L-fuzzy Relations Research Articles

On compositions of (L-fuzzy) automata: A categorical approach

This paper studies different compositions of (L-fuzzy) automata by using the notions from category theory, where L is a complete residuated lattice. Specifically, we introduce four different categories of (L-fuzzy) automata with morphisms as crisp relations and L-fuzzy relations, and show that each category is a symmetric monoidal with a product, coproduct, internal monoid, and comonoid. Such categorical study advances the existing categories because of the choice of morphisms. Also, among these compositions, the monoidal description of these categories further enriches the more established fuzzy automata theory.

Alexandrov L-quasi-G-filters and Alexandrov L-fuzzy pre-uniformities

This paper presents the idea of Alexandrov L-quasi-G-filter space and examines its relationship with L-fuzzy relations. It is proved that every Alexandrov L-quasi-G-filter induces an L-fuzzy relation and vice-versa. By identifying certain functors, the study has brought out the connections between the categories of Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces. Further, the study has explored and thereby establishes the scope of Alexandrov L-quasi-G-filter spaces in mathematical modeling and decision-making processes.

On L-fuzzy approximation operators and L-fuzzy relations on residuated lattices

We consider properties of L-fuzzy relations and L-normal operators for a residuated lattice L in detail and show that the class RL(U) of all L-fuzzy relations on U and the class NL(U) of all L-normal operators are residuated lattices and they are isomorphic as lattices. Moreover, we prove that for any L-normal operators F, it is reflexive (or transitive) if and only if the L-fuzzy relation RF induced by F is reflexive (or transitive) respectively.

The Single Axiomatization on CCRL-Fuzzy Rough Approximation Operators and Related Fuzzy Topology

It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For L=(L,⊛), a complete co-residuated lattice (CCRL), Qiao recently defined an L-fuzzy lower approximation operator (LFLAO) on the basis of the L-fuzzy relation. In this article, we give a further study on Qiao’s LFLAO around the axiomatic characterization and induced L-topology. Firstly, we investigate and discuss three new LFLAO generated by ⊛-transitive, ⊛-Euclidean and ⊛-mediated L-fuzzy relations. Secondly, we utilize a single axiom to characterize the LFLAO generated by serial, symmetric, reflexive, ⊛-transitive and ⊛-mediate L-fuzzy relations and their compositions. Thirdly, we present a method to generate Alexandrov L-topology (ALTPO) from LFLAO and construct a bijection between ALTPO and ⊛-preorder (i.e., reflexive and ⊛-transitive L-fuzzy relation) on the same underlying set.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

Axiomatic characterizations of L-fuzzy rough sets by L-fuzzy unions and L-fuzzy intersections

In this paper, considering L being a complete residuated lattice, we first introduce the concept of mediate L-fuzzy relations and give its characterizations by L-fuzzy upper and lower rough approximation operators. Then we provide an axiomatic approach to characterize L-fuzzy rough sets by L-fuzzy unions and L-fuzzy intersections. Concretely, we present axiomatic characterizations of L-fuzzy rough sets with respect to serial, reflexive, symmetric, transitive, mediate and Euclidean L-fuzzy relations. Finally, we give characterizations of L-fuzzy upper (lower) rough approximation operators by L-closure (interior) operators from a categorical aspect.

Topological structures induced by L-fuzzifying approximation operators

This paper further studies topological structures induced by L-fuzzifying approximation operators, where L denotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudo-similarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations.

L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations

For L a complete co-residuated lattice and R an L-fuzzy relation, an L-fuzzy upper approximation operator based on co-implication adjoint with L is constructed and discussed. It is proved that, when L is regular, the new approximation operator is the dual operator of the Qiao–Hu L-fuzzy lower approximation operator defined in 2018. Then, the new approximation operator is characterized by using an axiom set (in particular, by single axiom). Furthermore, the L-fuzzy upper approximation operators generated by serial, symmetric, reflexive, mediate, transitive, and Euclidean L-fuzzy relations and their compositions are characterize through axiom set (single axiom), respectively.

(1909-5496) Topological structures induced by L-fuzzifying approximation operators

This paper further studies topological structures induced by L-fuzzifying approximation operators, where Ldenotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudo-similarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations.

L-fuzzifying approximation operators derived from general L-fuzzifying neighborhood systems

For a completely distributive De Morgan algebra L, we develop a general framework of L-fuzzy rough sets. Said precisely, we introduce a pair of L-fuzzy approximation operators, called upper and lower L-fuzzifying approximation operators derived from general L-fuzzifying neighborhood systems. It is shown that the proposed approximation operators are a common extension of the L-fuzzifying approximation operators derived from L-fuzzy relations (INS 2019) and the approximation operators derived from general neighborhood systems (KBS 2014). Furthermore, we investigate the unary, serial, reflexive, transitive and symmetric conditions in general L-fuzzifying neighborhood systems, and then study the associated approximation operators from both a constructive method and an axiomatic method. Particularly, for transitivity (resp., symmetry), we give two interpretations, one is an appropriate generalization of transitivity (resp., symmetry) for L-fuzzy relations, and the other is a suitable extension of transitivity (resp., symmetry) for general neighborhood systems. In addition, for some special L-fuzzifying approximation operators, we use single axiom to characterize them, respectively. At last, the proposed approximation operators are applied in the research of incomplete information system, and a three-way decision model based on them is established. To exhibit the effectiveness of the model, a practical example is presented.

Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined.

General L-fuzzy aggregation functions based on complete residuated lattices

As a vital tool in data analysis, aggregation functions have been widely studied in many papers. In particular, one of the recent research topics for aggregation functions is the study of the various extension forms of those useful functions. This paper continues to research this topic from the theoretical point of view. First, we introduce the notions of L-fuzzy aggregation functions and general L-fuzzy aggregation functions based on complete residuated lattices. Then we present the upper and lower general L-fuzzy aggregation approximation functions of the general L-fuzzy aggregation functions, which are the pointwise extension of an L-fuzzy aggregation function. Moreover, we consider some vital properties of those aggregation approximation functions and investigate the relationship between those aggregation approximation functions and the corresponding L-fuzzy relations. Finally, we show that the approach of axiomatizations of the upper and lower general L-fuzzy aggregation approximation functions ensures the existence of corresponding L-fuzzy relations which generate the functions.

L-fuzzy rough approximation operators via three new types of L-fuzzy relations

Considering L being a frame with an order-reversing involution, three new types of L-fuzzy relations are introduced, which are called mediate, Euclidean and adjoint L-fuzzy relations, respectively. By means of these L-fuzzy relations, three types of L-fuzzy rough approximation operators are constructed and their connections with those three L-fuzzy relations are examined, respectively. An axiomatic approach is adopted to deal with L-fuzzy rough approximation operators. It is shown that each type of L-fuzzy rough approximation operators corresponding to mediate, Euclidean and adjoint L-fuzzy relations as well as their compositions can be characterized by single axioms.

New results on single axioms for L-fuzzy rough approximation operators

This paper further studies axiomatic characterizations of L-fuzzy rough sets, where L denotes a residuated lattice. Single axioms for upper L-fuzzy rough approximation operators are investigated in two different approaches, which describe upper L-fuzzy approximation operators with ordinary L-fuzzy operations and L-fuzzy product operation, respectively. We characterize upper L-fuzzy rough approximation operators by only one axiom with ordinary L-fuzzy operations, when the L-fuzzy relation is Euclidean as well as the composition of Euclidean with serial (resp. reflexive). We also discuss the single axioms for upper L-fuzzy rough approximation operators with L-fuzzy product operation on arbitrary L-fuzzy relations that are not limited as either general L-fuzzy relations or symmetric L-fuzzy relations. The single axioms for lower L-fuzzy rough approximation operators are discussed with ordinary L-fuzzy operations regardless of the regularity of residuated lattice L.

L-fuzzifying approximation operators in fuzzy rough sets

In rough set theory, lower and upper approximation operators are two primitive notions. Various fuzzy generalizations of lower and upper approximation operators have been introduced over the years. Considering L being a completely distributive De Morgan algebra, this paper mainly proposes a general framework of L-fuzzifying approximation operators in which constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper L-fuzzifying approximation operators is defined. The connections between L-fuzzy relations and L-fuzzifying approximation operators are examined. In the axiomatic approach, various types of L-fuzzifying rough sets are proposed and L-fuzzifying approximation operators corresponding to each type of L-fuzzy relations as well as their compositions are characterized by single axioms. Moreover, the relationships between L-fuzzifying rough sets and L-fuzzifying topological spaces are investigated. It is shown that there is a one-to-one correspondence between reflexive and transitive L-fuzzifying approximation spaces and saturated L-fuzzifying topological spaces.

L-fuzzy relational mathematical morphology based on adjoint triples

We propose an alternative to the standard structure of L-fuzzy Mathematical Morphology (MM) by, on the one hand, considering L-fuzzy relations as structuring elements and, on the other hand, by using adjoint triples to handle membership values. Those modifications lead to a framework based on set-theoretical operations where we can prove a representation theorem for algebraic morphological erosions and dilations. In addition, we also present some new results concerning duality and transformation invariance. Concerning duality, we show that duality and adjointness can coexist in this L-fuzzy relational MM. Concerning transformation invariance, we show sufficient conditions to guarantee the invariance of morphological operators under arbitrary transformations.

A comparative study of variable precision fuzzy rough sets based on residuated lattices

Although Qiao and Hu intended to propose granular variable precision L-fuzzy rough sets based on residuated lattices, L actually denotes a doubly residuated lattice with an involutive negation. However, there exist doubly residuated lattices without involutive negations. This paper presents a comparative study of variable precision fuzzy rough sets based on residuated lattices, in which we compare variable precision L-fuzzy rough sets with the preexisting models. A variable precision upper L-fuzzy rough approximation operator is proposed with L-fuzzy granules based on L-fuzzy implication on a residuated lattice L instead of a doubly residuated lattice. The variable precision upper L-fuzzy rough approximation operator is investigated with respect to different L-fuzzy relations regardless of proposing an involutive negation on a residuated lattice.

Alexandroff L-fuzzy topological spaces and reflexive L-fuzzy relations

Galois connection in category theory palys an important role in establish the relationships between different spatial structures. In this paper, we prove that there exist many interesting Galois connections between the category of Alexandroff $L$-fuzzy topological spaces and the category of reflexive $L$-fuzzy relations.

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.