Timeline for Properly "transfinitely" Euclidean domains
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2017 at 18:24 | vote | accept | Sridhar Ramesh | ||
| Mar 20, 2014 at 23:23 | answer | added | user46855 | timeline score: 6 | |
| Oct 21, 2012 at 18:53 | comment | added | Sridhar Ramesh | 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). | |
| Oct 21, 2012 at 3:30 | comment | added | Feldmann Denis | As the floor function is defined in the surreals (floor(x)=\{x−1|x+1\}) is an omnific integer, Euclidian division is indeed possible | |
| Oct 21, 2012 at 3:01 | comment | added | Joel David Hamkins | Or the polynomials over a field, but allowing transfinite ordinal degrees, which add by symmetric ordinal addition. | |
| Oct 21, 2012 at 2:18 | comment | added | Joel David Hamkins | 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. | |
| Oct 21, 2012 at 1:52 | history | asked | Sridhar Ramesh | CC BY-SA 3.0 |