Quasigroups, units and Belousov’s problem # 18

Abstract

We investigate {left, right, middle} units of quasi- groups and families of identities which might imply their ex- istence. A prominent role is played by the newly introduced notion of derivative operation, generalizing Belousov’s notions of left/right derivative operations for quasigroups. Partial solu- tions of the Belousov’s Problem # 18 and its generalizations are obtained. Several related problems remain open.