Monoid
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:
- Closure:
x + y
∈ M - Associativity:
(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)
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