Fuzzifying Topology Research Articles

Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line

In this paper, three natural fuzzifying topologies are presented on the fuzzy real line. Then the notion of fuzzifying pseudo-quasi-metrics is introduced. It is proved that the three fuzzifying topologies can be induced respectively by three fuzzifying pseudo-quasi-metrics. Our definition of fuzzifying pseudo-metric is slightly different from that of KM-fuzzy metric. A fuzzifying pseudo-metrics can be regarded as a weak form of a KM fuzzy metric.

Fuzzifying bornologies induced by fuzzy pseudo-norms

The main purpose of this paper is to introduce two new fuzzifying bornologies induced by fuzzy pseudo-norms. It is proved that these new fuzzifying bornologies are compatible with the vector structure. With the help of concrete examples, it is presented that these special kinds of fuzzifying bornologies are different from an example (introduced by Šostak and Uļjane, 2016 [23]) of a fuzzifying bornology. It is shown that the degree to which a sequence is convergent to a point bornologically is equal to the degree to which this sequence is convergent to above point topologically in fuzzy pseudo-normed linear spaces. Moreover, it is presented that V. Neumann fuzzifying bornology is separated in fuzzy pseudo-normed linear spaces if and only if the fuzzifying topology determined by a fuzzy pseudo-norm is Hausdorff.

Intuitionistic Fuzzy Topology Based on Intuitionistic Fuzzy Logic

There are many symmetries in intuitionistic fuzzifying topology. In the present paper, the notion of intuitionistic fuzzifying topology as an extension of fuzzifying topology and a preliminary of the research on bi-intuitionistic fuzzy topology is introduced. A theory of intuitionistic fuzzy topology with the semantic method of intuitionistic fuzzy logic is established.

Fuzzifying topology induced by Morsi fuzzy pseudo-norms

The purpose of this paper is to endow the fuzzy pseudo-norm in the sense of Morsi with many-valued topological structures. It is shown that there exists a one-to-one correspondence between any fuzzy pseudo-norm given by Morsi and a family of left continuous and non-ascending pseudo-norms. Then a fuzzifying topology induced by Morsi fuzzy pseudo-norms is introduced, it is proved that this fuzzifying topology is compatible with the vector structure. Based on this many-valued topological structure, the degree to fuzzy normed space which is Hausdorff, the degree to a sequence which is convergent, and the degree to a set which is bounded are studied. The layered characterizations of them are presented. Finally, the conclusion which a linear operator is fuzzy continuous if and only if it is fuzzy bounded is proved.

A new definition of fuzzy k-pseudo metric and its induced fuzzifying structures

In this paper, a new definition of a fuzzy $k$-pseudo metric is introduced and its induced fuzzifying structures are constructed, such as a fuzzifying neighborhood system, a fuzzifying topology, a fuzzifying closure operator, a fuzzifying uniformity.Besides, it is shown that there is a one-to-one correspondence between fuzzy $k$-pseudo metrics and nests of crisp $k$-pseudo metrics.

Linear fuzzifying uniformities

The main purpose of this paper is to study Hutton type fuzzifying uniformities on linear spaces. Firstly, we show that if a base of a fuzzifying uniformity defined over a linear space is translation-invariant, balanced and absorbed, then it generates a linear fuzzifying topology. From this linear fuzzifying topology, we can construct a new linear fuzzifying uniformity (i.e., a fuzzifying uniformity compatible with the linear structure) which is equivalent to the original fuzzifying uniformity. Secondly, the Hausdorff separation and complete boundedness in linear fuzzifying uniformities are investigated. In addition, as an example, the linear fuzzifying uniformity induced by a fuzzy norm is also discussed.

A construction of a fuzzy topology from a strong fuzzy metric

<p>After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems, 6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology ${\mathcal T}:2^X \to [0,1]$ induced by a fuzzy metric $m: X\times X \times [0,\infty)$ was constructed. In this paper we extend this construction to get the fuzzy topology ${\mathcal T}: [0,1]^X \to [0,1]$ and study some properties of this fuzzy topology.54A</p>

Fuzzifying topology induced by a strong fuzzy metric

A construction of a fuzzifying topology induced by a strong fuzzy metric is presented. Properties of this fuzzifying topology, in particular, its convergence structure are studied. Our special interest is in the study of the relations between products of fuzzy metrics and the products of the induced fuzzifying topologies.

