L-fuzzy Subsets Research Articles

A Novel Approach to the Fuzzification of Fields

There are many symmetries in L-fuzzy algebras. In this paper, a novel approach to the fuzzification of a field is introduced. We define a mapping F:LX→L from the family of all the L-fuzzy sets on a field X to L such that each L-fuzzy set is an L-fuzzy subfield to some extent. Some equivalent characterizations F(μ) are given by means of cut sets. It is proved that F is L-fuzzy convex structure on X, hence (X,F) forms an L-fuzzy convexity space. A homomorphism between fields is exactly an L-fuzzy convexity preserving mapping and an L-fuzzy convex-to-convex mapping. Finally, we discuss some operations of L-fuzzy subsets.

Observability in locally vague environments

In this paper, we introduce observable L-fuzzy subsets of an L-fuzzy set in a locally vague environment, and give their axiomatization. In addition to this, lower and upper observability operators that enable us to approximate non-observable L-fuzzy sets within some observable bounds are studied. In particular, we deal with their topological characterization, and expose that they can be identified with the stratified L-interior and the stratified L-closure operators of a stratified Alexandroff L-topology on an L-fuzzy set.

On the representations of L-equivalence relations on L-fuzzy sets with applications to locally vague environments

ABSTRACTL-equivalence relations on L-fuzzy sets, equivalent to L-valued equalities on ordinary sets, are modelling fuzzy similarity of two objects that are partial members of a fuzzily defined set. This paper deals with construction and representation properties of L-equivalence relations on L-fuzzy sets. To this end, we set up a method for constructing them by means of quasi-pseudo-metrics, and then point out how L-equivalence relations on L-fuzzy sets are represented by families of L-fuzzy subsets of L-fuzzy sets. The present results are applied to locally vague environments to elucidate conceptual structures of locally vague environments, and to show how the points in a locally vague environment are described.

Metric-based [formula omitted]-fuzzy rough sets: Approximation operators and definable sets

Binary relations, coverings and neighborhood systems/operators are useful tools to study rough set theory. In this paper, we use the notion of ⊕-hemimetric, a weak version of the standard metric in topology and analysis, as the basic structure to study L-fuzzy rough set theory, where L is a complete residuated lattice. We define a pair of L-fuzzy upper and lower approximation operators and then investigate their properties and relations. It is shown that both operators are monotone with respect to the L-fuzzy order of fuzzy inclusion relation between L-fuzzy subsets. The L-fuzzy upper approximation operator has more nice properties than the lower one, and if L is regular and the hemimetric is symmetric, then they are dual to each other. We then study the upper and lower definable sets in this model. The family of upper definable sets forms an Alexandrov stratified L-topology while that of lower definable ones does not necessarily. If L is regular (even if the hemimetric is not symmetric), the upper definability coincides with the lower definability. We finally present an application of metric-based L-fuzzy set theory to fuzzy clustering for weighted graphs.

On the measure of M-rough approximation of L-fuzzy sets

We develop an approach allowing to measure the “quality” of rough approximation of fuzzy sets. It is based on what we call “an approximative quadruple” \(Q=(L,M,\varphi ,\psi )\) where L and M are complete lattice commutative monoids and \(\varphi : L \rightarrow M\), \(\psi : M \rightarrow L\) are mappings satisfying certain conditions. By realization of this scheme, we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with an M-preoder \(R: X\times X \rightarrow M\). In case R is symmetric, these measures coincide. Basic properties of such measures are studied. Besides, we present an interpretation of measures of rough approximation in terms of LM-fuzzy topologies.

On strongly convex L-fuzzy subsets of an ordered semigroup

On strongly convex L-fuzzy subsets of an ordered semigroup

L-valued bornologies on powersets

In M. Abel and A. Šostak (2011) [1], the concept of an L-fuzzy bornology was introduced. Actually, an L-fuzzy bornology on a set X is a certain ideal in the family LX of L-fuzzy subsets of a set X. Here we propose an alternative approach to fuzzification of the concept of bornology. We define an L-valued bornology on a set X as an L-fuzzy subset B of the powerset 2X satisfying L-valued analogues of the axioms of a bornology. Basic properties of L-valued bornological spaces are studied. Our special interest concerns L-valued bornologies induced by fuzzy metrics and relative compactness-type L-valued bornologies in Chang–Goguen L-topological spaces.

