A new view of fuzzy hypernear-rings

In this survey article, we first introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. This concept is a generalized concept of quasi-coincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval valued (∈,∈∨q)-fuzzy sub-hypernear-rings (hyperideals) of a hypernear-ring, and hence, a generalization of a fuzzy sub-near-ring (ideal) is given. Some related properties of fuzzy hypernear-rings are described. Finally, we consider the concept of implication-based interval valued fuzzy sub-hypernear-rings (hyperideals) in a hypernear-ring, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Extensions of fuzzy hyperideals in <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-semigroups

The object of the present paper is the investigation and study of (fuzzy) hyperideals in H _v - semigroups. Regular equivalence relations play in H _v - semigroup theory a role analogous to congruences in semigroup theory, so we introduce fuzzy regular equivalence relations on H _v -semigroups and then we study fuzzy Rees regular relations on H _v -semigroups. Using this ideas, we establish a relation between fuzzy hyperideal of an H _v -semigroup H and fuzzy hyperideal of a quotient H _v -semigroup of H. Some characterizations of them are then shown.      

Fuzzy hypersemigroups

We introduce the notions of fuzzy hypersemigroup, fuzzy hypergroup, fuzzy hyperideal, homomorphism, hyper congruence, fuzzy homomorphism, fuzzy hypercongruence. The purpose of this note is the study of some characterization of fuzzy hypersemigroup, fuzzy hyperideal of a fuzzy hypersemigroup and homomorphism and hypercongruence on a hypersemigroup.     

Fuzzy hyperrings (Hv-rings) based on fuzzy universal sets

The aim of this paper is to define and study the concept of a fuzzy hyperring, which depends on the concept of a fuzzy universal set. A correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of Davvaz is obtained.     

On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.     

Fuzzy hyperalgebras

We introduce and study the notion of fuzzy multialgebras on the basis of a new definition of a fuzzy hyperoperation. We introduce the notion of fuzzy regular (resp., fuzzy strongly regular) relations of hyperalgebras and obtain their basic properties. Finally, we prove that with every fuzzy hyperalgebra we can associate a unique hyperalgebra via a regular (resp., strongly regular) relation.     

Fuzzy independence and extended conditional probability

In many applications, the use of Bayesian probability theory is problematical. Information needed to feasibility calculate is unavailable. There are different methodologies for dealing with this problem, e.g., maximal entropy and Dempster-Shafer Theory. If one can make independence assumptions, many of the problems disappear, and in fact, this is often the method of choice even when it is obviously incorrect. The notion of independence is a 0–1 concept, which implies that human guesses about its validity will not lead to robust systems. In this paper, we propose a fuzzy formulation of this concept. It should lend itself to probabilistic updating formulas by allowing heuristic estimation of the “degree of independence.” We show how this can be applied to compute a new notion of conditional probability (we call this “extended conditional probability”). Given information, one typically has the choice of full conditioning (standard dependence) or ignoring the information (standard independence). We list some desiderata for the extension of this to allowing degree of conditioning. We then show how our formulation of degree of independence leads to a formula fulfilling these desiderata. After describing this formula, we show how this compares with other possible formulations of parameterized independence. In particular, we compare it to a linear interpolant, a higher power of a linear interpolant, and to a notion originally presented by Hummel and Manevitz [Tenth Int. Joint Conf. on Artificial Intelligence, 1987]. Interestingly, it turns out that a transformation of the Hummel-Manevitz method and our “fuzzy” method are close approximations of each other. Two examples illustrate how fuzzy independence and extended conditional probability might be applied. The first shows how linguistic probabilities result from treating fuzzy independence as a linguistic variable. The second is an industrial example of troubleshooting on the shop floor.     

Possibilistic Information Fusion Using Maximal Coherent Subsets

When multiple sources provide information about the same unknown quantity, their fusion into a synthetic interpretable message is often a tricky problem, especially when sources are conflicting. In this paper, we propose to use possibility theory and the notion of maximal coherent subsets (MCSs), often used in logic-based representations, to build a fuzzy belief structure that will be instrumental both for extracting useful insight about various features of the information conveyed by the sources and for compressing this information into a unique possibility distribution. Extensions and properties of the basic fusion rule are also studied.      

