Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Congruence relation
Summary
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.
The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer , two integers and are called congruent modulo , written
if is divisible by (or equivalently if and have the same remainder when divided by ).
For example, and are congruent modulo ,
since is a multiple of 10, or equivalently since both and have a remainder of when divided by .
Congruence modulo (for a fixed ) is compatible with both addition and multiplication on the integers. That is,
if
and
then
and
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.
The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.
For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If is a group with operation , a congruence relation on is an equivalence relation on the elements of satisfying
and
for all . For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the other cosets of this subgroup.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.