An algebraic system consisting of 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 (θ0, θ1, …, θn) of n + 1 permutations of Σ such that f(θ1(x1), …, θn(xn)) ≡ θ0(f(x1, …, xn)). We show that every n-ary quasigroup of order 4 has at least 2[n/2]+2 and not more than 6 · 4n autotopies. We characterize the n-ary quasigroups of order 4 with 2(n+3)/2, 2 · 4n, and 6 · 4n autotopies.

Original languageEnglish
Pages (from-to)227-250
Number of pages24
JournalQuasigroups and Related Systems
Issue number2
Publication statusPublished - 1 Jan 2019


  • Autotopy group
  • Latin hypercube
  • Multiary quasigroup

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