If satisfies only implies (resp. ) then is called a right congruence (resp. left congruence).
Suppose is a semigroup homomorphism. Define by iff . Then it is easy to see that is a congruence.
If is a congruence, defined on a semigroup , write for the equivalence class of under . Then it is easy to see that 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 by and is written .
Thus semigroup are related to homomorphic images of semigroups in the same way that normal subgroups are related to homomorphic images of groups. More precisely, in the group case, the congruence is the coset relation, rather than the normal subgroup itself.
|Date of creation||2013-03-22 13:01:08|
|Last modified on||2013-03-22 13:01:08|
|Last modified by||mclase (549)|