IJPAM: Volume 18, No. 1 (2005)
PART I. MULTIPLICATIONS, FINITE
QUASIGROUPS AND BILINEAR
University for Economics Activities
Ul. abiszynska, 03-204 Warsaw, POLAND
Abstract.The bilinear multiplication in is a bilinear mapping . If it is associative, then becomes a ring. Some of bilinear multiplications are induced by multiplications defined on a basis of . 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 -element quasigroups for and 5 and examples for . In this first part of the study of bilinear multiplications in we give several examples and we show that the complex numbers multiplication is not induced by any multiplication in any basis of . The quaternion multiplication also is not induced by any multiplication in any basis of . 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