Semigroups

Part of speech: noun

Definitions

1. A type of algebraic structure consisting of a set combined with an associative binary operation
2. a mathematical concept that generalizes the notion of groups without requiring the existence of identity elements
3. a collection of elements that can be combined through a specific operation while preserving an associative property

Etymology: Semigroups is simply the plural of semigroup, and the word is a plain compound: "semi-", meaning half or partly, joined to "group". The name reflects a deliberate omission. A group in algebra needs a set, an associative operation, an identity element, and inverses for every element. A semigroup keeps the first two conditions but drops the requirements for identity and inverses, so it is in a sense a "half-group." The term is therefore a technical label built from ordinary English parts rather than a word with a dramatic history. Its meaning comes entirely from the mathematical structure it names: the set-and-operation pairing is the core, and each further condition that is removed moves the structure further from a full group.

Synonyms: subgroups, partial groups