In this paper, a new definition of fuzzy bounded sets and totally fuzzy bounded sets is introduced and properties of such sets are studied. Then a relation between totally fuzzy bounded sets and N-compactness is discussed. Finally, a geometric characterization for fuzzy totally bounded sets in I- topological vector spaces is derived.

Through this study, we are going to introduce a new aspect of cut sets for hesitant fuzzy sets. Compared to the rare researches in regard of cut sets for hesitant fuzzy sets, not only this study does not deal with extra parameters, but also the proposed concept of cut set conforms with the accepted principles, and further, it is extension of existing cut set concept of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets and three-dimensional fuzzy sets. In the sequel, using proposed concept of cut sets for hesitant fuzzy sets, we extend the corresponding theorems of decomposition and representation for hesitant fuzzy sets. The current research can be a basis for the future work on the application of cut sets in multiple criteria decision making problems under hesitant fuzzy environment.

Given a uniform space (X, U), we introduce, from the uniformity U, some uniformities on the set F(X) of all normal, upper semicontinuous with compact support fuzzy sets on X: the Skorokhod uniformity, the level-wise uniformity, the endograph uniformity and the sendograph uniformity. For metric spaces we prove that these uniformities coincide with the uniformities induced by the Skorokhod metric (the level-wise metric, the endograph metric and the sendograph metric, respectively). We study completeness of this class of uniform spaces.

In this paper, we take a GL-quantale as the truth value table to study a new rough set model|L-valued fuzzy rough sets. The three key components of this model are: an L-fuzzy set A as the universal set, an L-valued relation of A and an L-fuzzy set of A (a fuzzy subset of fuzzy sets). Then L-valued fuzzy rough sets are completely characterized via both constructive and axiomatic approaches.

In this paper, two related quotient structures are investigated utilizing the concept of coset. At first, a new hypervector space F/V = (F/V,\circ,\circledcirc,K) is created, which is composed of all cosets of a bipolar fuzzy soft set (F;A) over a hypervector space V . Then it will be shown that dim F/V = dim V/W, where the quotient hypervector space V/W includes all cosets of an especial subhyperspace W of V. Also, three bipolar fuzzy soft sets over the quotient hypervector space V/W are presented and in this way some new bipolar fuzzy soft hypervector spaces are defined.

This paper presents a rock mass classification system development based on the theory of fuzzy sets. This classification system, which is a generalization of the conventional method of rock mass rating, allows us to introduce uncertainties as well as evaluating overall reliability. Appropriate fuzzy sets are assigned to die rock mass parameters and the procedure required to find the final fuzzy-based rock mass rating (FRMR) is discussed in detail. The proposed classification system provides the rock mass rating, FRMR, together with its reliability, in addition to the corresponding rock mass class aid estimates of cohesion and friction angle. Comparison of FRMR with die conventional rock mass rating, RMR, indicates that different combinations of rock mass parameters leading to the same RMR can have different FRMR values. However, on average there is very good agreement between the two classification systems.

Based on the point of view of geometrical representation of an intuitionistic fuzzy set, we take into account all three parameters describing intuitionistic fuzzy set, propose a kind of new method to calculate correlation and correlation coefficient of intuitionistic fuzzy sets which is similar to the cosine of the intersectional angle in finite sets and probability space, respectively. Further, we discuss some of their properties and give three numerical examples to illustrate our proposed method reasonable.

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.