Ordinal multiplication ($\alpha \beta$)

Ordinal multiplication is defined recursively byfor nonzero limit ordinal $\lambda$. It is associative and left-distributive over ordinal addition, but generally neither commutative nor right-distributive.

Two ordinals commute under ordinal multiplication if and only if their squares commute. One implication follows from associativity. For the converse, squaring is strictly increasing: $A<B$ implies $A^2\le BA<B^2$. If $\alpha \beta <\beta \alpha$, monotonicity givesa contradiction to commuting squares. Swap the factors to exclude the reverse inequality; zero factors cause no exception.

Ordinal multiplication is distributive in its right argument:Indeed a sequence of copies of $\alpha$ indexed by the concatenation of $\beta$ and $\gamma$ is the concatenation of the two sequences of copies. The other distributive law fails: $(1+1)\omega =\omega$, while $\omega +\omega >\omega$.