Comments on the permutation property for semigroups
Let S be a semigroup and n an integer (n\(\geq 2)\). We say that S has the property \(P_ n\) if for any \(x_ 1,x_ 2,...,x_ n\) in S there exists a permutation \(\sigma\) of \(\{\) 1,2,...,n\(\}\), \(\sigma\neq id\), such that \(x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\). We say that S has the property P (the ``permutation property'') if S satisfies \(P_ n\) for some n. Similarly, we say that S has the property \(P^*_ n\) if for any \(x_ 1,x_ 2,...,x_ n\) in S, there exist two distinct permutations \(\sigma\), \(\tau\) such that \[ x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}=x_{\tau (1)}x_{\tau (2)}...x_{\tau (n)}, \] and we say that S has the property \(P^*\) if it satisfies \(P^*_ n\) for some n. In this paper we prove the following propositions: Proposition 1. The bicyclic semigroup B has the property \(P^*_ 5\) but it does not have the property P. Also, any direct product of a finite number of copies of the bicyclic semigroup has the property \(P^*\). - Proposition 2. Let S and D be semigroups, with D finite, and let \(\phi\) be a morphism from S into D. If for every idempotent e of \(\phi\) (S), \(\phi^{-1}(e)\) has the property P (resp. \(P^*)\), then S has the property P (resp. \(P^*)\).
scientific article; zbMATH DE number 4087924