88 research outputs found
On the number of n-ary quasigroups of finite order
Let be the number of -ary quasigroups of order . We derive a
recurrent formula for Q(n,4). We prove that for all and the
following inequalities hold: , where does not depend on . So, the upper
asymptotic bound for is improved for any and the lower bound
is improved for odd . Keywords: n-ary quasigroup, latin cube, loop,
asymptotic estimate, component, latin trade.Comment: english 9pp, russian 9pp. v.2: corrected: initial data for recursion;
added: Appendix with progra
On irreducible n-ary quasigroups with reducible retracts
An n-ary operation q:A^n->A is called an n-ary quasigroup of order |A| if in
x_0=q(x_1,...,x_n) knowledge of any n elements of x_0,...,x_n uniquely
specifies the remaining one. An n-ary quasigroup q is permutably reducible if
q(x_1,...,x_n)=p(r(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)}) where p and
r are (n-k+1)-ary and k-ary quasigroups, s is a permutation, and 1<k<n. For
even n we construct a permutably irreducible n-ary quasigroup of order 4r such
that all its retracts obtained by fixing one variable are permutably reducible.
We use a partial Boolean function that satisfies similar properties. For odd n
the existence of a permutably irreducible n-ary quasigroup such that all its
(n-1)-ary retracts are permutably reducible is an open question; however, there
are nonexistence results for 5-ary and 7-ary quasigroups of order 4.
Keywords:n-ary quasigroups, n-quasigroups, reducibility, Seidel switching,
two-graphsComment: 8 p., 1 fig., ACCT-10. v2: revised, the figure improve
On reducibility of n-ary quasigroups
An -ary operation is called an -ary quasigroup of order
if in the equation knowledge of any elements
of , ..., uniquely specifies the remaining one. is permutably
reducible if
where
and are -ary and -ary quasigroups, is a permutation, and
. An -ary quasigroup is called a retract of if it can be
obtained from or one of its inverses by fixing arguments. We prove
that if the maximum arity of a permutably irreducible retract of an -ary
quasigroup belongs to , then is permutably reducible.
Keywords: n-ary quasigroups, retracts, reducibility, distance 2 MDS codes,
latin hypercubesComment: 13 pages; presented at ACCT'2004 v2: revised; bibliography updated; 2
appendixe
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