Sn-Normal Semigroups of Partial Transformations

For an integer n ≥ 1, let [Formula: see text] and Sn be, respectively, the semigroup of partial transformations and the symmetric group on the set X = {1,…,n}. Then Sn is 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].

R-Unipotent Congruences on Eventually Regular Semigroups

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.

The Structure of Normal Inverse Semigroups

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.