Halfgroups

A halfgroup is a mathematical structure consisting of a set equipped with a single binary operation that is associative. It is also known as a semigroup. The defining characteristic of a halfgroup is that for any elements a, b, and c in the set, the equation (a b) c = a (b c) holds true, where '' denotes the binary operation.

The concept of halfgroups is fundamental in abstract algebra and has numerous applications in various fields of mathematics and computer science. For example, halfgroups are used in the study of automata theory, formal languages, and combinatorics. Unlike groups, halfgroups are not required to have an identity element, nor do they necessarily have inverse elements for each of their members.

Examples of halfgroups include the set of natural numbers with addition, where the operation is associative. Another example is the set of all finite strings over an alphabet with string concatenation. The set of all functions from a set to itself under function composition also forms a halfgroup. If a halfgroup has an identity element, it is called a monoid. If every element in a halfgroup has an inverse, it is a group. Therefore, groups and monoids are special cases of halfgroups.