The Post-Gluskin-Hosszu theorem for finite n-quasigroups and self-invariant families of permutations

摘要

We study finite n-quasigroups (n > 3) with the following property of additional invertibility: if the quasigroup operation gives the same results on some two tuples of n arguments with the same first components, then the tuples of the other n - 1 components effect the same left translations. We prove an analogue of the Post-Gluskin-Hosszu theorem for such n-quasigroups. This has been proved previously, but only in the associative case. The theorem reduces the operation of the n-quasigroup to a group operation. The main tool used in the proof is a two-parameter self-invariant family of permutations on an arbitrary finite set. This is introduced and studied in the paper.