Some remarks on the permutation property for semigroups (Q921125)

A semigroup S has property P (resp. \(P^*)\) if for some \(n\geq 1\), and for any \(s_ 1,...,s_ n\) in S, there exists a permutation \(\sigma\neq id\) (resp. there exist permutations \(\sigma\neq \alpha)\) in \(S_ n\), such that \(s_ 1...s_ n=s_{\sigma (1)}...s_{\sigma (n)}\) (resp. \(s_{\alpha (1)}...s_{\alpha (n)}=s_{\sigma (1)}...\sigma_{\sigma (n)})\). If \(S=E\cup F\), where E is a subsemigroup and F is finite, and if E has property P or \(P^*\), then so has S. If S is the union of k subsemigroups each having P (resp. \(P^*)\), then S has P (resp. \(P^*)\). If \(S\to D\) is a homomorphism with bounded fibers, and D has \(P^*\), then so has S. If M is a Rees matrix semigroup (S;I,J;P), and if S is finite, then M has \(P^*\); if I and J are finite, and S has P (resp. \(P^*)\), then so has M.