Transversals, near transversals, and diagonals in iterated groups and\n quasigroups

2020/06/06 by Anna A. Taranenko, Taranenko, Anna A.
Engineering · Materials Science · Mathematics · #05A16 #05B15 #05D15 #05E15 #20N05 #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Quasicrystal Structures and Properties #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2006.03786

openalex publication_date 2020/06/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Given a binary quasigroup G of order n, a d-iterated quasigroup G[d]\nis the (d+1)-ary quasigroup equal to the d-times composition of G with\nitself. The Cayley table of every d-ary quasigroup is a d-dimensional latin\nhypercube. Transversals and diagonals in multiary quasigroups are defined so as\nto coincide with those in the corresponding latin hypercube.\n We prove that if a group G of order n satisfies the Hall--Paige\ncondition, then the number of transversals in G[d] is equal to fracn!\n|G'| n^n-1 \⋅ n!^d (1 + o(1)) for large d, where G' is the\ncommutator subgroup of G. For a general quasigroup G, we obtain similar\nestimations on the numbers of transversals and near transversals in G[d] and\ndevelop a method for counting diagonals of other types in iterated quasigroups.\n

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