On inverse semigroups the closure of whose set of idempotents is a clifford semigroup (original) (raw)

O ct 2 01 8 CONJUGACY IN INVERSE SEMIGROUPS

In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a ∼ i b, if there exists g ∈ S 1 such that g −1 ag = b and gbg −1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.

Inverse semigroups with idempotent-fixing automorphisms

Semigroup Forum, 2014

A celebrated result of J. Thompson says that if a finite group G has a fixed-point-free automorphism of prime order, then G is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixedpoint-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

Inverse semigroup homomorphisms via partial group actions

Bulletin of the Australian Mathematical Society, 2001

This paper constructs all homomorphisms of inverse semigroups which factor through an S-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation fia with a preserving the maximal group image, fi idempotent separating, and the domain I of /3 E-unitary; moreover, the P-representation of / is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is ^-unitary. Stronger results are obtained for the case of F-inverse monoids.

Conjugacy in inverse semigroups

Journal of Algebra

In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a ∼ i b, if there exists g ∈ S 1 such that g −1 ag = b and gbg −1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.

A Munn Type Representation for a Class of E-Semiadequate Semigroups

Journal of Algebra, 1999

Munn's construction of a fundamental inverse semigroup T E from a semilattice E provides an important tool in the study of inverse semigroups. We present here a semigroup F E that plays for a class of E-semiadequate semigroups the role that T E plays for inverse semigroups. Every inverse semigroup with semilattice of idempotents E is E-semiadequate. There are however many interesting E-semiadequate semigroups that are not inverse; we consider various such examples arising from Schützenberger products.

Inverse semigroups with zero: covers and their structure

Journal of the Australian Mathematical Society, 1999

We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroup is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all £*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical £*-unitary semigroups recently considered by Gomes and Howie. 1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18.

Expansions of inverse semigroups

Journal of the Australian Mathematical Society, 2006

Abstract. We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes preflx expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms,which we term Fmorphisms. These play the same role in the theory of idempotent-pure homomorphisms,that F-inverse monoids play in the theory of E-unitary inverse semigroups.