SEMIGROUPS Meaning and
Definition
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A semigroup is a mathematical structure that consists of a set and an associative binary operation defined on that set. More formally, a semigroup is a set S equipped with a binary operation •, such that for any three elements a, b, and c in S, the operation satisfies the associative property: (a • b) • c = a • (b • c).
In other words, a semigroup is a closed set with an operation that combines any two elements from that set, and the order of how the elements are combined does not matter. This means that the result of the operation between three elements will be the same regardless of the grouping of the operation.
Additionally, a semigroup does not necessarily require the existence of an identity element, which is an element that, when combined with any other element, leaves the other element unchanged.
Semigroups commonly appear in various areas of mathematics, such as algebra, geometry, and functional analysis. They provide a fundamental framework for studying abstract algebraic structures. Many important mathematical structures, such as groups and rings, are built upon or extended from the concept of a semigroup.
The study of semigroups involves investigating their properties, including structure, homomorphisms, subsemigroups, and classification. Furthermore, semigroup theory plays a significant role in the development of theories and applications in other areas of mathematics and computer science, such as automata theory, graph theory, and formal language theory.
Etymology of SEMIGROUPS
The word "semigroup" is derived from two components: "semi-" and "-group".
The prefix "semi-" comes from the Latin word "semis", which means "half". In this context, "semi-" suggests that semigroups share certain similarities with groups, but not all of them. Specifically, semigroups do not necessarily have an identity element or inverse elements for every element in the set, which are fundamental properties of groups.
The term "group" comes from the late 19th-century mathematical language and is generally attributed to the work of the French mathematician Évariste Galois. It refers to a set of elements together with a binary operation that satisfies four conditions: closure, associativity, identity, and invertibility. However, since semigroups lack the invertibility property, they are referred to as "semigroups" to highlight this distinction.
