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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.:

Generalizations of monoids feature in Category theory, for example: