n-Ary quasigroups of order 4

Abstract

We characterize the set of all N-ary quasigroups of order 4: every N-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an N-ary quasigroup can be represented as a composition of K-ary and (N-K+1)-ary quasigroups for some K from 2 to N-1, where the order of arguments in the representation can differ from the original order. The set of semilinear N-ary quasigroups has a characterization in terms of Boolean functions. Keywords: Latin hypercube, n-ary quasigroup, reducibility

10pp. V2: revised

References in corpus (4)

• Asymptotics for the number of n-quasigroups of order 4
• On reconstructing reducible n-ary quasigroups and switching subquasigroups
• On irreducible n-ary quasigroups with reducible retracts
• On reducibility of n-ary quasigroups

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• Distance-2 MDS codes and latin colorings in the Doob graphs
• Constructions of transitive latin hypercubes
• MDS codes in the Doob graphs
• On the number of frequency hypercubes $F^n(4;2,2)$
• On a connection between the switching separability of a graph and that of its subgraphs
• On a test on switching separability of graphs modulo $q$