On Orders in Normal Cryptogroups

A semigroup S is an order in a semigroup Q if every element of Q is of the form q?=?a ^?1 b?=?cd ^?1 for some ${a,b,c,d\in S}$ , where a ^?1 and d ^?1 are inverses within group ${{\mathcal H}}$ -classes of Q. We study orders in normal cryptogroups based on considering the restrictions of Green’s relations of Q to S. This produces certain relations which help us to gain an overview of all normal cryptogroups Q in which S is an order. Distinguished orders play an important role in this context.     

拟正则密码群并半群的一些等式(英文)

本文在拟正则密码群并半群范围内给出了局部纯正的拟正则密码群并半群及纯正的拟正则密码群并半群等的等式。     

A Construction of Cryptogroups

Abstract In this paper, a cryptogroup is constructed by a band and a family of groups and homomorphisms satisfying some conditions.     

密码群的若干性质

In this paper, we give some characterizations of(regular, normal) cryptogroups with Green's relations, left(right) translations and homomorphisms.     

A Construction of Regular Cryptogroups

In this paper, regular cryptogroups are constructed by left and right regular bands and a family of groups and homomorphisms satisfying some conditions.     

Cryptogroups with an associate subgroup

A cryptogroup is a completely regular semigroup S on which Green’s relation $\mathcal{H}$ is a congruence. For a,x∈S, x is an associate of a if a=axa. A subgroup G of S is an associate subgroup of S if it contains precisely one associate of each element of S. Further, S is a regular (respectively normal) cryptogroup if $S/\mathcal{H}$ is a regular (respectively normal) band. We provide a construction of a general (respectively regular or normal) cryptogroup in terms of groups and functions. On this model of S, we find several conditions equivalent to S containing an associate subgroup G. We characterize several varieties of completely regular semigroups, provided with the unary operation s?s ^?, where s ^? is the associate of s in G. They include completely regular semigroups, (regular, normal) cryptogroups, completely simple semigroups, and their monoid and/or overabelian members.     

A Lattice of Quasivarieties of Normal Cryptogroups

A normal cryptogroup S is a completely regular semigroup in which is a congruence and is a normal band. We represent S as a strong semilattice of completely simple semigroups, and may set For each we set and represent by means of an h-quintuple These parameters are used to characterize certain quasivarieties of normal cryptogroups. Specifically, we construct the lattice of quasivarieties generated by the (quasi)varieties and This is the lattice generated by the lattice of quasivarieties of normal bands, groups and completely simple semigroups. We also determine the B-relation on the lattice of all quasivarieties of normal cryptogroups. Each quasivariety studied is characterized in several ways.     

Some Normal Cryptogroups as Semigroups of Isomorphisms

Given a Clifford semigroup G, we construct special G-operands L and R which we term conformai. Certain suboperands of L and R we call threads and fix some special G-isomorphisms, which we term coherent, of threads in R onto threads in L. On the set of all coherent G-isomorphisms of threads in L onto threads in R we define a sandwich-type multiplication. When we restrict our threads to be cyclic suboperands of L and R, this construction produces a normal cryptogroup which we represent as $ S=[Y;S_{\alpha},\chi_{{\alpha},{\beta}}] $ -Without any restriction on the threads this produces a semigroup isomorphic with a remarkable ideal of the translational hull of S. Conversely, given a strong semilattice of completely simple semigroups, satisfying certain conditions, we can represent it isomorphically as indicated above.     

A Problem on Central Cryptogroups

Using a construction theorem of cryptogroups, we positively answer an open problem which is proposed by Petrich and Reilly in [1]: ${\cal OBG}\vee{\cal BA}={\cal CBG}?Using a construction theorem of cryptogroups, we positively answer an open problem which is proposed by Petrich and Reilly in [1]:     

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 )).     

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．     

局部左正则纯正密群的一个等式

本文给出局部左正则纯正密群的一个等式.证明了一个完全正则半群是左半正规密群当且仅当它是局部左正则纯正密群.     

关于l－群的半单结构

令Ｇ是一个ｌ－群，Ｇ的一个凸ｌ－子群Ａ叫做多余的，如果对Ｇ的任－凸ｌ－子群Ｗ，只要Ａ∨Ｗ＝Ｇ，就有Ｗ＝Ｇ．复令Ｒ（Ｇ）为Ｇ的所有多余凸ｌ－子群的集合并生成的凸ｌ－子群．我们证明了Ｒ（Ｇ）是ｌ－群Ｇ的一种报并且是在Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ意义下的根．进一步我们得到了有限值ｌ－群的半单结构定理即Ｒ（Ｇ）＝０当且仅当Ｇｌ－同构于具有半单性的单ｌ－群的亚直积，同时我们还得到了一系列有意义的推论．     

关于l－群的半单结构

令Ｇ是一个ｌ－群，Ｇ的一个凸ｌ－子群Ａ叫做多余的，如果对Ｇ的任－凸ｌ－子群Ｗ，只要Ａ∨Ｗ＝Ｇ，就有Ｗ＝Ｇ．复令Ｒ（Ｇ）为Ｇ的所有多余凸ｌ－子群的集合并生成的凸ｌ－子群．我们证明了Ｒ（Ｇ）是ｌ－群Ｇ的一种报并且是在Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ意义下的根．进一步我们得到了有限值ｌ－群的半单结构定理即Ｒ（Ｇ）＝０当且仅当Ｇｌ－同构于具有半单性的单ｌ－群的亚直积，同时我们还得到了一系列有意义的推论．     

关于NF-环的构造

研究了无限的ＮＦ－环及有限的ＮＦ－环，并且给出了有限ＮＦ－环的构造．     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

On the Structure of Specht Modules

Most of our notation is taken from James [7], where furtherdetails of the representation theory of the symmetric groupsmay be found; note, however, that we write functions on theleft. Let n be a non-negative integer, and a partition of n. Saythat two -tableaux are row equivalent if one can be obtainedfrom the other by permuting the entries within each row, anddefine column equivalence similarly. Let _row and _col denotethese relations.