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Transversals, plexes, and multiplexes in iterated quasigroups. / Taranenko, Anna.

в: Electronic Journal of Combinatorics, Том 25, № 4, #P4.30, 02.11.2018.

Результаты исследований: Научные публикации в периодических изданияхстатьяРецензирование

Harvard

Taranenko, A 2018, 'Transversals, plexes, and multiplexes in iterated quasigroups', Electronic Journal of Combinatorics, Том. 25, № 4, #P4.30. https://doi.org/10.37236/7304

APA

Vancouver

Taranenko A. Transversals, plexes, and multiplexes in iterated quasigroups. Electronic Journal of Combinatorics. 2018 нояб. 2;25(4):#P4.30. doi: 10.37236/7304

Author

Taranenko, Anna. / Transversals, plexes, and multiplexes in iterated quasigroups. в: Electronic Journal of Combinatorics. 2018 ; Том 25, № 4.

BibTeX

@article{907592cba9374c0fa724f3fc7e3adb32,
title = "Transversals, plexes, and multiplexes in iterated quasigroups",
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.",
author = "Anna Taranenko",
note = "Publisher Copyright: {\textcopyright} The author.",
year = "2018",
month = nov,
day = "2",
doi = "10.37236/7304",
language = "English",
volume = "25",
journal = "Electronic Journal of Combinatorics",
issn = "1077-8926",
publisher = "Electronic Journal of Combinatorics",
number = "4",

}

RIS

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.

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DO - 10.37236/7304

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JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

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M1 - #P4.30

ER -

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