Fuzzifying topologies on the space of linear operators

We introduce the notion of fuzzifying topologies on the space of linear operators. The concepts of fuzzifying topologies of pointwise convergence, bounded convergence, and complete bounded convergence are presented. We prove a sufficient and necessary condition for which the fuzzifying topologies are compatible with the structure on the space of linear operators. The Hausdorff separation of a fuzzifying topology on the space of linear operators is discussed. Specially, the net convergence, fuzzy boundedness, and fuzzy continuity of evaluation mappings in the fuzzifying topological linear space of pointwise convergence are investigated.

Locally α-compact spaces based on continuous valued logic

Abstract This paper is a continuation of [1] . That is, it considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [2] . It investigates topological notions defined by means of α-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]). Other characterizations of fuzzifying α-compactness are given, including characterizations in terms of nets and α-subbases. Several characterizations of locally α-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

α-Irresoluteness and α-compactness based on continuous valued logic

Abstract This paper considers fuzzifying topologies, a special case of I -fuzzy topologies (bifuzzy topologies), introduced by Ying [1] . It investigates topological notions defined by means of α -open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]) . The concept of α -irresolute functions and α -compactness in the framework of fuzzifying topology are introduced and some of their properties are obtained. We use the finite intersection property to give a characterization of fuzzifying α -compact spaces. Furthermore, we study the image of fuzzifying α -compact spaces under fuzzifying α -continuity and fuzzifying α -irresolute maps.

A Unified Theory (I) for Neighborhood Systems and Basic Concepts on Fuzzifying Topological Spaces

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying. It investigates topological notions defined by means of -open sets when these are planted into the frame-work of Ying’s fuzzifying topological spaces (by Lukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying operations from P(X) to , where (X, τ) is a fuzzifying topological space. By making use of them we contract neighborhood structures, derived sets, closure operations and interior operations.

On Fuzzifying Nearly Compact Spaces

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies) introduced by Ying [16, (I)]. It investigates topological notions defined by means of regular open sets when these are planted into the frame-work of Ying's fuzzifying topological spaces (in <TEX>${\L}$</TEX>ukasiewwicz fuzzy logic). The concept of fuzzifying nearly compact spaces is introduced and some of its properties are obtained. We use the finite intersection property to give a characterization of fuzzifying nearly compact spaces. Furthermore, we study the image of fuzzifying nearly compact spaces under fuzzifying completely continuous functions, fuzzifying almost continuity and fuzzifying R-map.

Intuitionistic I-fuzzy topological spaces

The main purpose of this paper is to introduce the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic I-fuzzy topological spaces and the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic I-fuzzy topology is proposed, and the connections among generated intuitionistic I-fuzzy topological spaces, fuzzifying topological spaces and I-fuzzy topological spaces are discussed. Finally, the properties of the operators Iω, ι are obtained.

