WLR-regular orthogroups

A regular orthogroup S with the property that D _e =R _e or D _e =L _e for any idempotent e∈S is called a WLR-regular orthogroup. In this paper, we give constructions of such semigroups in terms of spined products of left and right regular orthogroups with respect to Clifford semigroups. WLR-cryptogroups and its special cases are also investigated. Research supported by General Scientific Research Project of Shanghai Normal University No. SK200707.     

Ortho-u-monoids

In this paper, we study the class of ortho-u-monoids which are generalized orthogroups within the class of E(S)-semiabundant semigroups. After introducing the concept of (∼)-Green’s relations, and obtaining some important properties of (∼)-Green’s relations and super E(S)-semiabundant semigroups, we have given the semilattice decomposition of ortho-u-monoids and a structure theorem for regular ortho-u-monoids. The main techniques that we used in the study are the (∼)-Green’s relations, and the semi-spined product of semigroups.     

The structure of superabundant semigroups

A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.     

On (*, ∼)-Green's Relations and Ortho-lc-Monoids

An ortho-lc-monoid is a generalized orthogroup within the class of rpp semigroups. In this article, after giving some important properties of r-ample semigroups and super-r-ample semigroups, some structure theorems of ortho-lc-monoids are established, and several special cases are considered. The main techniques that we use in the study are the (*, ～)-Green's relations and semispined product of semigroups.     

Yetter–Drinfel'd Modules for Group-Cograded Multiplier Hopf Algebras

We give a representation-theoretic and a categorical interpretation of the Drinfel'd double into the framework of group-cograded multiplier Hopf algebras. The Drinfel'd double as constructed by Zunino for a finite-type Hopf group-coalgebra is an example of this construction in the sense that the components of the group-cograded multiplier Hopf algebras are unital and finite-dimensional algebras and the admissible action is related with the adjoint action of the group on itself.     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.     

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.     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids

Let π be a group. In this article, we introduce the notions of a weak Doi–Hopf π-module and a weak π-twisted smash product. We show that the Yetter–Drinfel'd π-modules over a weak crossed Hopf π-coalgebra (WT-coalgebra) are special cases as these new weak Doi–Hopf π-modules, generalizing the main result by Caenepeel et al. (1997 Caenepeel , S. , Militaru , G. , Zhu , S. ( 1997 ). Crossed modules and Doi–Hopf modules . Israel J. Math. 100 : 221 – 248 .[Crossref] , [Google Scholar]) and that the Drinfel'd double for WT-coalgebras (Van Daele and Wang, 2008 Van Daele , A. , Wang , S. H. ( 2008 ). New braided crossed categories and Drinfel'd quantum double for weak Hopf group-coalgebras . Comm. Algebra 36 : 2341 – 2386 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) appears as, a type of such a weak π -twisted smash product, respectively. Finally, starting from a weak Hopf algebra endowed with an action of a group π by weak Hopf automorphisms, we construct a quasitriangular weak Hopf π -coalgebra by a twisted double method, generalizing the main result in Virelizier (2005 Virelizier , A. ( 2005 ). Graded quantum groups and quasitriangular Hopf group-coalgebras . Comm. Algebra 33 ( 9 ): 3029 – 3050 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This method allows us to obtain nontrivial examples of quasitriangular weak Hopf π-coalgebras.     

The structure of ?*-inverse semigroups

The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.     

Coquasitriangular Hopf Group Algebras and Drinfel'd Co-Doubles

A ring R is called “semicommutative” if any right annihilator over R is an ideal of R. We show that special subrings of upper triangular matrix rings over a reduced ring are maximal semicommutative. Consequently, new families of semicommutative rings are presented.     

Ideal extensions of lattices

Following the well-known Schreier extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford-Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice L and a lattice K having a least element, and construct (all) the lattices V which have an ideal L′ which is isomorphic to L and the Rees quotient V|L′ is isomorphic to K. Conversely, we prove that each lattice which is an extension of L by K can be so constructed. An illustrative example is given at the end. The text was submitted by the author in English.     

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．     

The Structure of Ortho-LC-Monoids

The super-r-wide semigroups (in [6 Guo , Y. Q. , Shum , K. P. , Gong , C. M. ( 2011 ). On (*, ～)-Green's relations and ortho-lc-monoids . Communications in Algebra 39 : 5 – 31 . [Google Scholar]], called super-r-ample semigroups) [ortho-lc-monoids] within the class of rpp semigroups form a kind of generalized completely regular semigroups [orthogroups]. In this article, some structure theorems of ortho-lc-monoids are established and some special ortho-lc-monoids such as orthocrypto-lc-monoids, lc-Clifford semigroups, which we defined, are considered; the semilattice decomposition of super-r-wide semigroups is given. As direct corollaries of the results that we obtained, some new structure theorems for ortho-c-monoids and orthogroups, different from [10, 13], are given, and hence, the structure theorem for ortho-lc-monoids that we established is not the direct generalization of the results in [10 Petrich , M. ( 1987 ). A structure theorem for completely regular semigroups . Proc. Amer. Math. Soc. 99 : 617 – 622 .[Crossref], [Web of Science ®] , [Google Scholar], 13 Ren , X. M. , Shum , K. P. ( 2007 ). On superabundant semigroups whose set of idempotents form a subsemigroup . Algebra Colloquium 14 : 215 – 228 .[Crossref] , [Google Scholar]]; the structure of orthocryptogroups is reobtained; the well-known Clifford theorem is further generalized.     

LR-normal orthogroups

A new kind of regular orthogroups, namely, the LR-normal orthogroups is introduced and investigated. In particular, we will introduce the gearing techniques for fitting semigroup components on a semilattice in the construction of LR-normal orthogroups.     

Left Abundant Semigroups

Abstract

The aim of this paper is to study some special lpp-semigroups, namely, the left GC-lpp semigroups. After obtaining some properties and characterizations of such semigroups, we establish some structure theorems of this class of semigroups. In addition, we also consider some special cases. As an application, we describe the structure theorems of IC quasi-adequate semigroups whose idempotent band is a regular band.     

Constructing New Braided T-Categories Over Regular Multiplier Hopf Algebras

Let A be a regular multiplier Hopf algebra, and let Aut(A) denote the set of all isomorphisms α from A to itself that are algebra maps satisfying (Δ ○ α)(a) = (α ? α) ○ Δ(a) for all a ∈ A. Let G be a certain crossed product group Aut(A) × Aut(A). The main purpose of this article is to provide a class of new braided T-categories in the sense of Turaev [\citealp9]. For this, we introduce a class of new categories _A 𝒴𝒟 ^A (α, β) of (α, β)-Yetter–Drinfel'd modules with α, β ∈Aut(A), and we show that the category ?𝒴𝒟(A) = { _A 𝒴𝒟 ^A (α, β)}_{(α, β)∈G} becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic [6 Panaite , F. , Staic , M. D. ( 2007 ). Generalized (anti) Yetter–Drinfel'd modules as components of a braided T-category . Israel J. Math. 158 : 349 – 365 .[Crossref], [Web of Science ®] , [Google Scholar]].