Definition
A monoid is a set, S, together with a binary operation "•" (pronounced "dot" or "times") that satisfies the following three axioms:
- Closure
- For all a, b in S, the result of the operation a • b is also in S.
- Associativity
- For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds.
- Identity element
- There exists an element e in S, such that for all elements a in S, the equation e • a = a • e = a holds.
And in mathematical notation we can write these as
- Closure: ,
- Associativity: and
- Identity element: .
More compactly, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. A monoid with invertibility property is a group.
The symbol for the binary operation is commonly omitted; for example the monoid axioms require and . This does not necessarily mean the variables are numbers being multiplied, any operation or elements may be used if they are well defined.
Read more about this topic: Monoid
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