Classification of Continuous Fractional Binary Operations on the Real and Complex Fields

Abstract

In this paper, we consider a classification problem for continuous fractional binary operations on $\mathbf K$, where $\mathbf K$ denotes the real field $\mathbf R$ or the complex field $\mathbf C$. 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 $\mathbf C$ 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 $\mathbf K$ up to isomorphism.