A Note on Euclidean Order Types

TL;DR

This paper revisits and extends the theory of Euclidean functions with ordinal values, introducing the Euclidean order type as an invariant for Euclidean rings and analyzing its behavior under ring products.

Contribution

It introduces the Euclidean order type as a new ordinal invariant for Euclidean rings and studies its properties, especially in finite products, using ordinal arithmetic.

Findings

Established bounds on Euclidean order types of product rings.

Extended previous Euclidean function theory with ordinal arithmetic techniques.

Revisited classical results with simplified proofs and new insights.

Abstract

Euclidean functions with values in an arbitrary well-ordered set were first considered in a 1949 work of Motzkin and studied in more detail in work of Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are revisited, simplified, and extended. The two main themes are (i) consideration of Ord-valued functions on an Artinian poset and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. In particular, to any Euclidean ring we associate an ordinal invariant, its Euclidean order type, and we initiate a study of this invariant. The main new result gives upper and lower bounds on the Euclidean order type of a finite product of Euclidean rings in terms of the Euclidean order types of the factor rings.