Considering left translations to be pairwise distinct, we study a left conjugacy closed (LCC) left quasigroup Q which has a set of left translations closed under the operation of conjugation. The relation of conjugacy on the left translations induces the structure of a groupoid on the underlying set called the left conjugation groupoid. We show it to be a left distributive idempotent left quasigroup. We investigate the case that Q is left distributive and show it to be equal to its left conjugation groupoid whenever the left translations are pairwise distinct. We also explore the conditions making the left translations of the left conjugation groupoid pairwise distinct. Finally, we show that there are left distributive left quasigroups that appear as the left conjugation groupoid of a LCC left quasigroup but cannot appear as that of a group.

algebra, non-associative, groupoid, conjugation.