on factorizing transfinites (I)

For some reasons, we should consider "factorization" of transfinite numbers.

Turing(1949) gave a interpretation of transfinite numbers as natural numbers as far as amounts of each orders are bounded

for example: 9w+35 => 9 x 100^2 + 35

we are interested in existence and uniqueness of such "factorization"

refs:
http://en.wikipedia.org/wiki/Ordinal_arithmetic
http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf