具有左简纯正断面的正则半群

本文引入并研究了左简纯正断面, 得到了与之相关的若干刻画; 推广并丰富了Blyth 和AlmeidaSantos 于1996 年得到的关于左简逆断面及Kong 于2007 年得到的关于纯正断面的相关结果; 同时,给出了具有左简纯正断面的正则半群的结构定理.     

具有Clifford断面的正则纯正半群

给出了具有Clifford断面的右正规纯正半群的等价刻画,得到了具有Clifford断面的正则纯正半群的次直积分解,证明了具有Clifford断面的正则纯正半群一定是正则纯正群．     

Free orthodox semigroups and free bands with inverse transversals

It is well known that the subclass of inverse semigroups and the subclass of completely regular semigroups of the class of regular semigroups form the so called e-varieties of semigroups. However, the class of regular semigroups with inverse transversals does not belong to this variety. We now call this class of semigroups the ist-variety of semigroups, and denote it by IST. In this paper, we consider the class of orthodox semigroups with inverse transversals, which is a special ist-variety and is denoted by OIST. Some previous results given by Tang and Wang on this topic are extended. In particular, the structure of free bands with inverse transversals is investigated. Results of McAlister, McFadden, Blyth and Saito on semigroups with inverse transversals are hence generalized and enriched.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.     

Good congruences on abundant semigroups with PSQ-adequate transversals

In this paper, we will characterize good congruences on abundant semigroups with \(PSQ\) -adequate transversals by using the so called good congruence triple, which generalizes some results on abundant semigroups with multiplicative adequate transversals.     

Semilattices of bisimple orthodox semigroups

A structure theorem for bisimple orthodox semigroups was given by Clifford [2]. In this paper we determine all homomorphisms of a certain type from one bisimple orthodox semigroup into another, and apply the results to give a structure theorem for any semilattice of bisimple orthodox semigroups with identity in which the set of identity elements forms a subsemigroup. A special case of these results is indicated for bisimple left unipotent semigroups.     

E-unitary almost factorizable orthodox semigroups

It is established that an E-unitary almost factorizable orthodox semigroup need not be isomorphic to a semidirect product of a band by a group, and a necessary and sufficient condition is given for an E-unitary almost factorizable orthodox semigroup to be isomorphic to such a semidirect product. Moreover, the structure of every E-unitary almost factorizable orthodox semigroup is described by means of bands and groups.     

Dually perfect orthodox dubreil-jacotin semigroups

It is known that for a strong Dubreil-Jacotin semigroup the propeq of being perfect is equivalent to that of being naturally ordered. Here we consider the existence of a smallest idempotent and the analogous notion of being dually perfect, whch we show is not equivalent to being dually naturally ordered. The main objective of the paper is to obtain a suucture theorem for dually perfect orthodox Dubreil-Jacotinsemigroups on which Green's relations are regular.     

On congruence lattices of regular semigroups with Q-inverse transversals

Communicated by F. Pastijn