Transversals, plexes, and multiplexes in iterated quasigroups

TL;DR

This paper investigates the asymptotic behavior of transversals and multiplexes in iterated quasigroups, establishing formulas for their counts and characterizing typical structures as the dimension grows.

Contribution

It proves the existence of a constant governing the asymptotic number of multiplexes in iterated quasigroups and derives formulas for transversals in such structures.

Findings

Asymptotic formulas for the number of $k$-multiplexes in iterated quasigroups.

Asymptotic count of transversals in $d$-iterated quasigroups.

Characterization of typical $k$-multiplexes and estimates for partial multiplexes.

Abstract

A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$. A transversal is a 1-plex. In this paper we prove that there exists a constant $c(G,k)$ such that if a $d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$ the number of its $k$-multiplexes is asymptotically equal to $c(G,k) \left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of order $n$ is nonzero then asymptotically it is $c(G,1) n!^{d-1}$. In addition, we provide limit constants and recurrence formulas for the numbers of transversals in two iterated quasigroups of order 5, characterize a typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and transversals in $d$-iterated quasigroups.