关于欧氏序型的一个注记
A Note on Euclidean Order Types
摘要：
欧几里德函数的值在任意良序集首先被认为是在1949年的工作Motzkin和研究更详细的工作弗莱彻，塞缪尔和永田在20世纪70年代和80年代。在这里，这些结果被重新审视，简化和扩展。两个主要主题是（i）考虑Artin偏序集上的序值函数和（ii）使用序数算术，包括Hessenberg-Brookfield序数和。特别是，任何欧几里德环，我们关联一个序数不变，其欧几里德序型，我们开始研究这个不变量。主要的新结果是利用因子环的欧氏序型给出了欧氏环的有限积的欧氏序型的上、下界.
英文摘要：
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.
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