Normal transformation semigroups

Abstract

Many transformation semigroups S over a set X have the inner-automorphism property : their automorphisms are precisely those mappings of the form g -1 · g : α → g -1 αg(α∈ S ) where g is a fixed permutation of S . For example, the full transformation semigroup, any two sided ideal of this semigroup, and, with some exceptions for small cardinals, the alternating group and the symmetric group itself. See Malcev (1952) and Scott (1965, Chapter 11). In this paper we determine all transformation semigroups over finite X with this property. In fact, we do more: we characterize all transformation semigroups invariant under the mappings g -1 · g . We refer to these as -normal transformation semigroups.