Fuzzy Power Set Research Articles

Fuzzy Algebras of Concepts

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.

On Some Theoretic Aspects of Fuzzy Subsets

Abstract: This article proposes some important theoretic aspects of L -fuzzy power set of a finite set. Here, we take a non-empty finite set E and an ordered subset M of the closed interval 0,1, then the set of mappings from E to M denoted by F is defined as L -fuzzy power set. Considering the disjunctive union ‘ ’ operations between fuzzy subsets of F , the structure F,forms a groupoid, which is defined as special fuzzy groupoid. Later, we try to introduce the product of special fuzzy groupoids and their properties

Powerset operators induced by fuzzy relations as a basis for fuzzification of various mathematical structures

We propose a unified approach to the theory of forward and backward powerset operators induced by fuzzy relations with special attention to the algebraic and topological properties of these operators. On the basis of this theory, we reconsider previous studies done by various authors in the fields of fuzzy rough sets, fuzzy concept analysis and fuzzy mathematical morphology as well as obtain some new results in these areas. In this way we contribute to the development of the theory of fuzzy rough sets, fuzzy concept analysis and fuzzy mathematical morphology on the basis of fuzzy powerset operators induced by fuzzy relations.

γ– algebra of Sets and Some of its Properties

The main objective of this work is to generalize the concept of fuzzy algebra by introducing the notion of fuzzy algebra. Characterization and examples of the proposed generalization are presented, as well as several different properties of fuzzy algebra are proven. Furthermore, the relationship between fuzzy algebra and fuzzy algebra is studied, where it is shown that the fuzzy algebra is a generalization of fuzzy algebra too. In addition, the notion of restriction, as an important property in the study of measure theory, is studied as well. Many properties of restriction of a nonempty family of fuzzy subsets of fuzzy power set are investigated and it is shown that the restriction of fuzzy algebra is fuzzy algebra too.

Partially ordered fuzzy power set monads on the category of L-sets and their associated categories of topological space objects

For a GL-monoid L with a complete lattice L underneath, this paper considers the L-unbalanced power object monad on the category Set(L) of L-sets, and introduces a new monad on Set(Lop) called the L-power L-set monad. The Kleisli and Eilenberg-Moore categories of these monads are studied. Both monads are enhanced to partially ordered monads that allow us to apply the theory of topological space objects developed by Ulrich Höhle. We give the explicit descriptions of the category of topological space objects in Set(L) with respect to the partially ordered L-unbalanced power object monad and the category of topological space objects in Set(Lop) with respect to the partially ordered L-power L-set monad. Furthermore, it is shown that the L-power L-set monad on Set(Lop) determines a monad on Set(L) in a one-to-one manner, and all existing results can be directly converted into this monad.

On the mathematical foundations of fuzzy set theory

In this paper we provide arguments supporting the idea that module theory in the symmetric, monoidal closed category of complete lattices and join-preserving maps is a mathematical framework for fuzzy set theory as developed by Zadeh. In this context, quantale-enriched category theory is a well established mathematical technique allowing an internal development of various concepts from fuzzy set theory – e.g. the fuzzy power set, Zadeh's forward operator or the fuzzy inclusion order.

New Methods of finding support planes of cut-level fuzzy power sets and Geometry of the convex fuzzy power sets

The concepts of the convex fuzzy power sets and crisply defined membership functions in fuzzy set theory are important to deeply analyze to look further for more precise solutions of the fuzzy mathematical programming problems. The Extension principle was defined in terms of the level cuts of the membership functions. The theorem of the decomposition proves the existence of the fuzzy power set as the union of all level cuts of membership function. The Convexity of the level cut fuzzy set was determined at the points of the universe, where the membership function is approximated. The Theorems of the convex and quasi-convex fuzzy sets extend and expand the extension principle to the higher quality to look for more precise solutions of fuzzy problems beyond the support planes. The Fuzzy geometry and topological categories researched further to arrive to the newest Lemma for the cyclic quadrilateral convex fuzzy sets. The analytical and graphical representation of modeling with membership functions type I through types 6 are investigated to the point to be highly recommended in engineering problems by using heuristic methods and models.

POWERSET OPERATORS OF EXTENSIONAL FUZZY SETS

Powerset structures of extensional fuzzy sets in sets with similarity relations are investigated. It is proved that extensional fuzzy sets have powerset structures which are powerset theories in the category of sets with similarity relations, and some of these powerset theories are defined also by algebraic theories (monads). Between Zadeh's fuzzy powerset theory and the classical powerset theory there exists a strong relation, which can be represented as a homomorphism. Analogical results are also proved for new powerset theories of extensional fuzzy sets.

A graded approach to cardinal theory of finite fuzzy sets, part I: Graded equipollence

In this article, we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent, which indicates that they have the same number of elements. The concepts of fuzzy sets and fuzzy classes in the class of all sets (in ZFC) are introduced, and several standard relations and constructions, such as the fuzzy power set and exponentiation, are defined. A functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown, and a relation to generalized cardinals and Wygralak's cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated.

Fuzzy sets as two-sorted algebras

This paper makes the attempt to explain the foundations of fuzzy sets from the point of view of universal algebra. The free fuzzy set and the fuzzy power set are constructed. Moreover, fuzzy power sets give rise to a monad. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets.

