On Inverse Transversals of Ordered Regular Semigroups

We first consider an ordered regular semigroup S in which every element has a biggest inverse and determine necessary and sufficient conditions for the subset S ^○ of biggest inverses to be an inverse transversal of S. Such an inverse transversal is necessarily weakly multiplicative. We then investigate principally ordered regular semigroups S with the property that S ^○ is an inverse transversal. In such a semigroup we determine precisely when the set S ^☆ of biggest pre-inverses is a subsemigroup and show that in this case S ^☆ is itself an inverse transversal of a subsemigroup of S. The ordered regular semigroup of 2 × 2 boolean matrices provides an informative illustrative example. The structure of S, when S ^☆ is a group, is also described.     

完全单半群及完全正则半群的逆断面

指出完全单半群Ｓ的任何一个Ｆ－类是逆断面,且为Ｑ－逆断面,而Ｓ的任何一个逆断面必是一个Ｆ－类,因而所有逆断面同构。并且给出完全正则半群的逆断面存在的充要条件。     

Finitely Generated Commutative Regular Semigroups

Abstract

In this paper we give a construction that allows us to describe up to isomorphisms all finitely generated regular commutative semigroups.     

序半群的正则同余类

本文的目的是给出一个序半群的子集能成为某个正则同余的同余类的刻面,同时我们可以容易看出[5]中的关于一般半群(没有序关系)的相应的结论仅是本文的结论的应用.     

具有逆断面的正则半群的同余的表示

具有道断面S°的正则半群可表示为有Saito's结构的半群W（I，S°，Λ，*，α，β）.我们利用由I，S°和Λ上的同余构成的所谓同余聚抽象地表示这类半群上的同余，进而给出了这类半群的同态象的构造法．     

On Left Regular Ordered Semigroups

The main result of the paper is a decomposition theorem of the left regular ordered semigroups into left regular and left simple semigroups.     

序半群的一类正则同余

设S是有向序半群,本文给出了S上的一类正则同余,称为强序同余的定义及性质.证明了S的强序同余是强正则同余,但反之不成立.同时证明了强序同余格SOC(S)是S的同余格C(S)关于通常集合的交和传递积的V-完备的分配子格.     

自然序Dubreil-Jacotin富足半群

本文考虑Ｄｕｂｒｅｉｌ－Ｊａｃｏｔｉｎ富足半群,给出若干特征后,建立了自然序Ｄｕｂｒｅｉｌ－Ｊａｃｏｔｉｎ富足半群的结构,作为应用,给出了具有正则性条件的自然序Ｄｕｂｒｅｉｋｌ－Ｊａｃｏｔｉｎ富足半群的结构,并将结果推广到具有最大幂等元的自然序的偏序富足半群上。     

Regular F-Abundant Semigroups

ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.     

E-酉正则半群

本文定义了PO-六元组的概念,给出了E-酉正则半群的一种结构.     

Some Characterizations of Left Weakly Regular Ordered Semigroups

In this paper, the notion of left weakly regular ordered semigroups is introduced. Furthermore, left weakly regular ordered semigroups are characterized by the properties of their left ideals, right ideals and (generalized) bi-ideals, and also by the properties of their fuzzy left ideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.     

Some Characterizations of Left Weakly Regular Ordered Semigroups

In this paper, the notion of left weakly regular ordered semigroups is introduced. Furthermore, left weakly regular ordered semigroups are characterized by the properties of their left ideals, right ideals and （generalized） bi-ideals, and also by the properties of their fuzzy left ideals, fuzzy right ideals and fuzzy （generalized） bi-ideals.     

On E-Unitary Regular Semigroups

In this paper we define a concept of a PO-sextet and describe a structure of E-unitary regular semigroups. Received June 22, 1999, Revised May 9, 2000, Accepted June 1, 2000     

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.     

正则半群上的广义逆半群同余

利用同余的核与超迹描述正则半群上的广义逆半群同余。     

Flows on Regular Semigroups

We study the structure of the flow monoid of a regular semigroup. This arises from the approach of Nambooripad of considering a regular semigroup as a groupoid – a category in which every morphism is invertible. A flow is then a section to the source map in this groupoid, and the monoid structure of the set of all flows is determined in terms of the Green relations on the original semigroup.     

关于交换序半群的一个注记

给出了正则交换序半群的与素理想结构有关的若干性质，还给出了为有限个主理想的并的交换序半群类中的诺特性，阿基米德性，正则性以及有限生成性之间的一些关系。