The product of cardinals comes from the cartesian product.κ·0 = 0·κ = 0.κ·μ = 0 → (κ = 0 or μ = 0).
One is a multiplicative identity κ·1 = 1·κ = κ.
Multiplication is associative (κ·μ)·ν = κ·(μ·ν).
Multiplication is commutative κ·μ = μ·κ.
Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).
Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ.
Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy.
If either κ or μ is infinite and both are non-zero, then