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

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 $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-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 $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.