Lattice-valued Convergence Spaces Research Articles

On the continuous action of enriched lattice-valued convergence groups: Some examples

Starting with a category SL-CONVGRP, of stratified enriched cl-premonoid-valued convergence groups as introduced earlier, we present a category SL-CONVTGRP, of stratified enriched cl-premonoid-valued convergence transformation groups, the idea behind this category is crept in the notion of convergence transformation group - a generalization of topological transformation group. In this respect, we are able to provide natural examples in support to our endeavor; these examples, however, stem from the action of convergence approach groups on convergence approach spaces, and the action of probabilistic convergence groups under triangular norm on probabilistic convergence spaces. Based on the category of enriched lattice-valued convergence spaces, a Cartesian closed category that enjoys lattice-valued convergence structure on function space, we look into among others, the lattice-valued convergence structures on the group of homeomorphisms of enriched lattice-valued convergence spaces, generalizing a concept of convergence transformation groups on convergence spaces, obtaining a characterization.

Quantale-valued fuzzy Scott topology

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varepsilon)$ is called a generalized GL-monoid if$(L,*,varepsilon)$ is a commutative unital quantale and the operation $*$ is$otimes$-semi-distributive. For generalized GL-monoid $L$ as thetruth value table, we systematically propose the stratified$L$-generalized convergence spaces based on stratified $L$-filters,which makes various existing lattice-valued convergence spaces asspecial cases. For $L$ being a commutative unital quantale, wedefine a fuzzy Scott convergence structure on $L$-fuzzy dcpos anduse it to induce a stratified $L$-topology. This is the inducing wayto the definition of quantale-valued fuzzy Scott topology, whichseems an appropriate way by some results.

CONNECTING T AND LATTICE-VALUED CONVERGENCES

$top$-filters can be used to define $top$-convergence spaces in the lattice-valued context. Connections between $top$-convergence spaces and lattice-valued convergence spaces are given. Regularity of a $top$-convergence space has recently been defined and studied by Fang and Yue. An equivalent characterization is given in the present work in terms of convergence of closures of $top$-filters. Moreover, a compactification of a $top$-convergence space is constructed whenever $L$ is a complete Boolean algebra.

Modified Top-convergence spaces and their relationships to lattice-valued convergence spaces

Modified Top-convergence spaces and their relationships to lattice-valued convergence spaces

Lattice-valued spaces: ⊤-Completions

The concept of a ⊤-Convergence space has recently been introduced and studied. These spaces are related to the top level space of a lattice-valued convergence space. In order to consider completions, ⊤-Cauchy spaces are defined and investigated in the present work. Each ⊤-Cauchy space possesses a completion and, moreover, each topological ⊤-Cauchy space has a topological completion. Completions of ⊤-uniform limit spaces are also introduced.

Lattice-valued convergence associated with CNS spaces

In this paper, in the context of meet continuous quantale, we present a notion of conical topological lattice-valued convergence space, which provides the theory of lattice-valued convergence associated with such conical neighborhood spaces.

Connectedness for lattice-valued subsets in lattice-valued convergence spaces

In this paper, a connectedness in stratified L-generalized convergence spaces, is defined and discussed. This connectedness is different from Jäger’s connectedness, in particular, it is defined for L-subsets however Jäger’s connectedness is defined for crisp subsets. Thus this give a partly answer to Jäger’s question in 2016: define a connectedness for L-subsets in lattice-valued convergence spaces. Then the basic properties of this connectedness are discussed.

THE CATEGORY OF T-CONVERGENCE SPACES AND ITS CARTESIAN-CLOSEDNESS

In this paper, we define a kind of lattice-valued convergence spaces based on the notion of $top$-filters, namely $top$-convergence spaces, and show the category of $top$-convergence spaces is Cartesian-closed. Further, in the lattice valued context of a complete $MV$-algebra, a close relation between the category of$top$-convergence spaces and that of strong $L$-topological spaces is established. In details, we show that the category of strong $L$-topological spaces is concretely isomorphic to that of strong $L$-topological $top$-convergence spaces categorically and bireflectively embedded in that of $top$-convergence spaces.

Connectedness and local connectedness for lattice-valued convergence spaces

We generalize Preuss' E-connectedness to lattice-valued convergence spaces and prove the basic theory for connected sets, including the product theorem. We further give a suitable definition of local E-connectedness and study its properties.

