On reconstructing reducible n-ary quasigroups and switching subquasigroups
Combinatorics
2015-07-07 v4 Group Theory
Abstract
(1) We prove that, provided n>=4, a permutably reducible n-ary quasigroup is uniquely specified by its values on the n-ples containing zero. (2) We observe that for each n,k>=2 and r<=[k/2] there exists a reducible n-ary quasigroup of order k with an n-ary subquasigroup of order r. As corollaries, we have the following: (3) For each k>=4 and n>=3 we can construct a permutably irreducible n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>3 has double-exponential growth as n tends to infinity; it is greater than exp exp(n ln[k/3]) if k>=6, and exp exp(n (ln 3)/3 - 0.44) if k=5.
Cite
@article{arxiv.math/0608269,
title = {On reconstructing reducible n-ary quasigroups and switching subquasigroups},
author = {Denis Krotov and Vladimir Potapov and Polina Sokolova},
journal= {arXiv preprint arXiv:math/0608269},
year = {2015}
}
Comments
12pp. V.4: improved lower bound (last section), orders 5 and 7