Transversals, plexes, and multiplexes in iterated quasigroups

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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[d] 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. It is common to call 1-plexes transversals. 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((kn)!for)d−1.large d the number of its k-multiplexes is asymptotically equal to (Formula Presented) 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.

Original languageEnglish
Article number#P4.30
Number of pages17
JournalElectronic Journal of Combinatorics
Volume25
Issue number4
DOIs
Publication statusPublished - 2 Nov 2018

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