# NOTE: Algebra Structure

Magma: A set equipped with a single binary operation that must be closed by definition.

Definition: a set \(M\) matched with an operation \(*\). magma or closure axiom: \(\forall a, b \in M \implies a * b \in M\)

Semigroup: Magma + associativity

\(\forall a, b, c \in M, (a * b) * c \iff a * (b * c)\)

Monoid: Semigroup + identity

\(\exists e \in M, \forall a \in M, a * e = e * a = a\)

Group: Monoid + invertibility

\(\forall a \in M, \exists b \in M, a * b = b * a = e\)