Monoid

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A monoid is an algebraic structure with an associative binary operation that has an identity element. Examples include:

• lists under concatenation
• numbers under addition or multiplication
• Booleans under conjunction or disjunction
• sets under union or intersection
• functions from a type to itself, under composition

Note that in most of these cases the operation is also commutative, but it need not be; concatenation and function composition are not commutative.

A Monoid class is defined in Data.Monoid, and used in Data.Foldable and in the Writer monad.

The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.:

• An introduction: Haskell Monoids and their Uses
• The blog article Monoids and Finger Trees
• Monad.Reader issue 11, "How to Refold a Map." (PDF), and a follow up

Generalizations of monoids feature in Category theory, for example:

• Arrows, like Monads, are Monoids (PDF)