(￡)集合范畴的Cartesian闭性

定义(た)集合范畴中的一些基本概念,并研究了(た)集合范畴的格值函数空同,进一步指出了格值函数空间函子与格值积函子互为伴随.即(た)集合范畴满足Gartesian闭性,其为Cartesian闭范畴.     

范畴Ω-Cat的函数空间及其性质

Ω-范畴具有范畴论和序理论的双重意义,可为计算机程序语言的语义提供量化的模型,本文研究了范畴Ω-Cat的Ω-值函数空间,得出了Ω-值函数空间函子与Ω-值积函子互为伴随函子,证明了范畴Ω-Cat是Cartesian闭范畴。     

Z-连续格的函数空间

若 Z为并完备的子集系统 ,且 IZ( L)关于集合的包含关系构成完备格 ,则 :( 1 ) Z-连续格的函数空间仍为 Z-连续的 ;( 2 )对于 Z-连续格范畴 ZL ,定义了一函子 F:ZL× ZL→ ZL.     

模糊集层次结构的本质刻画

将Zadeh提出的模糊集的模糊结构提升到格值结构,引入赋予格值结构的集合概念,称之为格值集合,并给出了格值集合的表示定理.在此基础上,证明格值集合范畴可以嵌入到集合的层范畴,说明格值结构具有层结构这一特征,从而揭示格值集合具有层次结构,这一结果也刻画了Zadeh模糊集的层次结构的本质特征.     

双诱导型定向与逆向函子伴随性的研究

Ω-范畴具有范畴论和序理论的双重意义,可为计算机程序语言的语义提供量化的模型,给出了范畴(?)_(Ω_(1))(X)与范畴(?)_(Ω_(2))(Y)之间的双诱导型定向函子及双诱导型逆向函子的定义,同时证明了双诱导型定向函子与双诱导型逆向函子互为一对伴随函子.     

基于Ω-范畴的定向与逆向函子伴随性的研究

Ω-范畴具有范畴论和序理论的双重意义,可为计算机程序语言的语义提供量化的模型,本文给出了范畴RΩ(X)之间的Zadeh型定向函子与Zadeh型逆向函子的定义,同时证明了Zadeh型定向函子与Zadeh型逆向函子互为一对伴随函子。     

基于群范畴的L-fuzzy结构提升范畴及其表示

引入群范畴上L-fuzzy结构提升范畴与格值结构提升范畴概念,L-fuzz结构是在逻辑层面表达群理论多值语义的有点化描述,也是Zermelo-Fr(a)nkel公理集合理论和各种代数形式理论的格值模型的语义赋值,而格值结构是在范畴层面表达群理论多值语义的无点化描述,本文建立了L-fuzzy结构与格值结构这两种不同数学结构之间的联系,证明了在范畴层面上述两种结构是同构的.给出了基于群范畴的L-fuzzy结构的格值结构表示.     

集合范畴上超滤函子的余代数

定义了集合范畴上的超滤函子F_u(-),并研究了相关性质.包括函子F_u(-)在有限集上保拉回,一个集合的子集成为F_u-子余代数的充要条件,以及两个余代数之间的态射是F_u-余代数同态的充要条件,子集成为子余代数的充要条件,最后以拓扑空间作为F_u-余代数的具体实例,研究了拓扑空间的连续映射与超滤函子的余代数同态之间的关系.     

L-fuzzy集理论的范畴基础与层表示

鉴于L-fuzzy集在理论上的重要性和应用上的广泛性,旨在建立L-fuzzy集理论的范畴基础与它的层表示,提出完备范畴中对象上的格值结构概念,这一概念是L-fuzzy结构在范畴层面上的提升,进一步提出完备范畴上格值结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与格值结构是同构的.以集层、群层、环层和左R-模层以及Grothendieck层等概念为基础,提出完备范畴中对象上的层结构以及完备范畴上层结构提升范畴概念,证明了在集合范畴中L-fuzzy结构与层结构也是同构的.     

超滤函子余代数范畴

研究了超滤函子余代数范畴set_(F_u)的乘积和余积问题.首先构造了集合乘积上的超滤,讨论集合乘积上超滤的存在形式;接着利用超滤函子的性质给出了范畴set_(F_u)的有限乘积以及任意余积构造;最后证明了范畴set_(F_u)的终对象存在.改进了Gumm关于滤子函子的研究结果,深化了相关文献关于超滤函子余代数的研究.     

THE FORGETFUL FUNCTOR Cont → Prox IS TOPOLOGICAL

Abstract

It is shown that the forgetful functor from the category of contiguity spaces to the category of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetful functor from the category of totally bounded uniform spaces to the category of proximity spaces.     

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.     

几种格上拓扑空间范畴中乘积与上积运算的封闭性

本文引入四种格上拓扑空间范畴,分别讨论了其中的乘积与上积运算,以及相应的结构性、唯一性和存在性问题。通过讨论,给出了一种比较理想的格上拓扑空间的乘积空间,并指出目前使用的格上拓扑空间的乘积空间具有一定的局限性,其所属范畴的态射是Ｚａｄｅｈ型映射,这类映射保持Ｆｕｚｚｙ点的高度不变,隐含度量不变性。     

The Booleanization of an inverse semigroup

Semigroup Forum - We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what...     

Lattice-valued fuzzy interior operators

A category of lattice-valued fuzzy interior operator spaces is defined and studied. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces.     

一类满足可补性质的双重Stone代数

该文构造了一个从布尔代数范畴到满足可补性质的双重Ｓｔｏｎｅ 代数范畴的函子，并证明了这个函子有一个等价的左伴随函子．     

自由函子及其伴随函子

本文证明了由集合范畴到f-模范畴的自由函子的存在性，构造了自由函子的伴随函子。     

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.     

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.     

Sober拓扑分子格

A topological molecular lattice (TML) is a pair (L, T), where L is a completely distributive lattice and r is a subframe of L. There is an obvious forgetful functor from the category TML of TML‘s to the category Loc of locales. In this note,it is showed that this forgetful functor has a right adjoint. Then, by this adjunction,a special kind of topological molecular lattices called sober topological molecular lattices is introduced and investigated.