P-Topologicalness and p-regularity for lattice-valued convergence spaces

In this paper, p-topologicalness and p-regularity for Jägerʼs stratified L-generalized convergence spaces and those for Boustique et al.ʼs stratified L-convergence spaces are studied through two different methods. One method generalizes the well-known Fischerʼs (dual) diagonal condition, the other method uses the (neighborhood) closure of stratified L-filter. In addition, the p-topologicalness and p-regularity w.r.t. initial constructions and final constructions are also discussed.

A NOTE ON STRATIFIED LM-FILTERS

We develop a theory of stratified $LM$-filters which generalizes the theory of stratified $L$-filters. Our stratification condition explicitly depends on a suitable mapping between the lattices $L$ and $M$. If $L$ and $M$ are identical and the mapping is the identity mapping, then we obtain the theory of stratified $L$-filters. Based on the stratified $LM$-filters, a general theory of lattice-valued convergence spaces can be developed.

Modifications: Lattice-valued structures

Several important topological concepts such as regularity, local compactness, and local boundedness are investigated in the category of lattice-valued convergence spaces. These definitions coincide with the usual ones in familiar categories.

Diagonal conditions for lattice-valued uniform convergence spaces

We define diagonal conditions for lattice-valued uniform convergence spaces and show that these conditions are preserved under initial constructions. Further, the forgetful functor from the category of stratified lattice-valued uniform convergence spaces into the category of stratified lattice-valued limit spaces maps these conditions to corresponding conditions. We especially define uniform regularity and characterize this condition by certain closures of lattice-valued filters. As an application we generalize an extension theorem for uniformly continuous mappings.

Gähler's neighborhood condition for lattice-valued convergence spaces

We generalize a neighborhood condition from the category of convergence spaces to the category of lattice-valued convergence spaces. For a space in the category of convergence spaces, this condition is equivalent to being a topological space. It turns out that there are two meaningful generalizations of this condition to the category of lattice-valued convergence spaces. The first one guarantees that a space in a certain subcategory is a lattice-valued topological space. The other one is a levelwise condition which is equivalent to a generalization of Fischer's diagonal condition. The latter generalization is preserved under the embedding of the categories of convergence approach spaces and of probabilistic limit spaces into the category of lattice-valued convergence spaces.

Largest and smallest T2-compactifications of lattice-valued convergence spaces

We give necessary and sufficient conditions for a lattice-valued convergence space to possess a largest or a smallest T2-compactification.

A one-point compactification for lattice-valued convergence spaces

We define a lattice-valued one-point compactification for a non-compact lattice-valued T2-space and find criteria when it is smaller than another T2-compactification of the same space. Also the concept of local boundedness is introduced and characterized using the lattice-valued one-point compactification and the lattice-valued Richardson compactification.

LATTICE-VALUED CATEGORIES OF LATTICE-VALUED CONVERGENCE SPACES

We study L-categories of lattice-valued convergence spaces. Suchcategories are obtained by fuzzifying the axioms of a lattice-valued convergencespace. We give a natural example, study initial constructions andfunction spaces. Further we look into some L-subcategories. Finally we usethis approach to quantify how close certain lattice-valued convergence spacesare to being lattice-valued topological spaces.

A note on regularity

An alternative characterization of Gähler's definition of regularity of a lattice-valued convergence space is given. In particular, a space is regular exactly whenever convergence of filters remains invariant with respect to taking “closures.” Verification employs a slight modification of the work by Jäger.

Compactness in lattice-valued function spaces

We study conditions which ensure the compactness and weak relative compactness of a set of mappings between two lattice-valued convergence spaces. We consider lattice-valued pointwise convergence and lattice-valued continuous convergence. A suitable notion of even continuity for subsets of a function space is introduced which allows to pass from compactness with respect to the lattice-valued pointwise convergence to compactness with respect to lattice-valued continuous convergence. It is shown that with a regularity condition for the second space, we can deduce compactness conditions for sets of continuous mappings between two lattice-valued convergence spaces.

Lattice-Valued Cauchy Spaces and Completion

We define lattice-valued Cauchy spaces. The category of these spaces is topological over SET and cartesian closed. Special examples are lattice-valued uniform convergence spaces and probabilistic Cauchy spaces. We further define completeness and give a completion for these spaces which has the property that Cauchy continuous mappings between spaces can be extended to Cauchy-continuous mappings between their completions.