The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets

This paper provides a continuation of ideas presented by Davvaz and Mahdavipour [B. Davvaz, M. Mahdavipour, Roughness in modules, Inform. Sci. 176 (2006) 3658-3674]. The notion of hypermodule is a generalization of the notion of module. In this paper, we consider the quotient hypermodule M/A and interpret the lower and upper approximations as subsets of the quotient hypermodule M/A. Then, we introduce the concept of quotient rough sub-hypermodule. Also, using the concept of fuzzy sets, we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub-hypermodules and level rough sets. This relation is characterized as a necessary and sufficient condition.     

Computational methods for study of foldness of H-ideals in BCI-algebras

We study the notion of n-folds H-ideals in BCI-algebras as a natural generalization of H-ideals in BCI-algebras. Thanks to the concept of fuzzy point, we give some properties of n-folds and fuzzy n-folds H-ideals. Finally, we establish some algorithms for study of foldness theory of H-ideals in BCI-algebras.     

The minimum worst case error of fuzzy approximators

The approximation capability of fuzzy systems is an important topic of research when the systems are regarded as input-output maps. By using the notion of information-based complexity (IBC), we derive the minimum worst case error of a fuzzy approximator, which is independent of the detailed construction of the fuzzy rule bases     

On pseudo-fuzzy linear mappings

In this paper we study a more general class of linear maps on fuzzy subsets of vector spaces than those on fuzzy subspaces. We obtain a version of the fundamental theorem of homomorphisms which applies to these pseudo-fuzzy linear maps. In the case of fuzzy subspaces, the results become those for fuzzy linear maps. We introduce the notion of magnification of pseudo-fuzzy linear mappings. It is shown that this notion behaves well with respect to the fundamental theorem and with respect to the composition of mapping. We note that the class of distribution functions of random variables naturally produces pseudo-fuzzy linear mappings, which are not fuzzy linear mappings. We observe that pseudo-fuzzy linear mappings are examples of gain relations. We consider general properties of these relations as well as many other examples. From these results additional properties of pseudo-fuzzy linear mappings are obtained.     

Bipolar fuzzy graphs

In this paper, we introduce the notion of bipolar fuzzy graphs, describe various methods of their construction, discuss the concept of isomorphisms of these graphs, and investigate some of their important properties. We then introduce the notion of strong bipolar fuzzy graphs and study some of their properties. We also discuss some propositions of self complementary and self weak complementary strong bipolar fuzzy graphs.     

An Algebraic Approach to Shadowed Sets

After the introduction of shadowed sets and the investigation of their relation with fuzzy sets, we present BZMV^dM algebras as an abstract environment for both shadowed and fuzzy sets. Then, we introduce the weaker notion of pre-BZMV^dM algebra. This structure enables us to algebraically define a mapping from fuzzy sets to shadowed sets.     

Fuzzy hyperlattices

In this paper, we introduce the notion of fuzzy hyperlattices and establish connections between fuzzy hyperlattices and hyperlattices. Then we discuss the relations between the weak homomorphism of fuzzy hyperlattices and the weak homomorphism of corresponding hyperlattices. Moreover, we define and analyze the hypercongruence on hyperlattices and the fuzzy (strong) hypercongruence on fuzzy hyperlattices. And we study the interrelation between the (fuzzy) hypercongruence on (fuzzy) hyperlattices and the homomorphism of (fuzzy) hyperlattices. Also, a relation between the fuzzy hypercongruence on fuzzy hyperlattices and the hypercongruence on corresponding hyperlattices is obtained. Finally, we apply the hypercongruence and the fuzzy strong hypercongruence to construct quotient structures of hyperlattices and fuzzy hyperlattices, respectively.     

Product structure of the fuzzy n-ary factor polygroup

Fuzzy n-ary hypergroups were introduced as suitable generalizations of fuzzy polygroups and a special case of fuzzy n-ary hyper groups. The aim of this paper is to introduce the notion of fuzzy n-ary factor polygroups of a polygroup. Based on the fuzzy n-ary factor group, we also study the product structures of the generating fuzzy factor n-ary groups. At the end of the paper, we prove the fundamental theorem of isomorphism of fuzzy n-ary groups.     

Schauder bases and approximation property in fuzzy normed spaces

Our aim in this article is to introduce and study the notion of weak and strong Schauder bases in fuzzy normed spaces. Further, we introduce strong and weak fuzzy approximation properties and set a relationship between these two new notions which may provide an acceleration to the structural analysis of fuzzy normed spaces.