An extension formula for right Bol loops arising from Bol reflections (Q6636339)

The authors' purpose in this paper is to study a new extension formula for right Bol loops. They describe the most important invariants, namely right multiplicator group, nuclei and center. The authors deal with a new extension formula for loops. The authors prove the following result on the structure of the extension: \N\NTheorem $L''$ is a right Bol loop if and only if $L$ is a right Bol loop with $x^2\in Z(L)$ for every $x\in L$. Moreover, the following are equivalent: (i) $L''$ is Moufang; (ii) $L''$ is associative; (iii) $L$ is an abelian group. \N\NIn Section 4, the authors demonstrate on the center and nuclei, the Theorem: Let $L$ be a right Bol loop with central squares. (i) The right nucleus of the extension $L''$ is $N_p (L'')=\{t_z,v_z\mid z\in Z(L)\}$; (ii) For the left nucleus of $L''$ it holds that $N_\lambda (L'')\bigcap Τ=\{t_n\mid n\in N_\lambda (L)\}\cong N_\lambda (L)$. Furthermore, if $L$ is a non-abelian group, then $N_\lambda ((L))''= Τ\cong L$; If $L$ is an AIP loop, then $N_\lambda (L'')=\{t_n,v_n\mid n\in N_\lambda (L)\}$. The authors also demonstrate the following result and they derive further results on the structure group of the extension's core. \N\NTheorem. Let $L$ be a right Bol loop with central squares. The core of $L''$ decomposes to the disjoint union of two subquandles $Τ$ and $V$, both isomorphic to the core of $L$.