TY - JOUR

T1 - Transversals, plexes, and multiplexes in iterated quasigroups

AU - Taranenko, Anna

N1 - Publisher Copyright: © The author.

PY - 2018/11/2

Y1 - 2018/11/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85056278123&partnerID=8YFLogxK

U2 - 10.37236/7304

DO - 10.37236/7304

M3 - Article

AN - SCOPUS:85056278123

VL - 25

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 4

M1 - #P4.30

ER -