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Classification of continuous fractional binary operations on the real and complex fields. (English) Zbl 1364.30005

Summary: In this paper, we consider a classification problem for continuous fractional binary operations on \(\mathbb K\), where \(\mathbb K\) denotes the real field \(\mathbb R\) or the complex field \(\mathbb C\). We first show that there exist exactly two continuous fractional binary operations on \(\mathbf R\) up to isomorphism. In the complex case, we describe completely all continuous fractional binary operations on \(\mathbb C\) in terms of ordinary fraction. Applying this description, we give a partial solution to the classification problem in the complex case. Moreover we show that there exist exactly two homogeneous cancellative binary operations on \(\mathbb K\) up to isomorphism.

MSC:

30A99 General properties of functions of one complex variable
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References:

[1] L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Analysis and Operator Theory 7 (2013), 1049-1063. DOI: 10.1007/s11785-011-0154-1. · Zbl 1285.46024
[2] S. Saitoh, A natural interpretation on 100/0=0 and 0/0=0, and an open question,
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