×

Geometry of \(IM\)-quasigroups. (English) Zbl 0794.51010

A quasigroup with the properties \(a^ 2=a\) and \((ab)(cd)=(ac)(bd)\) is called an \(IM\)-quasigroup. Let \({\mathbb C}\) be the complex numbers. For some \(q\in{\mathbb C}\), \(q\neq 0\), define \(a*b=(1-q)a+qb\) for all \(a,b\in{\mathbb{C}}\). Then \(({\mathbb C},*)\) is an \(IM\)-quasigroup. Every identity of this \(IM\)- quasigroup expresses a geometric property in the complex plane.
The author derives a number of those identities and gives the corresponding geometric interpretations.

MSC:

51M05 Euclidean geometries (general) and generalizations
20N05 Loops, quasigroups
PDFBibTeX XMLCite