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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.
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: