Congruences on eventually regular semigroups

Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored.     

毕竟正则半群上的同余及■(■)关系

探讨了毕竟正则半群上的(～Γ)(～R)等价关系.通过毕竟正则半群的同态像得到了一些信息,并且利用已知半群的性质及同态象的信息构造了一类半群.我们的结果推广了Edwards和Hall的相应结果.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

拟正则半群的圈积嵌入与矩阵同余

给出了拟正则半群的圈积嵌入．研究了幂等元集 Ｅ Ｓ 生成的子半群〈 Ｅ Ｓ〉上的矩阵同余可扩张为 Ｓ上的矩阵同余的条件． 特别地,如果〈 Ｅ Ｓ〉上的最小矩阵同余可扩张为 Ｓ上的矩阵同余,那么〈 Ｅ Ｓ〉上的每个矩阵同余可扩张为 Ｓ上的矩阵同余     

The Green relations approach to congruences on completely regular semigroups

On the congruence lattice (2(5) of a completely regular semigroup S the following mappings are considered _p: P and _p: VP, whereP is any of the Green relationsH, L, R orD. The equivalence relationsP andP ^V induced by these maps represent the main object of study in the paper. The former is a complete -congruence whereas the latter is a complete congruence onC(S). In particularH ,H ^V,L ^V,R ^V coincide with the kernel, trace, left trace and right trace relations onC(S), respectively. All essential properties known for the latter relations carry over to the new relationsP andP ^V. In addition, some interesting interplays of these provide for more richness in the theory of congruences on completely regular than is the case for the kernel-trace approach to congruences on regular semigroups.     

Perturbations of eventually differentiable and eventually norm-continuous semigroups on Banach spaces

In this paper we discuss perturbations of eventually differentiable and eventually norm-continuous semigroups on a Banach space. Two kinds of new perturbation theorems are obtained.     

Clifford拟正则半群的半直积

给出了两个半群S和T的半直积是Clifford拟正则半群的充要条件，同时还讨论了S和T^e半直积的结构，其中T^e＝{t^e|Vt∈T,Vc∈E（S）}。     

Congruences on regular semigroups

The kernel of a congruence on a regular semigroup S may be characterized as a set of subsets of S which satisfy the Teissier-Vagner-Preston conditions. A simple construction of the unique congruence associated with such a set is obtained. A more useful characterization of the kernel of a congruence on an orthodox semigroup (a regular semigroup whose idempotents form a subsemigroup) is provided, and the minimal group congruence on an orthodox semigroup is determined.     

Completely regular semigroups

No abstract. August 22, 2001     

M-classification of regular semigroups

On any regular semigroup S, the least group congruence σ, the greatest idempotent pure congruence τ and the least band congruence β are used to give the M -classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ, K, T), where K and T are the restrictions of the K- and T-relations on {C(S) to Λ. Such triples are characterized abstractly and form the objects of a category M whose morphisms are surjective T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category M whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from M to M. Several properties of the classification of regular semigroups induced by this functor are established.     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.