Halbkgruppe

A Halbgruppe, also known as a semigroup, is a fundamental algebraic structure in abstract algebra. It is a set equipped with an associative binary operation, meaning that for any elements a, b, and c in the set, the equation (a * b) * c = a * (b * c) holds true. This property is known as associativity.

The concept of a Halbgruppe was introduced by the German mathematician Heinrich Weber in the late 19th century. It generalizes the notion of a group by relaxing the requirement for the existence of an identity element and inverses. In a Halbgruppe, the operation does not necessarily have an identity element, and elements may not have inverses.

Halbgruppen are studied in various branches of mathematics, including algebra, combinatorics, and theoretical computer science. They have applications in the study of formal languages, automata theory, and the theory of semirings. Some well-known examples of Halbgruppen include the set of non-negative integers under addition, the set of positive real numbers under multiplication, and the set of matrices under matrix multiplication.

In summary, a Halbgruppe is a set with an associative binary operation. It is a generalization of the concept of a group and has applications in various areas of mathematics. The study of Halbgruppen provides insights into the structure and properties of algebraic systems.