Products of quandles

We generalize the constructions of $Q$- and $G$-families of quandles introduced in the paper of A. Ishii et al. in [Ill. J. Math., 57, No. 3, 817—838 (2013)], and establish how they are related to other constructions of quandles. A composition of structures of quandles defined on the same set is specified, and conditions are found under which this composition yields a quandle. It is proved that under such a multiplication we obtain a group that will be Abelian. Also a direct product of quandles is examined.