halfgroup

A halfgroup is a mathematical structure consisting of a set and a single binary operation that is associative. This means 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. Halfgroups are a generalization of groups, differing in that they are not required to have an identity element or inverses for their elements.

The concept of a halfgroup is also known as a semigroup. In some contexts, the term "halfgroup" might be used informally to emphasize the fact that it possesses only half of the axioms typically required for a group (associativity, but not necessarily identity and inverses). Examples of halfgroups include the set of natural numbers with addition, or the set of strings with concatenation. In these examples, associativity holds, but there is no universal identity element for all operations (e.g., for addition of natural numbers, 0 is an identity, but it's not always included in the set depending on the definition of natural numbers). Similarly, inverses are not generally defined for these operations. The study of halfgroups, or semigroups, is a significant area within abstract algebra.

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