The bicyclic semigroup has \(P_ 4^*\) (Q2367903)

A semigroup \(S\) is said to have the property \(P^*_ n\), where \(n > 1\) is an integer, if for any \(s_ 1,\dots,s_ n \in S\) there exist distinct permutations \(\sigma\), \(\tau\) in the symmetric group of degree \(n\) such that \[ s_{\sigma(1)} \dots s_{\sigma(n)} = s_{\tau(1)} \dots s_{\tau(n)}. \] \textit{J. Justin} and \textit{G. Pirillo} [Semigroup Forum 39, No. 1, 109-112 (1989; Zbl 0665.20028)] proved that the bicyclic semigroup \(B\) has \(P^*_ 5\) but not \(P^*_ 3\). In the present note the authors prove that \(B\) has \(P^*_ 4\), too.