PlanetMath: congruence

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congruence

(Definition)

Let be a semigroup. An equivalence relation $\sim$ defined on is called a congruence if it is preserved under the semigroup operation. That is, for all $x, y, z \in S$ , if $x \sim y$ then $xz \sim yz$ and $zx \sim zy$ .

If $\sim$ satisfies only $x \sim y$ implies $xz \sim yz$ (resp. $zx \sim zy$ ) then $\sim$ is called a right congruence (resp. left congruence).

Example 1 Suppose $f: S \to T$ is a semigroup homomorphism. Define $\sim$ by $x \sim y$ iff . Then it is easy to see that $\sim$ is a congruence.

If $\sim$ is a congruence, defined on a semigroup , write for the equivalence class of under $\sim$ . Then it is easy to see that $[x] \cdot [y] = [xy]$ 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 by $\sim$ and is written $S/\sim$ .

Thus semigroup congruences 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.

"congruence" is owned by mclase.

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Also defines:

quotient semigroup

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Cross-references: relation, coset, groups, normal subgroups, homomorphic images, quotient, well-defined, equivalence class, easy to see, iff, semigroup homomorphism, right, implies, operation, equivalence relation, semigroup
There are 13 references to this entry.

This is version 4 of congruence, born on 2002-09-01, modified 2004-03-07.
Object id is 3403, canonical name is Congruence2.
Accessed 5467 times total.

Classification:

AMS MSC:

20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)

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