# Monoid (Math)
## Metadata
**Status**:: #x
**Zettel**:: #zettel/literature
**Created**:: [[2024-03-27]]
## Synopsis
Monoid is a [[Semigroup]] and there exists an identity element.
- Abelian group is a set U under the binary operator ⨁.
- ([[Magma]]) The binary operator is closed: $\forall a, b \in U: a \oplus b \in U$
- ([[Semigroup]]) The binary operator is associative: $(a \oplus b) \oplus c = a \oplus (b \oplus c)$
- (Monoid) There exists an identity element: $\exists 0 \in U: a \oplus 0 = 0 \oplus a = a$