L-fuzzy numbers and their properties

In this paper, the notions of L-fuzzy convex sets and L-fuzzy numbers are introduced where L is a completely distributive lattice. The notions of [0, 1]-fuzzy convex sets and [0, 1]-fuzzy numbers are generalized. Furthermore their properties and characterizations are presented in terms of cut sets of L-fuzzy sets.     

The axiomatic characterizations on L-generalized fuzzy neighborhood system-based approximation operators

The rough sets based on L-fuzzy relations and L-fuzzy coverings are the two most well-known L-fuzzy rough sets. Quite recently, we prove that some of these rough sets can be unified into one framework—rough sets based on L-generalized fuzzy neighborhood systems. So, the study on the rough sets based on L-generalized fuzzy neighborhood system has more general significance. Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations, L-fuzzy coverings that reproduce the approximation operators. In this paper, we shall give an axiomatic study on L-generalized fuzzy neighborhood system-based approximation operators. In particular, we will seek the axiomatic sets to characterize the approximation operators generated by serial, reflexive, unary and transitive L-generalized fuzzy neighborhood systems, respectively.     

Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice

This paper considers the relationships among L-fuzzy sets, rough sets and n-ary hypergroup theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy n-ary subhypergroup are introduced and studied. Then, a new algebraic structure called (invertible) L-fuzzy rough n-ary subhypergroups is defined, and the (strong) homomorphism of lower and upper L-fuzzy rough approximation operators is studied.     

Lattice-valued fuzzy frames

In this paper we aim to introduce the concept of lattice-valued fuzzy frame or L-fuzzy frame, related to traditional frames analogously to how L-fuzzy (resp. L-fuzzifying) topological spaces related to L-topological spaces (resp. traditional topological spaces). Some properties of L-fuzzy frames are introduced.     

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.     

The axiomatic characterizations on L-fuzzy covering-based approximation operators

Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.     

Similarity of binary relations based on <Emphasis Type="Italic">L</Emphasis>-fuzzy topologies

Binary relations play an important role in rough set theory. This paper investigates the similarity of binary relations based on L-fuzzy topologies, where L is a boolean algebra. First, rough approximations based on a boolean algebra are proposed through successor neighborhoods on binary relations. Next, L-fuzzy topologies induced by binary relations are investigated. Finally, similarity of binary relations is introduced by using the L-fuzzy topologies and the fact that every binary relation is solely similar to some preorder relation is proved. It is worth mentioning that similarity of binary relations are both originated in the L-fuzzy topology and independent of the L-fuzzy topology.     

Semigroup homomorphisms and fuzzy automata

A generalized Ω-fuzzy automaton over a complete residuated lattice Ω and a monoid (M,*) and with a set S of states is introduced as a monoid homomorphism F:(M,*)→(?,°), where (?,°) is a monoid of Ω-fuzzy sets in a set S×S. An extension principle depending of proper filters Φ in Ω is introduced which then enables to introduce morphisms between generalized Ω-fuzzy automata and to introduce the category ?_Φ of these automata. It is proved that categories of classical fuzzy automata, non-deterministic automata and some other systems are equivalent to subcategories of ?_Φ for a suitable filter Φ.     

Weakly semi-preopen and semi-preclosed functions in L-fuzzy topological spaces

A new class of functions called L-fuzzy weakly semi-preopen (semi-preclosed) functions in L-fuzzy topological spaces is introduced in this paper. Some characterizations of this class and its properties and the relationship with other classes of functions between L-fuzzy topological spaces are also obtained.     

Revising the link between L-Chu correspondences and completely lattice L-ordered sets

Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.     

On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)-fuzzy rough sets, where I is an implication and T is a triangular norm. In the present paper, by using a pair of implications (I,J), we define the so-called (I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)-fuzzy rough sets are investigated in detail.     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

Fuzzy semilattices

We discuss L-fuzzy semilattices and their ideals and establish some interesting properties. In particular we show that the lattice of all L-fuzzy ideals of a semilattice is an algebraic lattice if the valuation lattice L is algebraic.     

