Original sentence: A semigroup congruence is an equivalence relation that is compatible with the semigroup product.A semigroup congruence ˜ induces congruence classes.That is, it is a subset.The set of congruence classes thus is called the quotient semigroup, and is denoted.that is an equivalence relation and  and  implies  for every  in S.which makes the set of all congruence classes into a semigroup and  into a semigroup homomorphism, called the projection or quotient map
Edited sentence: A semigroup congruence  is an equivalence relation that is compatible with the semigroup operation.Because  is a congruence, the set of all congruence classes of  forms a semigroup with , called the quotient semigroup, and denoted.Like any equivalence relation, a semigroup congruence  induces congruence classes.That is, a subset  that is an equivalence relation and  and  implies  for every  in S.The mapping  is a semigroup homomorphism, called the projection or quotient map