# 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

$$\forall a \in M, \exists b \in M, a * b = b * a = e$$

Date: 2020-07-12 Sun 00:00