Q-fuzzy subsets on ordered semigroups

In this paper, we introduce the concept of a Q-fuzzy subset of an ordered semigroup where a quantale Q replaces the unit interval. We define a binary operation on the fuzzy power set that makes it a quantale. In order to discuss the relation between the referential set and the fuzzy power set, the notion of an ordered Q-fuzzy point of an ordered semigroup is introduced. In terms of ordered Q-fuzzy points, we prove that any ordered semigroup can be embedded into a quantale, and also build a functor between the category of ordered semigroups and the category of quantales.

ALGEBRAIC GENERATIONS OF SOME FUZZY POWERSET OPERATORS

In this paper, let $L$ be a completeresiduated lattice, and let {bf Set} denote the category of setsand mappings, $LF$-{bf Pos} denote the category of $LF$-posets and$LF$-monotone mappings, and $LF$-{bf CSLat}$(sqcup)$, $LF$-{bfCSLat}$(sqcap)$ denote the category of $LF$-completelattices and $LF$-join-preserving mappings and the category of$LF$-complete lattices and $LF$-meet-preserving mappings, respectively. It isproved that there are adjunctions between {bf Set} and $LF$-{bf CSLat}$(sqcup)$, between $LF$-{bfPos} and $LF$-{bf CSLat}$(sqcup)$, and between $LF$-{bf Pos} and$LF$-{bf CSLat}$(sqcap)$, that is, {bf Set}$dashv LF$-{bf CSLat}$(sqcup)$, $LF$-{bfPos}$dashv LF$-{bf CSLat}$(sqcup)$, and $LF$-{bf Pos}$dashv$$LF$-{bf CSLat}$(sqcap)$. And a usual mapping $f$ generates thetraditional Zadeh forward powerset operator $f_L^rightarrow$ andthe fuzzy forward powerset operators $widetilde{f}^rightarrow,widetilde{f}_ast^rightarrow, widetilde{f}^{astrightarrow}$defined by the author et al via these adjunctions. Moreover, it is also shownthat all the fuzzy powerset operators mentioned above can be generated by the underlying algebraic theories.

POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E. Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of L. A. Zadeh. On the next step, we provide a direct lift of the backward powerset operator using the notion of categorical biproduct. The obtained framework is readily extended to the variable-basis case, justifying the powerset theories currently popular in the fuzzy community. At the end of the paper, our general variety-based setting postulates the requirements, under which a convenient variety-based powerset theory can be developed, suitable for employment in all areas of fuzzy mathematics dealing with fuzzy powersets, including fuzzy algebra, logic and topology.

The Upgrade of Topological Group Based on a New Hyper-topology

Wang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Consequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, the paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations continues in the new hyper-topology after upgrading and a series of results are obtained creating hyper-topological group and fuzzy hyper-topological group. Accordingly, hyper-topological group and fuzzy hyper-topological group can be created, obtaining a breakthrough to upgrade topological group to its power set and fuzzy power set.

Relationships between L-ordered convergence structures and strong L-topologies

By making use of the intrinsic fuzzy inclusion orders on the fuzzy power set, the relationships between L-ordered convergence spaces and strong L-topological spaces are researched, where L is a commutative unital quantale. It is shown that the category of strong L-topological spaces can be embedded in the category of L-ordered convergence spaces as a reflective subcategory. As a result, we observe that there is a Galois correspondence between the category of L-ordered convergence spaces and the category of strong L-topological spaces. Further, it is proved that the class of all strong L-topological L-ordered convergence spaces precisely is the class of all strong L-topological spaces, and the class of spaces with non-idempotent L-ordered interior operators is characterized as a subclass of the class of L-ordered convergence spaces.

Stratified L-ordered convergence structures

In this paper, a new kind of lattice-valued convergence structures on a universal set, called stratified L-ordered convergence structures, are presented by modifying the axiom for stratified L-generalized convergence structures in the fuzzy setting so as to make use of the intrinsic fuzzy inclusion order on the fuzzy power set. The category of stratified L-ordered convergence spaces described here is shown to be a reflective full subcategory in the category of stratified L-generalized convergence spaces, and hence it is topological and Cartesian-closed. As preparation, a further investigation of stratified L-filters is presented from the viewpoint that latticed-valued filters should be compatible with the intrinsic fuzzy inclusion order on the fuzzy power set.

Fuzzy complete lattices

In this paper, based on L-fuzzy posets previously introduced by the third author, L-fuzzy complete lattices are defined, which are generalizations of usual complete lattices and coincide with Wagner's complete and cocomplete Ω -categories enriched over the frame L, and are consequently a special kind of complete Ω -lattices defined by Lai and Zhang. However, Tarski fixed-point theorem for the L-fuzzy complete lattices is proved in a different way from that by Lai and Zhang. Furthermore, some fuzzy powerset operators are suggested, they are not only generalizations of ordinary powerset operators, but also generalizations of L-valued Zadeh powerset operators, and their properties are discussed.

An enriched category approach to many valued topology

In this paper we discuss a basic question in fuzzy set theory: what is a many valued topology on a set X ? An answer is presented by interpreting the axioms (two logical formulas) for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of X . As preparation, a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory.

A note on a natural equivalence relation on fuzzy power set

The necessary and sufficient conditions under which two fuzzy sets with the same domain have equal families of cut sets are given. Consequently, there is a corresponding equivalence relation on the related fuzzy power set. The collection of classes under this relation can be ordered, and we give a necessary and sufficient condition under which it is a lattice.

Some remarks on fuzzy powerset operators

Fuzzy powerset operators related to morphisms of underlying ground categories of categories of fuzzy topological spaces are considered and some properties of such operators related to isomorphisms, fuzzy points and order reversing involutions are verified.