n -Ary Quasigroups of Order 4

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.