Fuzzy Closure Operators Research Articles

Fuzzy closure structures as formal concepts

Galois connections seem to be ubiquitous in mathematics. They have been used to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a complete residuated lattice. The existence of three fuzzy Galois connections (two antitone and one isotone) between three specific ordered sets is proved in this paper. The most interesting part is that fuzzy closure systems, fuzzy closure operators and strong fuzzy closure relations are formal concepts of these fuzzy Galois connections.

Topological structures and distance functions in complete co-residuated lattices

Most of the research in fuzzy rough sets and fuzzy topological structures have been studied on the basis of fuzzy partially ordered sets. Instead of fuzzy partially ordered sets, the concept of distance functions in complete co-residuated lattices is introduced. Using distance functions, we define Alexandrov pretopology, Alexandrov precotopology and fuzzy interior (fuzzy closure) operators in complete co-residuated lattices, and we investigate their properties. Moreover, we prove that there exist isomorphic categories and Galois correspondence between topological categories.

Regularity and normality of (L,M)-fuzzy topological spaces using residual implication

In this paper, the notions of regularity and normality of (L,M)-fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that (L,M)-fuzzy interior operator and (L,M)-fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an (L,M)-fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an (L,M)-fuzzy metric space.

Fuzzy closure operators and their applications

Fuzzy closure operators play a significant role in fuzzy order theory. This paper aims to further enrich and improve the study of fuzzy closure operators. Based on the work of Belohlavek and Yao, we shall continue to study the relative properties of fuzzy closure operators. First, we shall consider the extensions of $L$-subsets via fuzzy closure operators. Then we give an application of fuzzy closure operators, that is, by fuzzy closure operators we shall prove that the category CFPos of complete fuzzy posets and their fuzzy-join preserving maps is a reflective full subcategory of FPos, where FPos denotes the category of fuzzy posets and their fuzzy-existing-join preserving maps.

Fuzzy rough sets with a fuzzy ideal

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X,R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambdaR and the fuzzy ideal approximation closure operator text {cl}_{lambda }^{R}. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.

On Bipolar Fuzzy Gradation of Openness

The concept of bipolar fuzziness is of relatively recent origin where in addition to the presence of a property, which is done in fuzzy theory, the presence of its counter-property is also taken into consideration. This seems to be much natural and realistic. In this paper, an attempt has been made to incorporate this bipolar fuzziness in topological perspective. This is done by introducing a notion of bipolar gradation of openness and to redefine the bipolar fuzzy topology. Furthermore, a notion of bipolar gradation preserving map is given. A concept of bipolar fuzzy closure operator is also introduced and its characteristic properties are studied. A decomposition theorem involving our bipolar gradation of openness and Chang type bipolar fuzzy topology is established. Finally, some categorical results of bipolar fuzzy topology (both Chang type and in our sense) are proved.

𝒢Θτi⋆τj-Fuzzy Closure Operator

In this paper, a new class of fuzzy ideal sets, namely the [Formula: see text]-[Formula: see text]-[Formula: see text]-generalized fuzzy ideal closed sets, is introduced for fuzzy bitopological spaces in Šostak sense. This class falls strictly in between the class of [Formula: see text]-[Formula: see text]-[Formula: see text]-fuzzy ideal closed sets and the class of [Formula: see text]-[Formula: see text]-generalized fuzzy ideal closed sets. Furthermore, by using the class of [Formula: see text]-[Formula: see text]-[Formula: see text]-generalized fuzzy ideal closed sets we establish a new fuzzy closure operator which generates fuzzy bitopological spaces in Šostak sense. Finally, the [Formula: see text] strongly-[Formula: see text]-fuzzy ideal continuous, [Formula: see text]-[Formula: see text]-generalized fuzzy ideal continuous and [Formula: see text]-[Formula: see text]-generalized fuzzy ideal irresolute mappings are introduced, and we show the [Formula: see text]-[Formula: see text]-generalized fuzzy ideal continuous properly fuzzy ideal bitopological spaces in Šostak sense (for short, fibtss) in between [Formula: see text] strongly-[Formula: see text]-fuzzy ideal continuous and [Formula: see text]-generalized fuzzy continuous mappings.

Generation of proto-fuzzy concepts from fixpoints of fuzzy closure operators

It is well known in Formal Concept Analysis that computation of all formal concepts from data table with graded attributes can be reduced to the problem of computing fixpoints of two fuzzy closure operators, ↑↓ and ↓↑. It is also true that as the size of datasets grows, the fuzzy concepts generated from fuzzy context become larger in number. Therefore for large and complex datasets, it is very hard to deal with such a large number of fuzzy concepts. To handle large and complex datasets, several alternative approaches were proposed to the fuzzy concept lattice theory by researchers. The fuzzy concepts introduced by Kridlo et al (2008) are known as proto-fuzzy concepts. In point of view of applications in different domain, significance of proto-fuzzy concepts is very much effective. But as far as our knowledge is concerned, there is no general method to generate proto-fuzzy concepts. In this paper, we present an algorithm for finding proto-fuzzy concepts directly from the input data. The algorithm we present generates proto-fuzzy concepts form the fixpoints of the fuzzy closure operators, ↑↓ and ↓↑.

