L-fuzzy Topologies Research Articles

On L-fuzzy automata, coalgebras and dialgebras: Associated categories and L-fuzzy topologies

This paper is towards studying L-fuzzy automata, coalgebras, and dialgebras from categorical and L-fuzzy topological points of view, where L is a complete residuated lattice. Specifically, we introduce the categories of coalgebras and dialgebras and show that the category of L-fuzzy automata is categories of coalgebras and dialgebras, respectively. Further, we establish the relationships among the L-fuzzy languages recognized by L-fuzzy automata, coalgebras, and dialgebras and study bisimulation between coalgebras. Interestingly, L-fuzzy languages recognized by L-fuzzy automaton, coalgebra, and dialgebra are enriched with L-fuzzy co-topologies and L-fuzzy topologies.

L-fuzzy automata theory: Some characterizations via general fuzzy operators

This paper aims to study the L-fuzzy automata theory based on t-norm, t-conorm and general implicator, alongwith to investigate the algebraic and L-fuzzy topological properties of L-fuzzy automata. Specifically, we introduce L-fuzzy operators based on t-norm, t-conorm and implicator, and establish their relationships. Interestingly, we associate L-fuzzy co-topologies and L-fuzzy topologies for a given L-fuzzy automaton and use these to characterize some algebraic concepts. Further, we introduce the concepts of L-graded co-topologies, L-graded topologies and establish their relationships. We also demonstrate that the introduced L-graded co-topologies and L-graded topologies have some exciting consequences under a homomorphism.

Note on ‘Intuitionistic L-fuzzy Rough Sets, Intuitionistic L-fuzzy Preorders and Intuitionistic L-fuzzy Topologies’

In this note, we show that, in the paper [Zhong Y, Yan CH. Intuitionistic L-fuzzy rough sets, intuitionistic L-fuzzy preorders and intuitionistic L-fuzzy topologies. Fuzzy Inf Eng. 2016;8:255–279.], the conclusion regarding Proposition 3.3 of [Tiwari SP, Srivastava AK. Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 2013;210:63–68.] is not correct.

On a factorized L-fuzzy automaton and its L-fuzzy topological characterization

This paper is towards the study of L-fuzzy automata via theory of factorizations, where L is a complete residuated lattice. We study the concept of factorization of L-fuzzy sets and factorization of collection of L-fuzzy sets. Such factorizations are used (i) to study the concept of minimal L-fuzzy automaton, and (ii) to introduce L-fuzzy topologies for the study of concepts associated with an L-fuzzy automaton. Specifically, we introduce the notion of factorized L-fuzzy automaton corresponding to a given L-fuzzy automaton and study the minimality of such automaton. Further, we associate L-fuzzy topologies for factorized L-fuzzy automaton and use these to characterize some algebraic concepts.

L-fuzzy pre-proximities and application to L-fuzzy topologies

L-fuzzy pre-proximities and application to L-fuzzy topologies

Categories of Alexandrov L-Fuzzy Pre-uniformities, L-Approximation Operators and Alexandrov L-Fuzzy Topologies

Categories of Alexandrov L-Fuzzy Pre-uniformities, L-Approximation Operators and Alexandrov L-Fuzzy Topologies

Alexandrov L-Fuzzy Pre-Proximities

In this paper, we introduce the concepts of Alexandrov L-fuzzy pre-proximities on complete residuated lattices. Moreover, we investigate their relations among Alexandrov L-fuzzy pre-proximities, Alexandrov L-fuzzy topologies, L-fuzzy upper approximate operators, and L-fuzzy lower approximate operators. We give their examples.

Intuitionistic [formula omitted]-fuzzy Rough Sets, Intuitionistic [formula omitted]-fuzzy Preorders and Intuitionistic [formula omitted]-fuzzy Topologies

In this paper, by constructing an intuitionistic L-fuzzy triangle norm T and an intuitionistic L-fuzzy implicator I, lower and upper approximations of intuitionistic L-fuzzy sets are defined with respect to an intuitionistic L-fuzzy approximation space. Properties of intuitionistic L-fuzzy approximation operators are then given. This paper is devoted to discussing the relationship between intuitionistic L-fuzzy relations and intuitionistic L-fuzzy topologies. It proves that the set of all lower approximation sets based on a reflexive and transitive intuitionistic L-fuzzy relation consists of an intuitionistic L-fuzzy Alexandrov topology; and conversely, an intuitionistic L-fuzzy Alexandrov topology is just the set of all lower approximation sets under a reflexive and transitive intuitionistic L-fuzzy relation. That is to say, there exists a one-to-one correspondence between the set of all intuitionistic L-fuzzy preorders and the set of all intuitionistic L-fuzzy Alexandrov topologies.

Similarity of binary relations based on complete Boolean lattices and related results

Rough set theory is concerned with the lower and upper approximations through a binary relation on the universe. Binary relations play an important role in this theory. This paper investigates similarity of binary relations based on complete Boolean lattices. First, rough approximations based on complete Boolean lattices are proposed through the predecessor neighborhood. Second, L-fuzzy topologies induced by binary relations are researched where L is a complete Boolean lattice. Thirdly, similarity of binary relations is introduced by using L-fuzzy topology and the fact that every binary relation is solely similar to some preorder relation is showed. Finally, as its theoretical expansion, a topological problem "when can the given L-fuzzy topology be coincided with the L-fuzzy topology induced by some binary relation where L is a complete Boolean lattice?" is considered and this problem is answered by introducing L-fuzzy approximating spaces.

