Quotientenhalbgruppe

A Quotientenhalbgruppe, also known as a quotient semigroup or factor semigroup, is a mathematical structure formed by taking a semigroup and identifying elements that are equivalent with respect to a congruence relation. A semigroup is a set equipped with an associative binary operation. A congruence relation on a semigroup is an equivalence relation that is compatible with the semigroup operation, meaning that if a is congruent to b and c is congruent to d, then ac is congruent to bd and ca is congruent to db.

To construct the Quotientenhalbgruppe, we first define the set of equivalence classes of the semigroup under the congruence relation. Each equivalence class contains all elements that are considered equivalent. The binary operation from the original semigroup is then transferred to these equivalence classes. If [a] denotes the equivalence class containing element a and [b] denotes the equivalence class containing element b, then the product of these classes is defined as the equivalence class containing the product of a and b, i.e., [a] * [b] = [a * b]. This operation is well-defined because the congruence relation ensures that the result is independent of the choice of representatives a and b from their respective classes.

The resulting structure, consisting of the set of equivalence classes and the induced binary operation, forms a new semigroup, the Quotientenhalbgruppe. This construction is fundamental in abstract algebra for simplifying or analyzing semigroups by identifying and collapsing equivalent elements. It is analogous to the construction of quotient groups in group theory or quotient rings in ring theory.