Fuzzy roughness via ideals

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

Generalized closed fuzzy soft sets in Čech fuzzy soft closure spaces

In this paper, we present the concept of generalized closed fuzzy soft sets in the Čech fuzzy soft closure spaces, termed g-closed fss’s, and investigate their basic properties. By using g-closed fss’s, we define a new fuzzy soft closure operator which is generates a new fuzzy soft topology finer than that induced from the Čech fuzzy soft closure operator. In addition, we define generalized Čech fuzzy soft continuous (respectively, irresolute) mapping by means of g-closed fss’s. As application of these sets, -Čech fuzzy soft closure spaces are introduced and discuses.

L-fuzzy closure operators and L-fuzzy cotopologies

In this paper, we investigate $L$-fuzzy closure operators and $L$-fuzzy cotopologies in a complete residuated lattice. Also, we study the categorical relationship between $L$-fuzzy closure spaces and $L$-fuzzy cotopological space. Finally, we give their examples.

A categorical characterization of the least Q-quantale completion of Q-ordered semigroups

The main purpose of this paper is to describe the concrete form of the largest topological fuzzy closure operator on the power-set quantale of a Q-ordered semigroup. Meanwhile, we give a join- and meet-dense basis of the least Q-quantale completion of a Q-ordered semigroup. Moreover, we prove that the least Q-quantale completion is precisely an ϵ≤-injective hull of the Q-ordered semigroup in the category Q-Osg≤ of Q-ordered semigroups and submultiplicative Q-order-preserving mappings. Here ϵ≤ denotes the class of those morphisms h:S⟶T in Q-Osg≤ for which eT(h(a1)⋯h(an),h(a))≤eS(a1⋯an,a), where eS and eT are the Q-orders on the Q-ordered semigroups S and T, respectively.

L-fuzzy closure operators and L-fuzzy cotopologies

In this paper, we investigate $L$-fuzzy closure operators and $L$-fuzzy cotopologies in a complete residuated lattice. Also, we study the categorical relationship between $L$-fuzzy closure spaces and $L$-fuzzy cotopological space. Finally, we give their examples.

A reduction theorem to compute fixpoints of fuzzy closure operators

We present a reduction theorem which relates sets of fixpoints of fuzzy closure operators to sets of fixpoints of ordinary closure operators. As a result we obtain a method to compute sets of fixpoints of fuzzy closure operators by algorithms available for ordinary operators. We also provide explicit descriptions of selected algorithms which result from the presented approach.

A new definition of order relation for the introduction of algebraic fuzzy closure operators

In this paper, a new approach to order relation between fuzzy sets is provided, which is called well inclusion order between fuzzy sets. Based on this new order relation, the concept of algebraic fuzzy closure operators is introduced. It is shown that there is a categorical isomorphism between algebraic fuzzy closure operators and fuzzy convex structures. Also, the relationship between fuzzy closure systems and fuzzy convex structures is investigated. It is proved that the category of fuzzy convex spaces is a bicoreflective subcategory of the category of fuzzy closure system spaces.

Closure structures parameterized by systems of isotone Galois connections

We study properties of classes of closure operators and closure systems parameterized by systems of isotone Galois connections. The parameterizations express stronger requirements on idempotency and monotony conditions of closure operators. The present approach extends previous approaches to fuzzy closure operators which appeared in analysis of object-attribute data with graded attributes and reasoning with if-then rules in graded setting and is also related to analogous results developed in linear temporal logic. In the paper, we present foundations of the operators and include examples of general problems in data analysis where such operators appear.

SOME WEAKER FORMS OF SMOOTH FUZZY CONTINUOUS FUNCTIONS

In this paper we introduce various notions of continuous fuzzy proper functions by using the existing notions of fuzzy closure and fuzzy interior operators like 𝑅𝜏𝑟-closure, 𝑅𝜏𝑟-interior, etc., and present all possible relations among these types of continuities. Next, we introduce the concepts of α-quasi-coincidence, 𝑞𝛼𝑟-pre-neighborhood, 𝑞𝛼𝑟-pre-clo-sure and 𝑞𝛼𝑟- pre-continuous function in smooth fuzzy topological spaces and investigate the equivalent conditions of 𝑞𝛼𝑟- pre-continuity.

$r$-fuzzy regular semi open sets in smooth topological spaces

In this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. By using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. Also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Moreover, we investigate the relationship among fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Finally, we have given some counter examples to show that these types of mappings are not equivalent.

An efficient method to factorize fuzzy attribute-oriented concept lattices

Factorization by similarity is a mathematical technique used in formal concept analysis with fuzzy attributes for reducing the complexity of different types of fuzzy concept lattices. In this paper we find the structure of the factor lattice of a fuzzy attribute-oriented concept lattice, namely the intervals representing the blocks of this lattice. We provide a procedure for generating the infimum and the supremum concepts of these intervals as fixpoints of a fuzzy closure operator. This theoretical result allows to develop a more efficient algorithm for building the factor lattice of the fuzzy attribute-oriented concept lattice. A comparison between our approach and the existing algorithms is presented.

L-FUZZY APPROXIMATION SPACES AND L-FUZZY TOPOLOGICAL SPACES

The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.