Алгебралық құрылымдарды конгруенциялық қатынас арқылы бөлу нәтижесі

Result of partitioning the elements of an algebraic structure using a congruence relation
quotient associative algebras over a ring
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,
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 or as the first isomorphism theorem for universal algebra. 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.

For every algebra on the set A, the identity relation on A, and are trivial congruences. An algebra with no other congruences is called simple. Let be the set of congruences on the algebra Because congruences are closed under intersection, we can define a meet operation: by simply taking the intersection of the congruences
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 , such that it is a congruence, in the following way: Note that the closure of a binary relation is a congruence and thus depends on the operations in , not just on the carrier set. Now define as
For every algebra , with the two operations defined above forms a lattice, called the congruence lattice of .