余Frame范畴中的等化子和余等化子结构

本文对余Frame范畴(以余Frame为对象,以余Frame同态为态射)中的等化子和余等化子的结构进行了研究。首先,利用子余Frame和嵌入映射构造了两个态射(具有相同论域和相同余论域)的等化子;其次,利用商余Frame和典型映射构造了两个态射(具有相同论域和相同余论域)的余等化子。本文的结果是对余Frame研究工作的补充和完善。     

Abel范畴中态射乘积的性质

本文在Ａｂｅｌ范畴中讨论了态射α，β的积αβ的性质。作为应用的例子，利用这些性质证明了以域上矩阵为态射的范畴中矩阵乘积的秩的恒等式。     

态射的Draxin逆

本文研究了范畴中态射的Ｄｒａｚｉｎ逆，给出了一般范畴中态射的｛１＾ｍ，２，５｝逆的一个等价刻划。在Ａｂｅｌ范畴中，建立了指数与Ｄｒａｚｉｎ逆的概念，证明了有Ｄｒａｘｉｎ逆的态射必有柱心－幂零分解。     

具有广义分解态射的广义(i,…,j)逆

本文研究范畴中态射的广义(i，…，j)逆，利用态射广义分解的性质给出了态射广义(i，…，j)逆存在的一些充要条件，导出了态射的广义Moore-Penrose逆的表达式，推广了态射(i，…，j)逆的相应结果．     

形式背景的AE-仿紧性

利用拓扑学的思想定义了形式背景的AE-仿紧性,给出了AE-仿紧背景的充分条件,研究了AE-仿紧背景的若干性质.证明了AE-仿紧性被适当的信息态射所保持,对一类闭嵌入子背景是遗传的.在以形式背景为对象,信息态射为态射的范畴FCC中,给出了两个形式背景乘积对象的表示,证明了两个AE-仿紧背景的乘积对象还是AE-仿紧的.     

态射的Drazin逆

本文研究范畴中态射的Ｄｒａｚｉｎ逆．给出了一般范畴中态射的｛１ｍ，２，５｝逆的一个等价刻划．在Ａｂｅｌ范畴中，建立指数与Ｄｒａｚｉｎ逆的概念，证明了有Ｄｒａｚｉｎ逆的态射必有柱心－幂零分解．     

具有广义分解态射的加权Moore-Penrose逆

研究范畴中态射的加权Moore-Penrose逆,利用态射广义分解的性质给出了态射加权Moore-Penrose逆存在的一些充要条件,导出了态射的加权Moore-Penrose逆的表达式,推广了态射Moore-Penrose逆的相应结果.     

预加法范畴中态射集星型序的刻划

本文给出了预加法范畴中态射集左（右）星序，星序的显公式刻划，证得了素性，正性预加法范畴中态射集星型序的某些刻划，把Ｈａｒｔｗｉｇ，Ｓｔｙａｎ，Ｍｉｔｒａ，Ｂａｋｓａｌａｒｙ关于复矩阵星型序刻划的有关结果推广到这两类范畴中。     

分块态射的广义逆

ClineRE给出了分块矩阵的Moore-Penrose逆的表达式,PetrPeska引进了分块态射的记号且导出了分块态射的Moore-Penrose逆的表达式.本文中,我们推广了Cline型分块态射的记号并得到了Cline型分块态射的Moore-Penrose逆和Drazin逆以及群逆的表达式.     

Locale范畴的反射子范畴

本文研究ｌｏｃａｌｅ范畴的反射子范畴，给出反射子范畴的刻划定理，从一般的ｌｏｃａｌｅ出发，完全构造性地给出了ｌｏｃａｌｅ的正则反射、完全正则反射和零维反射的构造．     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

范畴Ω-Cat的完备性

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中的等值子和乘积,给出了范畴Ω-Cat中乘积的有点式和无点式刻画,证明了范畴Ω-Cat是完备范畴。     

Localization and Codensity Triples

u: A→H(I), I ε J], and let x: Q→S be the equalizer of ηS and Sη.     

半环的L-模糊理想的范畴性质

首先给出了半环的L-模糊理想同态的定义,在此基础上较系统地讨论了半环的L-模糊理想范畴的性质,证明了此范畴是半环范畴上的一个拓扑结构,并探讨了其中的等子、拉回和乘积等性质.另一方面,给出了半环的L-模糊理想范畴的逆系统的定义,建立了半环的L-模糊理想范畴中逆系统的逆极限结构.特别是在引入两个逆系统之间映射的基础上,得到了两个逆系统的逆极限之间的极限映射.     

基于个人超出值的模糊联盟合作博弈最小二乘预核仁

本文给出了基于个人超出值的无限模糊联盟合作博弈最小二乘预核仁的求解模型,得到该模型的显式解析解,并研究该解的若干重要性质。证明了:本文给出的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的相等解(The equalizer solution),基于个人超出值的字典序解三者相等。进一步证明了:基于Owen线性多维扩展的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的经典合作博弈最小二乘预核仁相等。最后,通过数值实例说明本文提出的无限模糊联盟合作博弈求解模型的实用性与有效性。     

Left Equalizer Simple Semigroups

We characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence \({\varrho}\) on a semigroup S, let \({{\mathbb F}[\varrho]}\) denote the ideal of the semigroup algebra \({{\mathbb F}[S]}\) which determines the kernel of the extended homomorphism of \({{\mathbb F}[S]}\) onto \({{\mathbb F}[S/\varrho]}\) induced by the canonical homomorphism of S onto \({S/\varrho}\). We examine the right colons (\({{\mathbb F}[\varrho] :_{r} {\mathbb F}[S]) = {a \epsilon {\mathbb F}[S] : {\mathbb F}[S]a \subseteqq {\mathbb F}[\varrho]}}\) in general, and in that special case when \({\varrho}\) has the property that the factor semigroup \({S/\varrho}\) is left equalizer simple.     

Homoflatness on Ideal Extensions

We consider monoids $S=G\;\dot{\cup}\; I$ where $G$ is a group and $I$ is an ideal of $S$ and show that if an $S$-act is principally weakly homoflat or weakly homoflat as an $I^1$-act, then it has these properties as an $S$-act. We also show that an $S$-act which is (weakly) pullback flat, equalizer flat, (principally) weakly kernel flat, translation kernel flat or satisfies Condition $(E)$ as an $I^1$-act may not generally have these properties as an $S$-act. The flatness notions considered in this paper were introduced in {\it V. Laan, Pullbacks and flatness properties of acts I, Comm. Alg. ${\bf 29}(2)$ (2001), 829--850}.     

Pre-torsors and Galois Comodules Over Mixed Distributive Laws

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street’s bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions (N _A ,R _A ) and (N _B ,R _B ) on one hand, and the category of regular comonad arrows (R _A ,ξ) from some equalizer preserving comonad \mathbb C{\mathbb C} to N _B R _B on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras. Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad \mathbb D{\mathbb D} and a co-regular comonad arrow from \mathbb D{\mathbb D} to N _A R _A , such that the comodule categories of \mathbb C{\mathbb C} and \mathbb D{\mathbb D} are equivalent.     

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)?{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.