Permutability Class of a Semigroup

Given a permutation σ Sn, a semigroup S is called σ-permutable if x1x2•••xn = xσ(1)xσ(2)•••xσ(n) for all n-tuples x1, x2,…,xn S. Various permutability conditions were considered for special classes of permutations. We define the permutability class of a semigroup S to be the infinite sequence (G2, G3, G4,…) of the invariance groups Gn = Gn(S) of the operation x1x2•••xn on S. The aim of this paper is to give a description of such sequences, which generalizes a number of old results on permutability conditions in semigroups.