## Operations

An operation on a set $S$ is a function $∘: S×S ⟶S (s1,s2 ) ⟼ s1∘s2$

An operation $\circ$ on a set $S$ is associative if $\circ :S×S\to S$ satisfies $if s1,s2, s3∈S then (s1∘ s2)∘ s3 =s1∘ (s2∘ s3) .$

An operation $\circ$ on a set $S$ is commutative if $\circ :S×S\to S$ satisfies $if s1,s2 ∈S then s1∘ s2 =s2∘ s1.$

## Examples

The function $+: ℤ×ℤ ⟶ ℤ (i,j) ⟼ i+j$ is an operation. This operation is both commutative and associative.

The function $-: ℤ×ℤ ⟶ ℤ (i,j) ⟼ i-j$ is an operation. This operation is both noncommutative and nonassociative.

## Notes and References

Since an operation is just a function, perhaps the term operation should be deprecated.

## References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.