单子和余单子的缠绕结构

研究单子和余单子的缠绕结构和缠绕模以及与代数和余代数的缠绕结构和缠绕模之间的关系,定义了余单子的类群元,得到了一些有意义的结论.最后构造了缠绕模范畴上的两个函子,并证明了它们是伴随函子.     

T-余单子可分的等价条件

文章类似于A-上环(coring)给出T-余单子(comonad)的一些性质(这里A是代数,T是单子(monad)).首先定义了实(firm)单子等相关概念,其次研究了与Frobenius函子等价的两个命题,最后给出了与余单子可分等价的五个命题.     

张量余单子与张量范畴

设是一个张量范畴,g和F均为上的张量余单子,p是一个余单子分配率.本文从FG的张量余单子结构和2-范畴的角度,描述了双余模范畴的张量结构,并给出了其做成张量范畴的一些充要条件. ’     

双单子分配律的R-矩阵

本文讨论了双单子分配律的表示及其R-矩阵结构.设F和G是给定的双单子,刻画了单子双模范畴,并给出了其为辫子范畴的充要条件,由此构造了量子YangBaxter方程的一组新解系.     

模糊理论与Kleisli范畴

运用范畴论的观点和语言,讨论了几种真值集不同的模糊集,得出它们都是特殊的模糊理论.更进一步,指出了模糊理论所对应的范畴与由模糊理论诱导的单子所构造的Kleisli范畴的等价关系.最后,通过一个实例,描述了伴随函子诱导的单子,并构造了相应的Kleisli范畴,指出了Kleisli范畴在模糊理论中的应用.     

超理想的单子及其应用

在扩大模型下,用超理想的单子对超理想进行刻画;进而用它给出了理想为超理想的条件;最后给出理想的单子与超理想的单子之间的关系.     

弱T余代数上的弱Doi-Hopf π-模和弱Yetter-Drinfeld π-模

本文揭示了弱Doi-Hopf π-模范畴和弱Yetter-Drinfeld π-模范畴之间的密切联系,并证明了弱Yetter-Drinfeld π-模范畴同构于一个T-范畴的中心.     

张量余单子的余半单性与余辫子结构

本文研究了张量余单子的余半单性和余表示范畴,给出了其余半单性和余可裂性的等价性定理.并证明了其余表示范畴是辫子范畴当且仅当该张量余单子是余辫子的.作为应用研究了张量型Hom-双代教的Hom-余模范畴的半单性和辫子结构.     

模糊拓扑空间中Moore-Smith收敛性的非标准刻画

应用了公理化非标准分析理论,将经典集合X上的模糊集扩张成*X上的模糊集.在非标准扩大模型中,定义了模糊邻近结构的单子,并讨论了这些单子的性质及其之间的关系.利用这些单子,刻画了模糊拓扑空间中模糊网的收敛性,进而得到了模糊拓扑空间中Moore-Smith收敛理论的非标准刻画.     

双扭HOPF代数的对偶

本文研究了一个双扭Hopf代数的分次对偶空间以及两个双扭Hopf代数的分次对偶关系.利用代数和余代数分次对偶空间的性质,得出一个局部有限的双扭(χ1,χ2)-Hopf代数的分次对偶空间是一个双扭(χ1T,χ2)-Hopf代数,并判定两个双扭Hopf代数的分次对偶可以简化为判定它们作为双扭双代数是分次对偶的.     

Galois theory in variable categories

The order-reversing bijection between field extensions and subgroups of the Galois group G follows from the equivalence between the opposite of the category of étale algebras and the category of discrete G-spaces [2]. We show that the basic ingredient for this equivalence of categories, and for various known generalizations, is a factorization system for variable categories.     

Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case

In proving the Fermionic formulae, a combinatorial bijection called the Kerov–Kirillov–Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial R matrices on rigged configurations.     

On Lattice Isomorphisms of Inverse Semigroups, II

A lattice isomorphism between inverse semigroups S and T is an isomorphism between their lattices of inverse subsemigroups. When S is combinatorial, it has long been known that a bijection is induced between S and T. Various authors have introduced successively weaker "archimedean" hypotheses under which this bijection is necessarily an isomorphism, naturally inducing the original lattice isomorphism. Since lattice-isomorphic groups need not have the same cardinality, extending these techniques to the non-combinatorial case requires some means of tying the subgroups to the rest of the semigroup. Ershova showed that if S has no nontrivial isolated subgroups (subgroups that form an entire D-class) then again a bijection exists between S and T. Recently, this technique has been successfully exploited, by Goberstein for fundamental inverse semigroups and by the author for completely semisimple inverse semigroups, under two different finiteness hypotheses. In this paper, we derive further properties of Ershova's bijection(s) and formulate a "quasi-connected" hypothesis that enables us to derive both Goberstein's and the author's earlier results as corollaries.     

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results.Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.     

Relative rigid objects in extriangulated categories

In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and τ-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with τ-rigid pairs, and also relative maximal rigid objects with support τ-tilting pairs under some assumptions. These results generalize the work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. In addition, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable complements. All results highlight new phenomena when they applied to exact categories.     

Interval Reductions and Extensions of Orders: Bijections to Chains in Lattices

We discuss bijections that relate families of chains in lattices associated to an order P and families of interval orders defined on the ground set of P. Two bijections of this type have been known:(1) The bijection between maximal chains in the antichain lattice A(P) and the linear extensions of P.(2) The bijection between maximal chains in the lattice of maximal antichains A_M(P) and minimal interval extensions of P.We discuss two approaches to associate interval orders with chains in A(P). This leads to new bijections generalizing Bijections 1 and 2. As a consequence, we characterize the chains corresponding to weak-order extensions and minimal weak-order extensions of P.Seeking for a way of representing interval reductions of P by chains we came upon the separation lattice S(P). Chains in this lattice encode an interesting subclass of interval reductions of P. Let S_M(P) be the lattice of maximal separations in the separation lattice. Restricted to maximal separations, the above bijection specializes to a bijection which nicely complements 1 and 2.(3) A bijection between maximal chains in the lattice of maximal separations S_M(P) and minimal interval reductions of P.     

套代数上Lie同态的一个局部刻画（英文）

In this paper,linear maps preserving Lie products at zero points on nest algebras are studied.It is proved that every linear map preserving Lie products at zero points on any finite nest algebra is a Lie homomorphism.As an application,the form of a linear bijection preserving Lie products at zero points between two finite nest algebras is obtained.     

Stable equivalence preserves representation type

Given two finite dimensional algebras and , it is shown that is of wild representation type if and only if is of wild representation type provided that the stable categories of finite dimensional modules over and $\Gamma$ are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over and , and the result follows from certain additional properties of this bijection. In the second part of this paper the Auslander-Reiten translation is extended to an operation on the category of all modules. It is shown that various finiteness conditions are preserved by this operation. Moreover, the Auslander-Reiten translation induces a homeomorphism between the set of non-projective and the set of non-injective points in the Ziegler spectrum. As a consequence one obtains that for an algebra of tame representation type every generic module remains fixed under the Auslander-Reiten translation. Received: July 24, 1996