<Emphasis Type="Italic">L</Emphasis>-fuzzy roughness of <Emphasis Type="Italic">n</Emphasis>-ary polygroups

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied.     

L-Fuzzy Contiguity Relations and L-Fuzzy Closure Operators in the Case of Completely Distributive,Complete Lattices L

In the case of completely distributive, complete lattices L the relations between L-fuzzy contiguity relations and L-fuzzy closure operators are investigated. In addition, if L is the real unit interval, L-fuzzy contiguity relations are represented by means of V_D-contiguity relations, PT spaces and random V_D-contiguity relations respectively.     

格的TL-模糊理想

本文研究了格的TL-模糊理想. 利用生成TL-模糊理想, 证明了一个模格的全体T_M-模糊理想形成一个完备的模格. 此外, 利用L-模糊集的投影和截影, 获得了将直积格的TL-模糊理想表示成分量格的TL-模糊理想的L-直积的一个充分必要条件. 所得结果进一步推广和发展了格的模糊理想的理论.     

L-fuzzy Q-convergence structures

In this paper, we construct a topological category of pretopological L-fuzzy Q-convergence spaces, which contains the category of topological L-fuzzy Q-convergence spaces as a bireflective full subcategory. Considering the connections with L-fuzzy topology, it is proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to the category of topological L-fuzzy quasi-coincident neighborhood spaces, and the latter is isomorphic to the category of L-fuzzy topological spaces. Moreover, we find that our pretopological L-fuzzy Q-convergence spaces can be characterized as a kind of L-fuzzy quasi-coincident neighborhood spaces, which is called strong L-fuzzy quasi-coincident neighborhood space.     

On L-fuzzy Ideals in Semirings I

In this paper we extend the concept of an L-fuzzy (characteristic) left (resp. right) ideal of a ring to a semiring R, and we show that each level left (resp. right) ideal of an .L-fuzzy left (resp. right) ideal of R is characteristic iff is L-fuzzy characteristic.     

On L-fuzzy topological spaces

L-fuzzy sets and, subsequently, L-fuzzy topological spaces are considered in this paper, where L is a complete and completely distributive non-atomic Boolean algebra. A few separation properties as well as subspace L-fuzzy topology are defined here, in the light of an earlier paper. These are more similar to ordinary topological spaces than to fuzzy topological spaces.     

Extensions of L-fuzzy closure spaces

In this paper, we study the extension theory to L-fuzzy closure spaces, where L is a strictly two-sided, commutative quantale lattice. We give new notions such as L-fuzzy stack, L-fuzzy c-grill and trace of a point. Also, we construct order relation and equivalence relation between two extensions. Also, We introduce the concept of a principal extension of L-fuzzy closure space and study some of its applications.     

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * _p , parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, * _p ). We show that: for everya,b εX the family {a*_pb} _pεX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * _p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX.     

L-topological vector spaces and families of L-fuzzy pseudo-norms

In this paper, the concept of a family of L-fuzzy pseudo-norms on vector spaces is proposed and the characterization of L-vector topologies in terms of a family of L-fuzzy pseudo-norms is presented. As applications of the characterization, the Hausdorff separation property, convergence of molecule nets and boundedness of L-sets in L-topological vector spaces are investigated.     

On L-fuzzy ideals in semirings II

We study some properties of L-fuzzy left (right) ideals of a semiring R related to level left (right) ideals     

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Generalizing the L-fuzzy unit interval

Given an arbitrary connected topological space, an L-fuzzy space is constructed. When the original space is the real line (unit interval) the constructed space is the L-fuzzy real line (L-fuzzy unit interval). For some spaces, including Rⁿ for all n, the original space is embedded as a subspace of the constructed space. Lastly, the construction yields a non-trivial fuzzy topology on certain classically important spaces of monotone mappings.     

Intuitionistic <Emphasis Type="Italic">I</Emphasis>-fuzzy topological spaces

The main purpose of this paper is to introduce the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic I-fuzzy topological spaces and the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic I-fuzzy topology is proposed, and the connections among generated intuitionistic I-fuzzy topological spaces, fuzzifying topological spaces and I-fuzzy topological spaces are discussed. Finally, the properties of the operators Iω, ι are obtained.     