A new view of L-fuzzy polygroups

In this paper, we introduce the concept of (weak) L-fuzzy polygroups and give a theorem to present the connection between the crisp polygroups and L-fuzzy polygroups. We also provide the notion of (normal) L-fuzzy subpolygroups of a (weak) L-fuzzy polygroup and investigate some of their properties. We show that the set of all the normal L-fuzzy subpolygroups is a modular lattice and obtain a kind of weak L-fuzzy quotient polygroup. Moreover, we define two kinds of operators on $${\fancyscript{L}(H)}$$, where $${\fancyscript{L}(H)}$$ is the set of all the L-fuzzy subsets in a weak L-fuzzy polygroup H, to characterize L-fuzzy subpolygroups and present some related theorems. Finally, we investigate the homomorphism properties of L-fuzzy polygroups.

Notes on L-fuzzy γ-open sets

The aim of this paper is to present a common approach allowing to obtain families of L-fuzzy sets in an L-topological space of Goguen type generalizing the class of all open L-fuzzy sets. In particular, we study the notion of an L-fuzzy ?-open set where ? is a monotone operator on the family of all L-fuzzy subsets of an L-topological space X and discuss some properties of L-fuzzy ?-open sets.

Very and more or less in non-commutative fuzzy logic

In this paper we consider fuzzy subsets of a universe as L-fuzzy subsets instead of [ 0, 1 ]-valued, where L is a complete lattice. We enrich the lattice L by adding some suitable operations to make it into a pseudo-BL algebra. Since BL algebras are main frameworks of fuzzy logic, we propose to consider the non-commutative BL-algebras which are more natural for modeling the fuzzy notions. Based on reasoning with in non-commutative fuzzy logic we model the linguistic modifiers such as very and more or less and give an appropriate membership function for each one by taking into account the context of the given fuzzy notion by means of resemblance L-fuzzy relations.

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.

P2-CONNECTEDNESS IN L-TOPOLOGICAL SPACES

In this paper, a certain new connectedness of L-fuzzy subsets in L-topological spaces is introduced and studied by means of preclosed sets. It preserves some fundamental properties of connected set in general topology. Especially the famous K. Fan's Theorem holds.

The minimal mathematical structure for a synchronic approach to fuzzy set theory

The paper gives a concrete presentation of the minimal structure on the monoid of an external subset classifier of a universe of usual sets with Zadeh–Goguen L-fuzzy subsets. The result was first investigated by the first author, with the aim of providing a new insight into the synchronic approach to fuzzy set theory. The extensive use of conceptual mathematics (i.e. category theory) in the original paper, restricts the audience of readers of this analysis to persons who are already acquainted with conceptual mathematics. The aim of this paper is to avoid as much as possible the use of functorial terminology in the presentation of the minimal mathematical structure on the set of truth values, for a synchronic approach to fuzzy sets. By doing so, we hope to widen the audience of scientists interested in the foundation of Zadeh–Goguen L-fuzzy sets and to provide a new way of looking at the synchronic approach to fuzziness.

L-FUZZY PS-COMPACTNESS

In this paper, the new compactness which is L-fuzzy PS-compactness in L-fuzzy topological spaces is introduced. It is defined for arbitrary L-fuzzy subsets. And some of its characteristic properties and fundamental properties are studied.

L-Fuzzy PS-Compactness

In this paper, the new compactness which is L-fuzzy PS-compactness in L-fuzzy topological spaces is introduced. It is defined for arbitrary L-fuzzy subsets. And some of its characteristic properties and fundamental properties are studied.

On the equivalence of four definitions of compactness of LF subsets

In this paper, the equivalence of four definitions of fuzzy compactness, strong fuzzy compactness, N-compactness and ultra-fuzzy compactness of L-fuzzy subsets is discussed systematically. A series of good results is obtained.

Operations and its applications on L-fuzzy bitopological spaces: Part I

In this and a subsequent paper, a new operation on the class of all L-fuzzy subsets of a universe endowed with a fuzzy topological space is introduced to generalize several characterization and properties of some fuzzy bitopological concepts such as a fuzzy interior operators and some new classes of fuzzy separation axioms. Also, the relation between these classes and the weaker and stronger forms in fuzzy bitopological spaces are investigated. Finally, we showed that these axioms are good extension in the sense of Lowen (1978) due to Lowen, in Part II this theory is applied on the mappings defined on L-fuzzy bitopological spaces.

Some important applications of nets of L-fuzzy sets

In this paper, some compactness, adherence point and accumulation point in L-fuzzy topological space are characterized by means of the nets of L-fuzzy subsets.

Paracompactness in L-fuzzy topological spaces

In this paper, we introduce a good definition of paracompactness in L-fuzzy topological spaces, where L is a fuzzy lattice. We define paracompactness on general L-fuzzy subsets and study its properties.

InsertingL-fuzzy Real-valued Functions

A new proof of the Katêtov-Tong Insertion Theorem for L-fuzzy topological spaces is given. All the proofs are performed in such a way that the presented results hold true in the more general case in which L-fuzzy topologies are replaced by σ-additive rings of L-fuzzy subsets.