L-fuzzy grammars

An L-fuzzy grammar is defined by assigning the element of lattice to the rewriting rules of a formal grammar. According to the kind of lattice, say, distributive lattice, lattice-ordered group, and lattice-ordered monoid, two type of L-fuzzy grammars are defined. It is shown that some context-sensitive languages can be generated by type 3 $\text{1-L-}\text{fuzzy}$ grammars with cut points. It is also shown that for type 2 L-fuzzy grammars, Chomsky and Greibach normal form can be constructed as an extension of corresponding notion in the theory of formal grammars.     

Application of the L-fuzzy concept analysis in the morphological image and signal processing

In this work we are going to set up a new relationship between the L-fuzzy Concept Analysis and the Fuzzy Mathematical Morphology. Specifically we prove that the problem of finding fuzzy images or signals that remain invariant under a fuzzy morphological opening or under a fuzzy morphological closing, is equal to the problem of finding the L-fuzzy concepts of some L-fuzzy context. Moreover, since the Formal Concept Analysis and the Mathematical Morphology are the particular cases of the fuzzy ones, the showed result has also an interpretation for binary images or signals.     

A thresholding method based on interval-valued intuitionistic fuzzy sets: an application to image segmentation

This paper proposes a new fuzzy approach for the segmentation of images. L-interval-valued intuitionistic fuzzy sets (IVIFSs) are constructed from two L-fuzzy sets that corresponds to the foreground (object) and the background of an image. Here, L denotes the number of gray levels in the image. The length of the membership interval of IVIFS quantifies the influence of the ignorance in the construction of the membership function. Threshold for an image is chosen by finding an IVIFS with least entropy. Contributions also include a comparative study with ten other image segmentation techniques. The results obtained by each method have been systematically evaluated using well-known measures for judging the segmentation quality. The proposed method has globally shown better results in all these segmentation quality measures. Experiments also show that the results acquired from the proposed method are highly correlated to the ground truth images.     

Partial ordering in L-underdeterminate sets

The purpose of this paper is to improve results on fuzzy partial orderings obtained by Zadeh in [9]. We overcome the difficulties connected with the axioms of antisymmetry and linearity. Moreover, if the underlying lattice L is a complete residuated lattice, we establish a Szpilrajn theorem, i.e., any (L-fuzzy) partial ordering has a linear extension. In opposition to Zadeh's, our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless. Therefore we first introduce the category LUS (cf. [2]), which can be considered as a mathematical model of fuzzy equality, and subsequently we specify the axioms of (L-fuzzy) partial orderings with respect to the frame given by LUS. The axioms we use clearly display the usefulness of having a Zadeh-like complementation and, as a consequence, the usefulness of a positivistic (and nonintuitionistic) frame of study. An example concerning L°(Rⁿ) which we give clearly shows that the LUS version of the Szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem, but provides us with additional information.     

fstds System: A fuzzy-set manipulation system

This paper describes an implementation of a system for fuzzy sets manipulation which is based on fstds (Fuzzy-Set-Theoretic Data Structure), an extended version of Childs's stds (Set-Theoretic Data Structure). The fstds language is considered as a fuzzy-set-theoretically oriented language which can deal, for example, with ordinary sets, ordinary relations, fuzzy sets, fuzzy relations, L-fuzzy sets, level-m fuzzy sets and type-n fuzzy sets. The system consists of an interpreter, a collection of fuzzy-set operations and the data structure, fstds, for representing fuzzy sets. fstds is made up of eight areas, namely, the fuzzy-set area, fuzzy-set representation area, grade area, grade-tuple area, element area, element-tuple area, fuzzy-set name area and fuzzy-set operator name area. The fstds system, in which 52 fuzzy-set operations are available, is implemented in fortran, and is currently running on a FACOM 230-45S computer.