 Commensurable ordinals

Two ordinals are commensurable when neither is absorbed by adding the other on its left. For positive ordinals, this is equivalent to equality of their leading Cantor normal form exponent. If leading exponents differ, the lower-leading ordinal is absorbed by the higher power of $\omega$. If they agree, either sum adds the two positive leading coefficients and therefore exceeds both original ordinals. Zero is commensurable with no ordinal.