On reconstructing reducible n-ary quasigroups and switching subquasigroups

Details On reconstructing reducible n-ary quasigroups and switching subquasigroups On reconstructing reducible n-ary quasigroups and switching subquasigroups

2006-08-11

arxiv.org/abs/math/0608269v4

Quasigroups Relat. Syst. 16(1) 2008, 55-67

Mathematics

Combinatorics

12pp. V.4: improved lower bound (last section), orders 5 and 7

Scientific paper

(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.

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