# 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: $\exists 0 \in U: a \oplus 0 = 0 \oplus a = a$