Effective codescent morphisms of n-quasigroups and n-loops

Effective codescent morphisms of n-quasigroups and of n-loops are characterized. To this end, it is proved that, for any n≥ 1, every codescent morphism of n-quasigroups (resp. n-loops) is effective. This statement generalizes our earlier results on qusigroups and loops. Moreover, it is shown that the elements of the amalgamated free products of n-quasigroups (resp. n-loops) have unique normal forms, and that the varieties of n-quasigroups and n-loops satisfy the strong amalgamation property. The latter two statements generalize the corresponding old results on quasigroups and loops by Evans.