Fuzzy Σ-algebra Research Articles

Some convergence theorems of fuzzy concave integral on fuzzy σ-algebra

Some convergence theorems of fuzzy concave integral on fuzzy σ-algebra

Study of Fuzzy σ-Ring and Some Related Concepts

This paper introduces the concept of fuzzy σ-ring as a generalization of fuzzy σ-algebra and basic properties; examples of this concept have been given. As the first result, it has been proved that every σ-algebra over a fuzzy set x* is a fuzzy σ-ring-over a fuzzy set x* and construct their converse by example. Furthermore, the fuzzy ring concept has been studied to generalize fuzzy algebra and its relation. Investigating that the concept of fuzzy σ-Ring is a stronger form of a fuzzy ring that is every fuzzy σ-Ring over a fuzzy set x* is a fuzzy ring over a fuzzy set x* and construct their converse by example. In addition, the idea of the smallest, as an important property in the study of real analysis, is studied as well. Finally, the main goal of this paper is to study these concepts and give basic properties, examples, characterizations and relationships between them.

The degree of (L, M)-fuzzy σ-algebra and its related mappings

The degree of (L, M)-fuzzy σ-algebra and its related mappings

Conditional logical entropy of fuzzy σ-algebras

Conditional logical entropy of fuzzy σ-algebras

Formula omitted]-intuitionistic fuzzy [formula omitted]-algebras

The main purpose of this paper is to generalize the concept of (L,M)-fuzzy σ-algebras defined by Shi [F.G. Shi, (L,M)-fuzzy σ-algebras, International Journal of Mathematics and Mathematical Sciences. doi:10.1155/2010/356581] from L-fuzzy sets to intuitionistic fuzzy sets; the notions of L-intuitionistic fuzzy σ-algebras and L-intuitionistic fuzzy measurable functions are introduced; the relationships among L-intuitionistic fuzzy σ-algebras, (I,I)-fuzzy σ-algebras and I-fuzzifying σ-algebras are studied. Then we give the definition of the product of L-intuitionistic fuzzy σ-algebras; some of their elementary properties are discussed. By use of the concept of I-fuzzifying σ-algebras, the notion of generated L-intuitionistic fuzzy σ-algebras is proposed. Finally, the properties of the generated functor Iζ from the category of I-fuzzifying σ-algebras and I-fuzzifying measurable functions to the category of L-intuitionistic fuzzy σ-algebras and L-intuitionistic fuzzy measurable functions are proved.

On Some Mathematical Structures of T-Fuzzy Rough Set Algebras in Infinite Universes of Discourse

In this paper, a general framework for the study of fuzzy rough approximation operators determined by a triangular norm in infinite universes of discourse is investigated. Lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space in infinite universes of discourse are first introduced. Essential properties of various types of T-fuzzy rough approximation operators are then examined. An operator-oriented characterization of fuzzy rough sets is also proposed, that is, T-fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. A comparative study of T-fuzzy rough set algebras with some other mathematical structures are presented. It is proved that there exists a one-to-one correspondence between the set of all reflexive and T-transitive fuzzy approximation spaces and the set of all fuzzy Alexandrov spaces such that the lower and upper T-fuzzy rough approximation operators are, respectively, the fuzzy interior and closure operators. It is also shown that a reflexive fuzzy approximation space induces a measurable space such that the family of definable fuzzy sets in the fuzzy approximation space forms the fuzzy σ-algebra of the measurable space. Finally, it is explored that the fuzzy belief functions in the Dempster-Shafer of evidence can be interpreted by the T-fuzzy rough approximation operators in the rough set theory, that is, for any fuzzy belief structure there must exist a probability fuzzy approximation space such that the derived probabilities of the lower and upper approximations of a fuzzy set are, respectively, the T-fuzzy belief and plausibility degrees of the fuzzy set in the given fuzzy belief structure. (This work was supported by grants from the National Natural Science Foundation of China (Nos. 61075120, 60673096 and 60773174), and the Natural Science Foundation of Zhejiang Province (No. Y107262).)

Formula omitted]-Equivalence, entropy and [formula omitted]-dynamical systems

The present paper deals with the theory of entropy of F-dynamical systems using the concept of atoms in a fuzzy σ-algebra. Having introduced the notions of m-equivalence and m-refinement, it is proved that entropy of an F-dynamical system with respect to each m-equivalence class is an m-isomorphism invariant. Results proved in this paper include the corresponding classical results as particular cases.

Fuzzy σ-algebras and conditional entropy

The purpose of the present paper is to introduce and study the concept of conditional entropy of fuzzy sub σ-algebras having finitely many atoms.

Fuzzy measurable functions

Abstract A generated fuzzy σ-algebra equipped by Giles fuzzy connectives is considered as a basis of this paper. Starting from a simple fuzzy measurable function, we introduce the notion of a fuzzy measurable function. A one-to-one correspondence between the fuzzy measurable functions and the random variables with values in the fuzzy real line is shown. An inverse of a fuzzy measurable function is proved to be an extended T ∞ -fuzzy observable and vice versa. Linear operations on fuzzy measurable functions are introduced.

