Associators, commutators and the linearity of a quasigroup

In a quasigroup $Q(\cdot)$ the associators of two types and the commutators with respect to an arbitrary fixed element $h\in Q$ ($h$-associators and $h$-commutators) are introduced. Their connection with the normal subsets (subquasigroups) is determined, and the characterization of the left (right) $h$-kernel, of the $h$-centre of quasigroups and of various linear (over group) quasigroups is given in terms of $h$-associators and $h$-commutators.