisms

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.