A monoid is a semigroup with an identity element. More formally, given a set M and an operation
+, we say that
(M, +) is a monoid if it satisfies the following properties for any x, y, z ∈ M:
x + y∈ M
(x + y) + z = x + (y + z)
- Identity: There exists an e ∈ M such that
e + x = x + e = x
We also say that M is a monoid under +.
Examples of Monoids
- The natural numbers N are monoids under addition
- N is a monoid under multiplication.
- Strings form a monoid under concatenation (
""is the identity element)
We’d love your help fleshing out this documentation! You can edit this page in your browser by clicking this link. These links might be helpful: