Commutative property
A binary operation * on a set S is commutative if:
\forall x,y (x * y = y * x)
We say that “x commutes with y“, or that “x and y commute under *“.
Associative property
A binary operation * on a set S is associative if:
\forall x,y,z ((x * y) * z = x * (y * z))
Distributive property
A binary operation * on a set S is distributive over a binary operation + on S if:
\forall x,y,z (x * (y + z) = (x * y) + (x * z))
Identity property
A binary operation * on a set S has an identity i if:
\forall x \exists i (x * i = x)