On the monotone complexity of the shift operator

We show that the complexity of minimal monotone circuits implementing a monotone version of the permutation operator on $n$ boolean vectors of length $q$ is $\mathrm{\Theta }(qn\log n)$. In particular, we obtain an alternative way to prove the known complexity bound $\mathrm{\Theta }(n\log n)$ for the monotone shift operator on $n$ boolean inputs.