Algebraic Geometry Over Completely Simple Semigroups

Algebraic geometry over completely simple semigroups is considered. We prove theorems that define coordinate semigroups and irreducible coordinate semigroups in several classes of completely simple semigroups.     

完全■-单半群

完全■-单半群是完全单半群和完全■~*-单半群在U-半富足半群类中的一个自然推广.本文证明了半群S是完全■-单半群,当且仅当S同构于幺半群T上的正规Rees矩阵半群■(T;I,A;P).这一结果不仅推广了完全单半群的著名Rees定理,而且推广了任学明和岑嘉评在2004年建立的完全■~*-单半群的一个结构定理.     

群的正则带的KG-强半格分解

推广了著名的Petrich的完全正则半群为群的正规带当且仅当它为完全单半群的强半格的结果，证明了完全正则半群为群的正则(或右拟正规)带当且仅当它是完全单半群的HG(LG)-强半格．     

On Existence Varieties of E-solid Semigroups

Within the class of regular E-solid semigroups, a theory of e-varieties including appropriate notions of biidentities and biinvariant congruences is presented, such that, together with bifree objects, these notions inherit the properties and interrelations well known from universial algebra. This theory generalizes the previously developed such theory for orthodox semigroups. As an application, the bifree objects in certain e-varieties of E-solid locally orthodox semigroups, which are constructed by means of Malcev products from a varities of bands, groups and completely simple semigroups, are described as subsemigroups in suitable Pastijn products of some bands by relatively bifree completely simple semigroups. As a consequence, it follows that every regular E-solid locally orthodox semigroup regularly divides a so-called solid Pastijn product of a band by a completely simple semigroup.     

关于序半群的模糊拟理想

In this paper,fuzzy quasi-ideals of ordered semigroups are characterized by the properties of their level subsets.Furthermore,we introduce the notion of completely semiprime fuzzy quasi-ideals of ordered semigroups and characterize strongly regular ordered semigroups in terms of completely semiprime fuzzy quasi-ideals.Finally,we investigate the characterizations and decompositions of left and right simple ordered semigroups by means of fuzzy quasi-ideals.     

群的正则带的拟强半格分解

推广了半群的强半格分解的定义,得到了半群的拟强半格分解,并证明了完全正则半群为群 的正则(或右拟正规)带当且仅当它是完全单半群的拟强半格(且 )).     

完全单的矩阵半群

引入矩阵型Rees矩阵半群的概念,证明完全单的矩阵半群等价于矩阵型Rees矩阵半群,进而给出矩阵拓扑半群的极小理想的刻画以及完全正则矩阵半群特别是一些重要类别的群带的刻画．     

Intuitionistic fuzzy quasi-ideals of ordered semigroups

In this paper we define intuitionistic fuzzy quasi-ideals of ordered semigroups. The main result of the paper is a characterization of quasi-ideals in terms of intuitionistic fuzzy quasi-ideals. We also characterize left simple, right simple, and completely regular ordered semigroups in terms of intuitionistic fuzzy quasi-ideals. We study the decomposition of left and right simple ordered semigroups using intuitionistic fuzzy quasi-ideals.     

Some conditions related to the exactness of Rees-Sushkevich varieties

This article presents some conditions, expressed in terms of the inclusion and exclusion of certain small semigroups, that are related to a Rees-Sushkevich variety being generated by completely 0-simple or completely simple semigroups.     

Completely Regular Semigroup Varieties Whose Free Objects Have Weakly Permutable Fully Invariant Congruences

We prove that a variety of completely regular semigroups has the property from the title if and only if it consists of either completely simple semigroups or semilattices of groups.     

Completely simple semigroups with nilpotent structure groups

In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing [x ₁,x ₂,…,x _c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case. The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia.     

完全单Γ-半群的结构定理和具有完全单Γ-核的半群的结构定理

本文给出了完全单Γ-半群的结构定理和具有完全单Γ-核的半群的结构定理     

Free Completely J~(e)-simple Semigroups

A semigroup is called completely J~((e))-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid.It is proved that completely J~((e))-simple semigroups form a quasivarr ity.Moreover,the construction of free completely J~((e))-simple semigroups is given.It is found that a free completely J~((e))-simple semigroup is just a free completely J~*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

Free completely <Emphasis Type="Italic">J</Emphasis><Superscript>(<Emphasis Type="Italic">ℓ</Emphasis>)</Superscript>-simple Semigroups

A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

Regularity of semigroup rings

Sufficient conditions are obtained in order that the semigroup rings of some semigroups be regular. For some inverse, orthodox and completely simple semigroups, the necessary and sufficient conditions for the regularity of the semigroup rings are found. The author wishes to thank Prof. Y. Q. Guo for his best help and advice. Key Words and Phrases: (von Neumann) regularity, semigroup ring, inverse semigroup, orthodox semigroup, completely [0-] simple semigroup.     

On E-solid Semigroups with Inverse Transversals

We characterize E-solid semigroups with inverse transversals and some special such semigroups in terms of congruences. Some properties of inverse congruences over completely simple semigroups are obtained.1991 MR Subject Classification: 20M10This work was supported by the Natural Science Foundation of Guangdong, China.     

强可收缩半群

称半群Ｓ强可收缩,若Ｓ的每个子半群都是它的一个缩回．本文逐次刻划了强可收缩的群、完全单半群、强可收缩的半格及正则半群的结构,在此基础上给出了任意强可收缩半群的结构定理．     

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.     

Near varieties of idempotent generated completely simple semigroups

Algebra universalis - Completely simple semigroups form a variety if we consider them as algebras with multiplication and inversion. A near variety of idempotent generated completely simple...