Quotient (universal algebra)

In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below. Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure. The idea of the quotient algebra abstracts into one common notion the quotient structure of quotient rings of ring theory, quotient groups of group theory, the quotient spaces of linear algebra and the quotient modules of representation theory into a common framework. Let A be the 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 implies for any with . An equivalence relation compatible with all the operations of an algebra is called a congruence with respect to this algebra. Any equivalence relation E in a set A partitions this set in equivalence classes. The set of these equivalence classes is usually called the quotient set, and denoted A/E. For an algebra , it is straightforward to define the operations induced on the elements of A/E if E is a congruence. Specifically, for any operation of arity in (where the superscript simply denotes that it is an operation in , and the subscript enumerates the functions in and their arities) define as , where denotes the equivalence class of generated by E ("x 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 via the kernel of the homomorphism, .