P-measures on soft fuzzy σ-algebras

Abstract We study P-measures on soft fuzzy σ-algebras as defined by Piasecki [46] and further investigated in [2–5], etc. We prove as main result that the space of P-measures of a soft fuzzy σ-algebra is ‘faithfully’ isomorphic to the space of probability measures of a Boolean σ-algebra. This result, we believe, may allow to employ the well-developed Boolean technique in certain ‘fuzzy’ investigations (especially in the stochastic ones) but, on the other hand, it lowers to some extent the chance for the soft fuzzy σ-algebras to be applied in quantum theories (see also [9] and [8]). (It should be noted that our result gives - as a by-product - a negative answer to the problem posed in [3].)

A note on measures in fuzzy topological spaces

Regularity of a fuzzy measure on a fuzzy topological space is considered. First it is proved that regular fuzzy sets form a fuzzy σ-algebra. Then the subadditive case is studied as well as a connection between additivity and subadditivity (the Alexandrov theorem).

The structural characteristics of the fuzzy number-valued fuzzy measure on the fuzzy σ-algebra and their applications

In this paper, some structural characteristics of the fuzzy number-valued fuzzy measure on the fuzzy σ-algebra are discussed. On the fuzzy number-valued fuzzy measure space, the concepts of ‘almost’ and ‘pseudo-almost’ are introduced, and Riesz's theorem, Lebesgue's theorem and Egoroff's theorem for sequences of fuzzy measurable functions and some convergence theorems for sequences of fuzzy number-valued fuzzy integrals on fuzzy sets are proved by some structural characteristics of the fuzzy number-valued fuzzy measure.

Fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy integral on the fuzzy set

In this paper, some new concepts — the fuzzy number-valued fuzzy measure, autocontinuity and null-additivity of fuzzy number-valued fuzzy measures on the fuzzy σ-algebra, and the fuzzy number-valued fuzzy integral on the fuzzy set — are introduced. Some properties of fuzzy number-valued fuzzy integrals and some convergence theorems for sequences of fuzzy number-valued fuzzy integrals on a fuzzy set are discussed.

Fuzzy integrals on L-fuzzy sets

The fuzzy measures and fuzzy integrals for Zadeh's fuzzy sets were studied by Qiao and Wang [5, 6, 7, 12]. In this paper we consider the real-valued fuzzy measures and fuzzy integrals for L-fuzzy sets (where L is a pseudocomplemented infinitely distributive complete lattice). On a fuzzy σ-algebra of L-fuzzy sets, the concepts of fuzzy measure and fuzzy integral are introduced, some equivalent forms of fuzzy integrals on L-fuzzy sets are given. Moreover we discuss several properties of fuzzy integrals on L-fuzzy sets, and prove some convergence theorems for a sequence of fuzzy integrals on L-fuzzy sets.

On fuzzy measure and fuzzy integral on fuzzy set

On a fuzzy σ-algebra of fuzzy sets, we discuss some structural characteristics of a fuzzy measure, the (pseudo-) autocontinuity and some other concepts for a fuzzy measure are introduced and the relations between these cre discussed. For the case of fuzzy sets, Riesz's theorems and Lebesgue's theorems for a sequence of measurable functions are proved. The concept of the fuzzy integral on a fuzzy set is given and some properties of fuzzy integrals on fuzzy sets are studied. Finally, we prove some convergence theorems of a sequence of fuzzy integrals on fuzzy sets.

On fuzzily measurable random variables

A fuzzy random variable is given as a ( σ − β ϱ )-measurable function transforming the sets of elementary events into the real line, where σ is a fuzzy σ-algebra and β ϱ is a soft fuzzy σ-algebra defined by means of the fuzzy relation ‘less or equal’ [8]. The main properties of a fuzzy random variable are proved. The connections between a fuzzy random variable, its distribution and its cumulative distribution are investigated, too.

Fuzzy probability spaces defined by means of the fuzzy relation ‘less than’

In this paper a fuzzy relation ‘less than’ and fuzzy probability spaces on the real line are defined. Fuzzy σ-algebras of events are introduced in an analogous way to the classical theory of probability by means of the above relation. Furthermore, specific properties of the fuzzy probability measures presented here are given.

Fuzzy random measure and its extension theorem

The goal of this paper is to introduce random measures into fuzzy σ-algebras. The structure of random measures of fuzzy sets is studied. Extension theorems of random measure on fuzzy algebras to the generated σ-algebras are founded. It shows that Bochner integrals with respect to measures of fuzzy sets can be used to express random measures of fuzzy sets. If it turns out that random measures have their fuzzy meanings then this paper presents a way to deal with fuzzy measures of fuzzy sets by use of a classical mathematical method, such as measure theory, functional analysis and topology.

Operations on fuzzy sets—an axiomatic approach

We introduce intersection and union of fuzzy sets, constructed by means of operations on the unit interval which may depend on the considered element of the universe of discourse. A similar approach for complementation was given by Lowen [7]. The main result is that element-dependent operations are not compatible with the preservation of the intersection and union under the inverse image of a function. Finally we consider fuzzy σ-algebras built with these set-theoretic operations.

Construction of Fuzzy σ-algebras using triangular norms

Generalizing the definitions given by the author [ Fuzzy Sets and Systems 4 (1980), 83–93] we introduce and study T-fuzzy σ-algebras, T being any triangular norm. The main result is that for a large class of triangular norms each T-fuzzy σ-algebra is generated, i.e., consists of all functions μ: X → [0, 1] being measurable with respect to some σ-algebra on X.