engleski

The meaning of computer search in the studying some classes of IM-quasigroups

A computer search allowing the discovery some special values for the complex number $q \in C \setminus \{;0, 1\};$ such that the quasigroup $C(q)$ is characterized by means of simple algebraic identities, which can be seen in the presentation of V. Kr\v{;c};adinac and V. Volenec. In this way we can find out simple formulae for defining the quaternary relation Par in such a quasigroup, where Par$(a, b, c, d)$ means that $a, b, c, d$ are the vertices of a parallelogram in this cyclical order. We are going to report on the IM--quasigroup $C(q)$ for $q=\frac{;1};{;2};(1+\sqrt{;5};)$ the so called GS--quasigroup which has nice geometric representation which justifies the research of this quasigroup. In general GS--quasigroup the relation Par can be defined by means of several equivalent formulae which can be discovered in $C(q)$ for $q=\frac{;1};{;2};(1+\sqrt{;5};)$. We will choose such a formula for Par which allows the definition of the addition in general GS--quasigroup by the equivalency $c=a+b \Leftrightarrow Par(0, a, c, b)$ where $0$ is the chosen element in GS--quasigroup. The theorem about characterization of GS--quasigroup will be proved by means of the above mentioned formula.

GS-quasigroup

nije evidentirano