On the weak permutation property (Q1318966)

Say that a semigroup \(S\) has the property \(P^*_ n\) if, for any \(x_ 1,\dots,x_ n\), there exist two distinct permutations \(\sigma\), \(\tau\) such that \(x_{\sigma_ 1},\dots,x_{\sigma_ n} = x_{\tau_ 1} \dots x_{\tau_ n}\). A semigroup \(S\) has the weak permutation property \(P^*\), if \(S\) has \(P^*_ n\) for some \(n\). It was shown by \textit{J. Justin, G. Pirillo} [Semigroup Forum 39, 109-112 (1989; Zbl 0665.20028)] that the bicyclic semigroup \(B\) does not have \(P^*_ 3\). In the present note it is proved that the bicyclic semigroup \(B\) has \(P^*_ 4\).