Medial and semimedial left quasigroups

Abstract: In this paper we investigate the class of semimedial left quasigroups, a class that properly contains racks and medial left quasigroups. We extend most of the results about commutator theory for racks collected in \cite{CP} and some of the results concerning Malt'sev conditions for quandles collected in \cite{Maltsev_paper} to the class of semimedial left quasigroups.