Congruence relation

In mathematics, more precisely, the algebra is called an equivalence relation on an algebraic structure is a congruence, if the operations of the algebraic structure of this equivalence relation are compatible. In its general form, as shown here, they are examined in the universal algebra.

Definition

Be a lot, one - digit operation (operational ) and an equivalence relation. One calls with compatible if for all with ever

Applies.

Let now an algebraic structure, then congruence is called on, if all are compatible with

Application

From an algebraic structure and a congruence to the algebraic structure can be obtained, the so-called factor structure factor algebra or quotient structure a new algebraic structure, besides the basic quantity of just the factor size and for each digit operation of a new operation on defined by

Examples

Homomorphism ( for algebras ): Are and two algebras of the same type (ie there is to every n- ary function exactly a " suitable" n- ary function ) and is an algebra homomorphism with kernel, then:

One could also formulate the isomorphism theorems, for which one suitable first the concept of Faktorkongruenz needed.

• Universal Algebra