Quasigroups arisen by right nuclear extension

Commentationes mathematicae Universitatis Carolinae

Péter T. Nagy¹ 

关键词: extension of quasigroups;    right nucleus;    quasigroup with right unit;    transversal;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma $ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists a normalized left transversal to $G$ in $Q$ contained in the commutant of~$G$.

【 授权许可】

CC BY