IJPAM: Volume 18, No. 1 (2005)


Jan Rogulski
University for Economics Activities
Ul. \Labiszynska, 03-204 Warsaw, POLAND
e-mail: mrogul@ibmer.waw.pl

Abstract.The bilinear multiplication $*$ in $\mathbb{R}^n$ is a bilinear mapping $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. If it is associative, then $(\mathbb{R}^n, +, *)$ becomes a ring. Some of bilinear multiplications are induced by multiplications defined on a basis of $\mathbb{R}^n$. This is the reason why we first study the multiplications in finite sets. A multiplication $*$ in a set is called a quasigroup multiplication if its left and right multiplications are bijections and $*$ has a neutral element ($*$ need not be associative). Thus an associative quasigroup is a group. We give here the description of isomorphism classes of $n$-element quasigroups for $n = 2, 3, 4$ and 5 and examples for $n = 6$. In this first part of the study of bilinear multiplications in $\mathbb{R}^n$ we give several examples and we show that the complex numbers multiplication is not induced by any multiplication in any basis of $\mathbb{R}^2$. The quaternion multiplication also is not induced by any multiplication in any basis of $\mathbb{R}^4$. For general multiplications we explain the independence of such properties as commutativity, associativity, and possessing a unit-element.

The forthcoming second part is devoted to spectral properties of bilinear multiplications, while the third deals with powers defined by non-associative multiplications.

Received: June 8, 2004

AMS Subject Classification: 08A05, 17A01, 20N02

Key Words and Phrases: non-associative multiplications, quasigroups non-associative rings, groupoids, binary operations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 18
Issue: 1