左$C$-Wrpp 半群的加细半格结构

本文研究了左$C$-wrpp半群的加细半格结构,证明了左$C$-wrpp半群是左-${\cal R}$可消带的加细半格当且仅当它是一个$C$-wrpp半群和一个左正则带的织积.     

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.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

Cancellative Orders

left order in Q and Q is a semigroup of left quotients of S if every q∈Q can be written as q=a^*b for some a, b∈S where a^* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the {\cal D}-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S. The authors are grateful to the ARC for its generous financial support.     

The New Structure Theorem of Right-e Wlpp Semigroups

Wlpp semigroups are generalizations of lpp semigroups and regular semi-groups. In this paper, we consider some kinds of wlpp semigroups, namely right-e wlpp semigroups. It is proved that such a semigroup S , if and only if S is the strong semilattice of L-right cancellative planks;also if and only if S is a spined product of a right-e wlpp semigroup and a left normal band.     

Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.     

Split Left GC-Lpp Semigroups

A left GC-lpp semigroup S is called split if the natural homomorphism γb of S onto S/γ induced by γ is split.It is proved that a left GC-lpp semigroup is split if and only if it has a left adequate transversal.In particular,a construction theorem for split left GC-lpp semigroups is established.     

具有中心幂等元的L-弱正则半群

本文定义了具有中心幂等元的(L)-弱正则半群,研究了这类半群的代数结构.利用半群上的右同余(L)+和左同余R+,证明了半群S是一个具有中心幂等元的(L)-弱正则半群,当且仅当S是H-左可消幺半群的强半格.这推广了Clifford半群的相应结果.     

The structure of left C-rpp semigroups

This paper studies the class of left Clifford-rpp semigroups and investigates the structure of their semi-spined products and semilattice decompositions. These semigroups are generalizations of left Clifford semigroups and Clifford-rpp semigroups. We also discuss some special cases such as when a semilattice decomposition becomes a strong semilattice decomposition and a semi-spined product becomes a spined product. Communicated by Boris Schein This research is jointly supported by a grant of National Natural Science Foundation of China and a small project grant #200.600.380 of CUHK.     

The direct sum of semigroups is of inner type

A semigroup S is said to be of left inner type if the monoid of left translations of S is canonically isomorphic with S¹. The semigroups of left inner type form a special class of semigroups whose simple characterization is not known. The main purpose of this paper is to show that the direct sum of at least three semigroups is of left inner type. A brief discussion is given on the exceptional case, when the semigroup is the direct sum of two summands.     

偏序半群的左基

引入偏序半群左基的的概念，给出了一个偏序半群左基的存在性与极大左理想之间的关系。最后还讨论了在什么情况下左基的存在能导出基的存在性。作为应用，本文中所有结论在一靓半群中均成立。     

Semigroups which contain magnifying elements are factorizable

A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma).

In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.     

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.     

序半群的阿基米德半群的半格分解

本文通过一个序半群S上的一些二元关系以及它的理想的根集的性质该序半群是阿基米德半群的半格,特别地是阿基米德半群的链的刻划,证明了S是阿基米德链当且仅当S是准素的.通过序半群的m-系的概念,证明了S的任意半素理想是含它的所有素理想的交,并通过该结论,证明了S是阿基米德半群的链当且仅当S是阿基米德半群的半格且S的所有素理想关于集包含关系构成链.作为应用,该结论在一般的半群（没有序）[1]中也成立.     

A generalized Clifford theorem of semigroups

A U-abundant semigroup S in which every H-class of S contains an element in the set of projections U of S is said to be a U-superabundant semigroup.This is an analogue of regular semigroups which are unions of groups and an analogue of abundant semigroups which are superabundant.In 1941,Clifford proved that a semigroup is a union of groups if and only if it is a semilattice of completely simple semigroups.Several years later,Fountain generalized this result to the class of superabundant semigroups.In this p...     

Quasi-reflexive rings with non-nilpotent minimal one-sided ideals

In this paper we show that a ring is a member of the class of rings named in the title if and only if the ring is quasi-reflexive and contains at least one idempotent canonical quasi-ideal.We also prove the latter criterion is equivalent to several other ones.To attain that, we introduce the concept of a left n-socle and dually that of a right n-socle for arbitrary rings.An example is displayed to show that the presence of e.g.a nonzero right n-socle in a ring does not ensure the existence of a nonzero left n-socle.But in the quasi-reflexive case, it turns out that the notion of a left n-socle coincides with the right one.Finally, we give decomposition results which mainly deal with nonzero n-socles of quasi-reflexive rings and their semigroups n-socles concordantly, thereby, generalizing corresponding work in the semiprime case.     

The structure of bisimple left unipotent semigroups as ordered pairs

Call a semigroup S left unipotent if eachℒ-class of S contains exactly one idempotent. A structure theorem for bisimple left unipotent semigroups is given which reduces to that of N. R. Reilly [8] for bisimple inverse semigroups. A structure theorem, alternative to one given by R. J. Warne [13], is given for the case when the band E_S of idempotents of S is an ω-chain of right zero semigroups, and two applications of it are made. This research was partially supported by a grant from the National Science Foundation.