Transversals, near transversals, and diagonals in iterated groups and quasigroups
Anna A. Taranenko

TL;DR
This paper investigates the structure and enumeration of transversals and diagonals in iterated quasigroups and groups, providing asymptotic formulas and methods for counting these configurations in high-dimensional Latin hypercubes.
Contribution
It establishes asymptotic formulas for the number of transversals in iterated groups satisfying the Hall–Paige condition and develops a general counting method for transversals and diagonals in iterated quasigroups.
Findings
Number of transversals in iterated groups with Hall–Paige condition asymptotically equals a specific formula.
Derived estimates for transversals and near transversals in general quasigroups.
Developed a method for counting diagonals of various types in iterated quasigroups.
Abstract
Given a binary quasigroup of order , a -iterated quasigroup is the -ary quasigroup equal to the -times composition of with itself. The Cayley table of every -ary quasigroup is a -dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group of order satisfies the Hall--Paige condition, then the number of transversals in is equal to for large , where is the commutator subgroup of . For a general quasigroup , we obtain similar estimations on the numbers of transversals and near transversals in and develop a method for counting diagonals of other types in iterated quasigroups.
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