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Monoid

Let T be a data type. A monoid is a triple (T, f, z) such that f is an associative binary operation on T, and z ∈ T has the property that for all a, b, c, x ∈ T it holds that

• f(x, z) = f(z, x) = x
• f(a, f(b, c)) = f(f(a, b), c)

Note that z is called an identity value

Simply, a monoid is defined as "an algebraic structure with a single associative binary operation and identity element."

A monoid must contain at least one element.

A monoid that is commutative is, known as a commutative monoid.

Example of Monoids

• Addition operation or a set of integers is a monoid and z is number 0.

• Multiplication operation or a set of integers is a monoid and z is number 1.

Example of Non-Monoids

• Average operation over a set of doubles data type is not a monoid since the means of means is not a mean.

• A division operation over a set of doubles data type is not a monoid since a / b is not the same as b / a

Sources and References

• Monoid

• Monoidify! Monoids as a Design Principle for Efficient MapReduce Algorithms by Jimmy Lin

• Monoids Piyush Mishra