# IJPAM: Volume 18, No. 1 (2005)

BILINEAR MULTIPLICATIONS IN
PART I. MULTIPLICATIONS, FINITE
QUASIGROUPS AND BILINEAR
MULTIPLICATIONS IN

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

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.