Similarity of binary relations based on L-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.

Relationships between L-fuzzy quasi-uniform structures and L-fuzzy topologies

In this paper, we investigated the properties of two L-fuzzy interior operators and two L-fuzzy topologies induced by L-quasi-uniformity in complete residuated lattices. We study the relations among L-fuzzy quasi-uniformities, L-fuzzy interior operators and L-fuzzy topologies. We give their examples.

L-FUZZY UNIFORMITIES INDUCED BY L-NEIGHBORHOOD SYSTEMS AND L-FUZZY TOPOLOGIES

In this paper, we obtain L-fuzzy uniformities induced by Lneighborhood systems and L-fuzzy topologies in complete residuated lattices. Moreover, every N -continuous maps are fuzzy uniformly continuous and every fuzzy continuous maps are fuzzy uniformly continuous. AMS Subject Classification: 03E72, 06A15, 06F07, 54A05

UPPER APPROXIMATION OPERATORS INDUCED BY MAPS

In this paper, we investigate the properties of Alexandrov fuzzy topologies and upper approximation operators induced by maps. We give their examples. Hohle (3) introduced L-fuzzy topologies and L-fuzzy interior operators. The relationship between rough set theory and topological spaces was investigated (4-12). Hajek (2) introduced a complete residuated lattice which is an algebraic structure for many valued logic. Kim (5-7) investigated the properties of join (resp. meet, meet join, join meet) preserving operators and Alexandrov fuzzy topology in complete residuated lattices. In this paper, we investigate the properties of Alexandrov fuzzy topologies and upper approximation operators induced by maps. We give their examples.

Some properties of L-fuzzy approximation spaces based on bounded integral residuated lattices

In this paper, we present a general framework for the study of L-fuzzy rough sets based on bounded integral residuated lattices. We introduce and discuss the notions of L-fuzzy topological spaces, left (right) lower and left (right) upper L-fuzzy approximation operators based on a complete bounded integral residuated lattice, investigate union, intersection, and composition operations of L-fuzzy approximation spaces, and study some properties of the L-fuzzy topologies generated by these L-fuzzy rough approximation operators and the L-relations induced by an L-fuzzy topology.

L-fuzzy syntopogenous structures, Part I: Fundamentals and application to L-fuzzy topologies, L-fuzzy proximities and L-fuzzy uniformities

We introduce the concept of an L-fuzzy syntopogenous structure where L is a complete lattice endowed with an implicator ↦:L×L→L satisfying certain properties (in particular, as L one can take an MV-algebra). As special cases our L-fuzzy syntopogenous structures contain classical Császár syntopogenous structures, Katsaras–Petalas fuzzy syntopogenous structures as well as fuzzy syntopogeneous structures introduced in the previous work of the second named author (A. Šostak, Fuzzy syntopogenous structures, Quaest. Math. 20 (1997) 431–461). Basic properties of the category of L-fuzzy syntopogenous spaces are studied; categories of L-fuzzy topological spaces, L-fuzzy proximity spaces and L-fuzzy uniform spaces are characterized in the framework of the category of L-fuzzy syntopogenous spaces.

L-fuzzy topogenous orders and L-fuzzy topologies

In this paper, we introduce the notions of L-fuzzy topoenous orders and investigate some of properties. We investigate the relationships among L-fuzzy topoenous orders, L-fuzzy topologies and L-fuzzy interior operators.

THE RELATIONSHIP BETWEEN L-FUZZY PROXIMITIES AND L-FUZZY QUASI-UNIFORMITIES

In this paper, we investigate the L-fuzzy proximities and the re- lationships between L-fuzzy topologies, L-fuzzy topogenous order and L-fuzzy uniformity. First, we show that the category of L-fuzzy topological spaces can be embedded in the category of L-fuzzy quasi-proximity spaces as a coreec- tive full subcategory. Second, we show that the category of L-fuzzy proximity spaces is isomorphic to the category of L-fuzzy topogenous order spaces. Fi- nally, we obtain that the category ofL-fuzzy proximity spaces can be embedded in the category of L-fuzzy uniform spaces as a bireective full subcategory.

Some topological properties of L-fuzzy normed spaces

In this paper, the L-topological structure of L-fuzzy normed linear spaces is discussed. Firstly, we prove that the L-fuzzy normed linear space is a special type of Hausdorff locally convex and locally bounded L-topological vector space by a new way. Secondly, we characterize the convergence of a sequence of L-fuzzy points in L-fuzzy normed linear space. Finally, we use the convergence of a sequence of L-fuzzy points to describe the containment relationship of two L-fuzzy topologies induced by two L-fuzzy norms; and we give a necessary and sufficient condition for two L-fuzzy norms to be equivalent.

L-FUZZY TOPOLOGICAL SPACES AND L-FUZZY QUASI-PROXIMITY SPACES

This paper studies the relationship between L-fuzzy proximities and L-fuzzy topologies by topological fuzzy remote neigh-borhood systems. We will prove that the category of L-fuzzy topo- logical spaces can be embedded in the category of L-fuzzy quasi-proximity spaces as a core ective full subcategory.

L-filters and L-filter convergence

In this paper, we study the relations between L-fuzzy topologies and L-filters on a strictly two-sided, commutative quantale lattice L. We define an L-fuzzy neighborhood filter and introduce the notion of L-filter convergence in L-fuzzy topological spaces.