Let S be a semigroupPlanetmathPlanetmath. An equivalence relationMathworldPlanetmath defined on S is called a congruencePlanetmathPlanetmathPlanetmathPlanetmath if it is preserved under the semigroup operationMathworldPlanetmath. That is, for all x,y,zS, if xy then xzyz and zxzy.

If satisfies only xy implies xzyz (resp. zxzy) then is called a right congruence (resp. left congruence).


Suppose f:ST is a semigroup homomorphism. Define by xy iff f(x)=f(y). Then it is easy to see that is a congruence.

If is a congruence, defined on a semigroup S, write [x] for the equivalence classMathworldPlanetmath of x under . Then it is easy to see that [x][y]=[xy] is a well-defined operation on the set of equivalence classes, and that in fact this set becomes a semigroup with this operation. This semigroup is called the quotient of S by and is written S/.

Thus semigroup are related to homomorphic imagesPlanetmathPlanetmathPlanetmath of semigroups in the same way that normal subgroupsMathworldPlanetmath are related to homomorphic images of groups. More precisely, in the group case, the congruence is the coset relationMathworldPlanetmathPlanetmath, rather than the normal subgroup itself.

Title congruence
Canonical name Congruence1
Date of creation 2013-03-22 13:01:08
Last modified on 2013-03-22 13:01:08
Owner mclase (549)
Last modified by mclase (549)
Numerical id 7
Author mclase (549)
Entry type Definition
Classification msc 20M99
Related topic Congruences
Related topic MultiplicativeCongruence
Related topic CongruenceRelationOnAnAlgebraicSystem
Defines quotient semigroup