Epicompletion in Frames with Skeletal Maps,I: Compact Regular Frames

A frame homomorphism h : A ⟶ B is skeletal if x ^⊥⊥ = 1 in A implies that h(x)^⊥⊥ = 1 in B. It is shown that, in , the category of compact regular frames with skeletal maps, the subcategory , consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in . Further, is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of , the boolean algebra of polars of A.      

DR0代数:由De Morgan代数导出的正则剩余格

首先讨论了De Morgan代数与剩余格的关系,并引入强De Morgan代数的概念,讨论了它的基本性质．随后,将著名的R0蕴涵拓广到De Morgan代数上,称为广义R0蕴涵;证明了添加广义凰蕴涵和相应 算子后的De Morgan代数L成为剩余格的充要条件是L为强De Morgan代数,并由此引入D‰代数的概念．接着,研究了DR0代数与‰代数的关系,证明了以下结论：Boole代数是DR0代数;全序DR0代数和全序R0代数等价;DR0代数是R0代数当且仅当它满足预线性条件;无中点的DR0代数是BL代数当且仅当它是Boole代数．最后,举例说明了非D兄D代数的RD代数、以及非R0代数的DR0代数都是存在的．     

Compact Hausdorff Approach Frames

In this paper we provide a Heine–Borel type characterization for 0-compactness in approach spaces (Lowen 1997). Since this requires making use of the so-called regular function frame the most natural setting to develop this in is approach frames (Banaschewski 1999; Banaschewski et al., Acta Math Hung 115(3):183–196, 2007, Topology Appl 153:3059–3070, 2006). We then go on to characterize Hausdorffness for approach frames which allows us to study some fundamental properties of compact Hausdorff approach frames.     

Regular Monomorphisms of Hausdorff Frames

Regular monomorphisms in the category of Hausdorff frames are characterized by means of a naturally defined closure operator; this is used also to characterize the epimorphisms. Further it is shown that for spatial (strongly) Hausdorff frames the regular monomorphisms do not generally coincide with the quotients, and do not generally compose. Also, an additional property (under which regular monomorphisms do compose) is briefly studied.     

弱Hopf代数与正则幺半群

本文定义了弱Ｈｏｐｆ代数并研究了弱Ｈｏｐｆ代数的弱对极与类群元幺半群的关系．首先,本文给出弱Ｈｏｐｆ代数的一些基本性质;然后,对类群元幺半群是逆半群或纯正半群的某些弱Ｈｏｐｆ代数,描述了其弱对极的一些性质;最后,给出一类其类群元幺半群为正则半群的弱Ｈｏｐｆ代数．     

Compact Composition Operators and Deddens Algebras

We consider the Deddens algebras associated to compact composition operators on the Hardy space \(H^2\) on the unit disk. When the compact composition operator corresponds to a function \(\varphi \) that satisfies \(\varphi (0)=0\) and \(\varphi '(0)\ne 0\), we show that the lattice of invariant subspaces of this algebra is \(\{0\}\cup \{z^n H^2: n=0,1,2,\ldots \}\). As a consequence, for this class of operators the associated Deddens algebra is weakly dense in the algebra of lower triangular matrices.     

A Generalization of De Vries Duality Theorem

Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of locally compact Hausdorff spaces and perfect maps.     

Regular Unital Finite-Dimensional Lattice-Ordered Algebras

The main result in this article is to show that a regular unital finite-dimensional lattice-ordered algebra over ? with zero ?-radical is isomorphic to a finite direct sum of lattice-ordered matrix algebras of lattice-ordered group algebras of finite groups over ?.     

Derivations in Commutative Regular Algebras

Mathematical Notes -     

分裂的正则Hom-Leibniz-Rinehart代数

作为Hom-Leibniz代数胚的代数类比, 本文引入Hom-Leibniz-Rinehart代数的概念. 证明了分裂的正则Hom-Leibniz-Rinehart代数$L$写成$L=U+\sum_{\gamma}I_\gamma$, 其中$U$为极大交换子代数$H$的子空间和$I_\gamma$为$L$的理想, 若$[\gamma]\neq[d]$, 满足$[I_\gamma, I_d]=0$. 随后分别发展了分裂Hom-Leibniz-Rinehart代数的根和权的连通技术.最后研究了紧致的正则Hom-Leibniz-Rinehart代数的结构.     

Prelinear Algebras in Relatively Regular Quasivarieties

Given a quasivariety of lattice-ordered algebras, the linearly ordered algebras therein generate the subquasivariety of prelinear algebras. In the case that there exist a constant 1 and binary term i such that the quasivariety satisfies: $1 \leq i(x,y) \Leftrightarrow x \leq y$ , we give an explicit axiomatization of the prelinear subquasivariety, relative to the original quasivariety. The existence of 1 and i with the above property is equivalent to the quasivariety being ‘relatively 1^?-regular’, by which we mean that each relative congruence is characterized by the negative cone of its 1-class. Dual results hold in the positive cone case.     

分裂的正则Hom-Poisson 着色代数结构

We introduce the class of split regular Hom-Poisson color algebras as the natural generalization of split regular Hom-Poisson algebras and the one of split regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show∑that such a split regular Hom-Poisson color algebras A is of the form A = U +αIα with U a subspace of a maximal abelian subalgebra H and any Iα, a well described ideal of A, satisfying[Iα, Iβ] + IαIβ = 0 if [α]≠[β]. Under certain conditions, in the case of A being of maximal length, the simplicity of the algebra is characterized.     

Partition Complete Boolean Algebras and Almost Compact Cardinals

For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated (e. g. every K-Suslin algebra is a K-partition complete Boolean algebra). It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ ≥ K. For strongly inaccessible K we show that K is K-almost compact iff K is weakly compact, and if K is 2^K-almost compact, then K is measurable. Further K is strongly compact iff it is γ-almost compact for all γ ≥ K.     

Free Algebras in Certain Varieties of Distributive Pseudocomplemented De Morgan Algebras

In this paper we characterize the join irreducible elements of the free algebras on n free generators in the subvarieties of the variety V₀ of pseudocomplemented De Morgan algebras satisfying the identity xx′* = (xx′*)′*.     

Epicompletion in Frames with Skeletal Maps, II: Compact Normal Joinfit Frames

An epireflection ψ is constructed of the category $\mathfrak{KNArS}$ of compact normal joinfit frames, with skeletal maps, in the subcategory $\mathfrak{SPArS}$ consisting of strongly projectable $\mathfrak{KNArS}$ -objects. The construction is achieved via a pushout in the category $\mathfrak{FrmS}$ of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ _A :A?→?ψA one obtains the extension of ρA to its absolute.