TY - JOUR

T1 - On the number of autotopies of an n-ary quasigroup of order 4

AU - Gorkunov, Evgeny V.

AU - Krotov, Denis S.

AU - Potapov, Vladimir N.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Autotopy group

KW - Latin hypercube

KW - Multiary quasigroup

UR - http://www.scopus.com/inward/record.url?scp=85078449851&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85078449851

VL - 27

SP - 227

EP - 250

JO - Quasigroups and Related Systems

JF - Quasigroups and Related Systems

SN - 1561-2848

IS - 2

ER -