On the Derivatives of Central Loops

TL;DR

This paper investigates derivatives and isotopes of central loops, demonstrating their properties as C-loops, their isotopic relationships with groups, and the structure of their centers, contributing to the algebraic theory of loops.

Contribution

It introduces new results on derivatives and isotopes of central loops, showing their isotopic relations to groups and characterizing their centers.

Findings

Derivatives of C-loops are also C-loops.

Certain isotopic systems of C-loops obey generalized distributive laws.

C-loops are isotopic to specific finite indecomposable groups.

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 $\big\{F,F^{a^{-1}},F_{a^{-1},e}\big\}$ and $\big\{F,F_{a^{-1}},F_{e,a^{-1}}\big\}$ 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 $(L,\theta ^*)$ 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 $\big\{(L,\theta),(L,\theta ^*),(L,\otimes),(L,\oplus)\big\}$ 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 ${\cal D}_i,i=1,2,3,4,5$ and that the center of such C-loops have a rank of 1,2 or 3.