On the number of autotopies of an n-ary qusigroup of order 4
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An algebraic system from a finite set Σ of cardinality k and an n-ary operation f invertible in each argument is called an n-ary quasigroup of order k. An autotopy of an n-ary quasigroup (Σ,f) is a collection (θ_0,θ_1,...,θ_n) of n+1 permutations of Σ such that f(θ_1(x_1),...,θ_n(x_n))≡θ_0(f(x_1,...,x_n)). We show that every n-ary quasigroup of order 4 has at least 2^[n/2]+2 and not more than 6· 4^n autotopies. We characterize the n-ary quasigroups of order 4 with 2^(n+3)/2, 2· 4^n, and 6· 4^n autotopies.
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