Projections of ordinal numbers [closed]

Let $p$ is a product of two non-zero ordinal numbers. Knowing $p$ we can restore each of these two ordinals, right?

How can we denote each of these two ordinals (supposing we know the value of $p$)? I suspect, these may be called "projections", right?

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closed as too localized by Andrés E. Caicedo, Asaf Karagila, Michael Greinecker♦, t.b., WilliamAug 20 '12 at 17:41

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No. $2 \times 3 = 1 \times 6$. – Zhen Lin Mar 18 '12 at 21:52

No, you can’t always reconstruct $\alpha$ and $\beta$ from $\alpha\cdot\beta$. For example, $n\cdot\omega=\omega$ for every non-zero finite ordinal $n$.

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