, Volume 32, Issue 2, pp 157–178 | Cite as

A Note on Euclidean Order Types

  • Pete L. Clark


Euclidean functions with values in an arbitrary well-ordered set were first considered by Motzkin in 1949 and studied in more detail by Samuel and Nagata in the 1970’s and 1980’s. Here these results are revisited, simplified, and extended. The main themes are (i) consideration of O r d-valued functions on an Artinian poset and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. To any Euclidean ring we associate an ordinal invariant, its Euclidean order type, and we initiate a study of this invariant, especially for Euclidean rings which are not domains.


Euclidean ring Euclidean function Artinian ordered set Ordinal invariant 


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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Boyd Graduate Studies Research CenterUniversity of GeorgiaAthensUSA

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