Left conjugacy closed left quasigroups with pairwise distinct left translations (Q2826267)

In this paper, the author considers a generalization of conjugacy closed loops. The author concentrates on a conjugacy closed (LCC) left quasigroup \(Q\) which has a set of left translations closed under conjugation. He investigates the left conjugation groupoid \((Q, *)\) of \(Q\) whose binary operation \(*\) is induced by the relation of conjugacy on the left translations. Such a groupoid is analogously to the conjugation groupoid \((G,*)\) of a group \(G\), where the binary operation \(*\) is given by the conjugation of group elements: \(a*b = aba-1\). The author shows that this is also the case for the unique left conjugation groupoid of a LCC left quasigroup with pairwise distinct left translations that is called effective. The author investigates in this paper the properties the original left quasigroup has to satisfy in order to have an effective left conjugation groupoid. This paper concentrates on left quasigroups, but according to the author, every result can be stated for right quasigroups using the right translation. Finally, in Section 6, the author shows that there are left distributive left quasigroups that appear as the left conjugation groupoid of an LCC left quasigroup but cannot appear as that of a group.