On the Derivatives of Central Loops

Abstract

The right(left) derivative, $a^{-1},e-$ and $e,a^{-1}-$ isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop $(L,F)$, it is shown that $\{F,F^{a^{-1}},F_{a^{-1},e}\}$ and $\{F,F_{a^{-1}},F_{e,a^{-1}}\}$ are systems of isotopic C-loops that obey a form of generalized distributive law. Quasigroup isotopes $(L,\otimes )$ and $(L,\ominus )$ of a loop $(L,\theta )$ and its parastrophe $(L,{\theta }^{\ast })$ respectively are proved to be isotopic if either $(L,\otimes )$ or $(L,\ominus )$ is commutative. If $(L,\theta )$ is a C-loop, then it is shown that $\{(L,\theta ),(L,{\theta }^{\ast }),(L,\otimes ),(L,\oplus )\}$ is a system of isotopic C-quasigroup under the above mentioned condition. It is shown that C-loops are isotopic to some finite indecomposable groups of the classes ${\mathcal{D}}_i,i=1,2,3,4,5$ and that the center of such C-loops have a rank of 1,2 or 3.