Effective codescent morphisms of $n$-quasigroups and $n$-loops

Abstract

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