On permutation properties in groups and semigroups (Q1091476)

The authors investigate semigroups S satisfying a permutation property PPn (n\(\geq 2)\), in which every product of n elements remains invariant under some nontrivial permutation of its factors. In particular the case \(n=3\) is studied. If S is a group some criteria are derived for S satisfying PP3, e.g. \(| S'| \leq 2\) is an equivalent property. If S is a regular PP3 semigroup, then S is a semilattice of right or left PP3 groups; further structure theorems for this case are shown. Finally the question is answered negatively whether the semigroup algebra of a PPn semigroup satisfies a polynomial identity.