Sn-Normal Semigroups of Partial Transformations

Abstract

For an integer n ≥ 1, let [Formula: see text] and Snbe, respectively, the semigroup of partial transformations and the symmetric group on the set X = {1,…,n}. Then Snis the group of units of [Formula: see text]. A subsemigroup S of [Formula: see text] is Sn-normal if for all a ∈ S and g ∈ Sn, g-1ag ∈ S. In 1976, Symons described the Sn-normal semigroups of full transformations of X. In 1995, Lipscomb and the second author determined the Sn-normal semigroups of partial injective transformations of X. In this paper, we complete the classification by describing all Sn-normal subsemigroups of [Formula: see text]. As a consequence of the classification theorem, we obtain a characterization of the automorphisms of any Sn-normal subsemigroup of [Formula: see text].