SEMIGROUP Meaning and
Definition
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A semigroup is a mathematical structure that consists of a set together with an associative binary operation. Specifically, it is an algebraic system that obeys the semigroup axioms.
In more detail, a semigroup is defined as a set S equipped with a binary operation * (denoted as S×S → S), such that for any three elements a, b, and c in S, the operation * satisfies the associative property. This property states that (a * b) * c = a * (b * c) holds for all a, b, and c in S.
The associative property allows for the composition of elements in a semigroup. This means that when applying the operation * to multiple elements in a sequence, the order in which the operations are performed does not affect the result. For example, (a * b) * c is always equivalent to a * (b * c).
Semigroups are generally studied in abstract algebra, as they provide a foundational structure for various mathematical entities. They can also be found in different fields, such as computer science, where semigroups are used to model operation sequences or string concatenation.
It is important to note that unlike a group, semigroups do not require the existence of an identity element or inverses for every element. Hence, a semigroup might not have a neutral element or allow for the cancellation of elements.
Etymology of SEMIGROUP
The word "semigroup" was coined by the mathematician Alfred Young in 1903. It is derived from the Latin word "semi" meaning "half" and the Greek word "graphein" meaning "to write" or "to draw". Young used the term to describe an algebraic structure that is more general than a group but still possesses certain properties. The choice of the name reflects the fact that semigroups have some, but not all, of the properties of groups.
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