On $4-T-$quasigroups with exactly $20$ distinct parastrophes

Tatiana Rotari^a, Parascovia Syrbu<sup>b

a</sup> Alecu Russo Balti State University, Chair of Mathematics and Science
^b Moldova State University, Department of Mathematics

Аннотация: The $T-$forms and the spectrum of $4-T-$quasigroups with exactly $20$ distinct parastrophes are considered in the present work. Characterizations of the spectra of finite binary quasigroups with a prescribed number of distinct parastrophes were given by C.C. Lindner and D. Steedly in [1]. Following the results of C.C. Lindner and D. Steedly, M. MacLeish proved that the maximum number of distinct parastrophes of an $n-$quasigroup $(Q, A)$ is a divizor of $(n+1)!,$ and obtained characterizations of the spectrum of finite ternary quasigroups with a prescribed (maximum) number of distinct parastrophes. Binary and ternary linear quasigroups over groups, with a given maximum number of distinct parastrophes have been studied by Belyavskaya, Rotari, Sokhatsky, Pirus, Fryz and others [4-8]. Characterizations of the general $T-$form of a $4-T-$quasigroup with exactly $20$ distinct parastrophes and some estimations of their spectrum are given in the present work.

Ключевые слова и фразы: $n$-quasigroup, $n-T-$quasigroup, $T-$form, parastrophe, spectrum.

MSC: 20N05, 20N15

Язык публикации: английский

DOI: 10.56415/basm.y2025.i1.p107