Continuity in quantitative domains

Based on the notion of an $L$-fuzzy partially ordered set constructed in (An L-fuzzy approach to quantitative domain (I)—generalized ordered set valued in frame and adjunction theory, Fuzzy Systems and Mathematics 14 (2000) 6–7), and by introducing the concepts of an $L$-fuzzy directed set and the join of an $L$-fuzzy set w.r.t. the $L$-fuzzy partial order, $L$-fuzzy domains are defined and the generalized Scott topology on an $L$-fuzzy domain is built. This approach is similar to Flagg's logic approach to quantitative domain theory (A Logical Approach to Quantitative Domain Theory, Elsevier, Amsterdam, 1996, submitted for publication). In addition, the concepts of stratified approximation and a basis for an $L$-fuzzy domain are proposed, and a notion of a continuous $L$-fuzzy domain is developed. It is proved that if $L$ is a completely distributive lattice in which 1 is $\vee$-irreducible and the well below relation is multiplicative, then the stratified interpolation property holds in a continuous $L$-fuzzy domain $(X,e)$, and $\{{\Uparrow }_{a}x\mid 0\ne a⋘1,x\in X\}$ is a base for the generalized Scott topology on $(X,e)$.     

Stratified lattice-valued neighborhood tower group

In this paper, by generalizing Höhle and ?ostak’s stratified L-fuzzy neighborhood system, the notion of stratified L-neighborhood tower space is introduced. Then by enriching a group structure on a stratified L-neighborhood tower space, the notion of stratified L-neighborhood tower group is proposed. It is proved that this notion can be regarded as a natural extension of stratified L-neighborhood group dis- cussed by Ahsanullah etal. Indeed, the category of stratified L-neighborhood tower groups includes the category of stratified L-neighborhood groups as a concretely reflective (resp., coreflective) full subcategory. Furthermore, it is shown that the group operations enrich a stratified L-neighborhood tower space to be topological (generally, stratified L-neighborhood tower space is not topological). This means that there is no di?erence between stratified L-neighborhood tower group and topologically stratified L-neighborhood tower group.     

Suitability in fuzzy topological spaces

An L-fuzzy topological space is said to be suitable if it possesses a nontrivial crisp closed subset. Basic properties of and sufficient conditions for suitable spaces are derived. Characterizations of the suitable subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line are obtained. Suitability is L-fuzzy productive; nondegenerate 1¹-Hausdorff spaces are suitable; the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line are not suitable; and no suitable subspace of the fuzzy unit interval, the fuzzy open unit interval, or the fuzzy real line is a fuzzy retract of the fuzzy unit interval, the fuzzy open unit interval, or the fuzzy real line, respectively. Without restrictions there cannot be a fuzzy extension theorem.     

Upper semicontinuous fuzzy sets and applications

Let (X, τ) be a generalized topological space of type $\text{V}$_α (see A. Appert and K. Fan, “Espaces topologiques intermédiares,” Herman, Paris, 1951) and (L, ?) be a complete Brouwerian lattice such that the dual lattice of (L, ?) is also Brouwerian. We prove that every upper semicontinuous L-fuzzy subset of X can be represented by a τ-closed random set. As an important application we obtain a fuzzification of measurable spaces as well as of topological spaces. In particular a concept of measurable (open) L-fuzzy sets is developed.     

On the level decompositions by the range of an <Emphasis Type="Italic">L</Emphasis>-fuzzy topological space

This paper is on the level decompositions corresponding to the elements in the range of an L-fuzzy topology on a given set and an investigation into the lattice structure of the same. In general, this lattice is not complete and distributive. However, certain necessary and sufficient conditions for it to be modular, distributive, complete and complemented are derived. Atoms and dual atoms of it along with the conditions for their existence are obtained. Certain related properties of it are also discussed here.     

Generalized fuzzy ideals of near-rings

The concept of（∈,∈∨q）-fuzzy subnear-rings（ideals） of a near-ring is introduced and some of its related properties are investigated.In particular,the relationships among ordinary fuzzy subnear-rings（ideals）,（∈,∈∨ q）-fuzzy subnear-rings（ideals） and（∈,∈∨q）-fuzzy subnear-rings（ideals） of near-rings are described.Finally,some characterization of [μ]t is given by means of（∈,∈∨ q）-fuzzy ideals.     

Redefined generalized fuzzy ideals of near-rings

With a new idea, we redefine generalized fuzzy subnear-rings(ideals) of a near- ring and investigate some of its related properties. Some new characterizations are given. In particular, we introduce the concepts of strong prime(or semiprime) (∈,∈∨q)-fuzzy ideals of near-rings, and discuss the relationship between strong prime(or semiprime) (∈,∈∨q)-fuzzy ideals and prime(or semiprime) (∈,∈∨q)-fuzzy ideals of near-rings.