Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?
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$\begingroup$ Nice question! A possible candidate: the omnific integers or some other ring associated closely with the ordinals. The ordinals themselves support division in the form $\forall\alpha,\beta\exists\gamma,r\ \beta=\alpha\gamma+r$, where $r\lt\alpha$, since one takes as many copies of $\alpha$ that fit into $\beta$, and $r$ is the leftover part. I'm not sure, however, if this feature extends fully to the Omnific integers with the symmetric arithmetic operations of the surreal numbers. $\endgroup$– Joel David HamkinsOct 21, 2012 at 2:18
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$\begingroup$ Or the polynomials over a field, but allowing transfinite ordinal degrees, which add by symmetric ordinal addition. $\endgroup$– Joel David HamkinsOct 21, 2012 at 3:01
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$\begingroup$ As the floor function is defined in the surreals (floor(x)=\{x−1|x+1\}) is an omnific integer, Euclidian division is indeed possible $\endgroup$– Feldmann DenisOct 21, 2012 at 3:30
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$\begingroup$ I don't think the omnific integers can admit an ordinal-valued (as opposed to omnific integer-valued) Euclidean function; an ordinal-valued Euclidean function should still lead to being a unique factorization domain, in the usual way, but the omnific integers lack unique factorization (e.g., 2, 3, 5, 7, ..., all remain prime, but $\omega$ is divisible by all infinitely many of them). $\endgroup$– Sridhar RameshOct 21, 2012 at 18:53
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1 Answer
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Yes, they exist. Even if the problem is left open in the the papers of T. Motzkin and P. Samuel cited in Comparing different Euclidean algorithms on a Euclidean domain
the problem is solved in
J. Hiblot, Des Anneaux euclidiens dont le plus petit algorithms n'est pas valeurs finies, C. R. Acad. Sci. Paris, 281 (1975), 411-414 with correction in vol 287.
M. Nagata, On Euclid algorithm. C. P. Ramanujama tribute, pp. 175 Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978.
See also:
Pag. 195 in P. M. Cohn, Free Ideal Rings and Localization in General Rings
Rod Downey, Euclidean Domains and Euclidean Functions, http://homepages.mcs.vuw.ac.nz/~downey/ntu_13.pdf
Pete L. Clark, A note on euclidean order types, http://arxiv.org/abs/1208.0977
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$\begingroup$ Ah, an answers years later, which I'd somehow missed till even more years later! Thank you very much! $\endgroup$ Jun 8, 2017 at 18:41
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$\begingroup$ Simpler examples are constructed in my recent paper with Conidis and Tombs entitled "Transfinitely valued Euclidean domains have arbitrary indecomposable order type" $\endgroup$ Sep 25, 2018 at 19:48