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A Note on Euclidean Order Types

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Abstract

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.

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  1. Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA, 30602 7403, USA

    Pete L. Clark

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  1. Pete L. Clark
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Correspondence to Pete L. Clark.

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Thanks to John Doyle, David Krumm and Robert Varley for inspiring this work.

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Clark, P.L. A Note on Euclidean Order Types. Order 32, 157–178 (2015). https://doi.org/10.1007/s11083-014-9323-y

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  • DOI: https://doi.org/10.1007/s11083-014-9323-y

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