A monoid is a set that is closed under an associative binary operation and has an identity element I ∈ Such that for all a ∈ S, Ia = aI = a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element.
A monoid is a set that is closed under an associative binary operation and has an identity element I ∈ Such that for all a ∈ S, Ia = aI = a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element. Put simply, 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.
