  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