CATEGORICAL RELATIONS AMONG MATROIDS, FUZZY MATROIDS AND FUZZIFYING MATROIDS

The aim of this paper is to study the categorical relations between matroids, Goetschel-Voxman's fuzzy matroids and Shi's fuzzifying matroids. It is shown that the category of fuzzifying matroids is isomorphic to that of closed fuzzy matroids and the latter is concretely coreflective in the category of fuzzy matroids. The category of matroids can be embedded in that of fuzzifying matroids as a simultaneously concretely reflective and coreflective subcategory. 1. Introduction and Preliminaries Matroids were introduced by Whitney in 1935 as a generalization of both graphs and metric spaces. It is well known that matroids play an important role in math- ematics and especially in applied mathematics. Matroids are precisely the struc- tures for which the very simple and ecient greedy algorithm works (4, 10). In 1988, matroid theory was generalized to fuzzy fields by Goetschel and Voxman (7). Their approach to the fuzzification of matroids preserves many basic properties of crisp matroids. Fuzzy circuits, fuzzy rank functions and fuzzy closure operators are widely studied (5, 6, 7, 8), while none of the corresponding axioms has been established. In (11), based on the idea of a fuzzifying topology (12), Shi intro- duced a fuzzifying matroid and established a one-to-one correspondence between all fuzzifying matroids and all fuzzy rank functions. The aim of this paper is to study the relations among matroids, fuzzy matroids and fuzzifying matroids from a categorical viewpoint. The paper is organized as follows. In Section 2, we study the relations between the category of fuzzy matroids and that of closed fuzzy matroids and show that the category of closed fuzzy ma- troids CFM is a concretely coreflective full subcategory of the category of fuzzy matroids FM. In Section 3, the properties of fuzzifying matroids are studied. We show that FYM, the category of fuzzifying matroids, is isomorphic to CFM. In Section 4, we show that the category of matroids M can be embedded in FYM as a simultaneously concretely reflective and coreflective subcategory. In summary, we show that

Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders

The present paper investigates the relations between fuzzifying topologies and generalized ideals of fuzzy subsets, as well as constructing generalized ideals and fuzzifying topologies by means of fuzzy preorders. Furthermore, a construction of generalized ideals from preideals, and vice versa, is obtained. As a particular consequence of the results in this paper, a construction of fuzzifying topology generated by generalized ideals of fuzzy subsets via a givenI-topology is given. The notion ofσ-generalized ideal is introduced and hence everyσ-generalized ideal is shown to be a fuzzifying topology induced by some fuzzy preorder.

Fuzzy orders and fuzzifying topologies

The paper presents the essential connections between fuzzy preorders and fuzzifying topologies. In detail, we establish systematic procedure indicating how fuzzifying topologies are induced from fuzzy preordered relations on a set X, and conversely, we will answer the problem that the specialization order can be induced by a fuzzifying topology on a set X. Moveover, a powerful method generating fuzzy preordered relations is obtained.

One-to-one correspondence between fuzzifying topologies and fuzzy preorders

This paper is devoted to the discussion of the relationship between fuzzifying topologies and fuzzy relations. A saturated fuzzifying topology can be generated by a reflexive fuzzy relation; and conversely, a fuzzy preorder can be generated by a fuzzifying topology. In fact, there exists a one-to-one correspondence between the set of all saturated fuzzifying topologies and that of all fuzzy preorders.

On [formula omitted]-fuzzy quasi-uniform spaces

An ( L , M ) -fuzzy topology is a graded extension of topological spaces handling M-valued families of L-fuzzy subsets of a referential, where L and M are completely distributive lattices. When M reduces to the set 2 = { 0 , 1 } , a ( 2 , M ) -fuzzy topology is called a fuzzifying topology after Ying. Šostak introduced the notion ( L , M ) -fuzzy uniform spaces. The aim of this paper is to study the relationship between ( 2 , M ) -fuzzy quasi-uniform spaces and ( L , M ) -fuzzy quasi-uniform spaces as well as the relationship between ( 2 , M ) -fuzzy quasi-uniform spaces and pointwise ( L , M ) -fuzzy quasi-uniform spaces—the extension of Shi's L-quasi-uniform space in a Kubiak–Šostak sense. It is shown that the category of ( 2 , M ) -fuzzy quasi-uniform spaces can be embedded in the category of stratified ( L , M ) -fuzzy quasi-uniform spaces as a both reflective and coreflective full subcategory; and the former category can also be embedded in the category of pointwise ( L , M ) -fuzzy quasi-uniform spaces.

Linear operators in fuzzifying topological linear spaces

In this paper, some of the properties of continuous linear operators between fuzzifying topological linear spaces are studied. Particularly, the equivalence between continuity and boundedness is investigated and the closed graph theorem is generalized to the fuzzifying setting. Finally some properties of the initial fuzzifying topologies determined by a family of linear operators are investigated.