Hessenberg natural sum

The Hessenberg natural sum aligns equal exponents in two Cantor normal forms and adds their finite coefficients; unlike ordinal addition it is commutative.

If a well-order is the union of suborders of types $\xi$ and $\eta$, its type is at most $\xi \#\eta$.

If an ordinal is partitioned into two copies of type $\beta$, its type is below $\beta +\beta +\beta$; two copies do not always give a strict bound.