Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup.

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A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms:

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• Associativity: for all a, b, c in M, (a*b)*c = a*(b*c)
• Identity element: there exists an element e in M, such that for all a in M, a*e = e*a = a.

One often sees the additional axiom

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• Closure: for all a, b in M, a*b is in M

though, strictly speaking, this isn't necessary as it is implied by the notion of a binary operation.

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Alternatively, a monoid is a semigroup with an identity element.

Monoid: Definition ►



 

Abstract algebra: :This article is about the branch of mathematics. For other uses of the term "algebra" see algebra (disambiguation)....

Algebraic structure: In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. When there are no ambiguities, mathematicians usually identify the set with the algebraic structure. For example, a group (G,*) is usually...

Associative: REDIRECT Associativity...