M-bornologies on L-valued Sets

We develop an approach to the concept of bornology in the framework of many-valued mathematical structures. It is based on the introduced concept of an M-bornology on an L-valued set (X, E), or an LM-bornology for short; here L is an iccl-monoid, M is a completely distributive lattice and \(E: X\times X \rightarrow L\) is an L-valued equality on the set X. We develop the basics of the theory of LM-bornological spaces and initiate the study of the category of LM-bornological spaces and appropriately defined bounded “mappings” of such spaces.

Fuzzy functions: a fuzzy extension of the category SET and some related categories

<p>In research Works where fuzzy sets are used, mostly certain usual functions are taken as morphisms. On the other hand, the aim of this paper is to fuzzify the concept of a function itself. Namely, a certain class of L-relations F : X x Y -> L is distinguished which could be considered as fuzzy functions from an L-valued set (X,Ex) to an L-valued set (Y,Ey). We study basic properties of these functions, consider some properties of the corresponding category of L-valued sets and fuzzy functions as well as briefly describe some categories related to algebra and topology with fuzzy functions in the role of morphisms.</p>

A fuzzification of the category of M-valued L-topological spaces

[EN] A fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FTOP(L,M) extending the category TOP(L,M) of M-valued L- topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak’s sense. Basic properties of the fuzzy category FTOP(L,M) and its objects are studied.

Bornological structures on many-valued sets

Towards the theory of M-approximate systems: Fundamentals and examples

The concept of an M-approximate system is introduced. Basic properties of the category of M-approximate systems and in a natural way defined morphisms between them are studied. It is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of M-approximate systems.

On a Category of Extensional Fuzzy Rough Approximation L-valued Spaces

We establish extensionality of some upper and lower fuzzy rough approximation operators on an L-valued set. Taking as the ground basic properties of these operators, we introduce the concept of an (extensional) fuzzy rough approximation L-valued space. We apply fuzzy functions satisfying certain continuity-type conditions, as morphisms between such spaces, and in the result obtain a category \(\mathcal{FRA}{} \mathbf{SPA}(L)\) of fuzzy rough approximation L-valued spaces. An interpretation of fuzzy rough approximation L-valued spaces as L-fuzzy (di)topological spaces is presented and applied for constructing examples in category \(\mathcal{FRA}{} \mathbf{SPA}(L)\).

On central algorithms of approximation under fuzzy information

We consider the problem of approximation of an operator by information described by n real characteristics in the case when this information is fuzzy. We develop the well-known idea of an optimal error method of approximation for this case. It is a method whose error is the infimum of the errors of all methods for a given problem characterized by fuzzy numbers in this case. We generalize the concept of central algorithms, which are always optimal error algorithms and in the crisp case are useful both in practice and in theory. In order to do this we define the centre of an L-fuzzy subset of a normed space. The introduced concepts allow us to describe optimal methods of approximation for lin…

Fuzzy Relational Mathematical Morphology: Erosion and Dilation

In the recent years, the subject if fuzzy mathematical morphology entered the field of interest of many researchers. In our recent paper [23], we have developed the basis of the (unstructured) L-fuzzy relation mathematical morphology where L is a quantale. In this paper we extend it to the structured case. We introduce structured L-fuzzy relational erosion and dilation operators, study their basic properties, show that under some conditions these operators are dual and form an adjunction pair. Basing on the topological interpretation of these operators, we introduce the category of L-fuzzy relational morphological spaces and their continuous transformations.

Variable-Range Approximate Systems Induced by Many-Valued L-Relations

The concept of a many-valued L-relation is introduced and studied. Many-valued L-relations are used to induce variable-range quasi-approximate systems defined on the lines of the paper (A. Sostak, Towards the theory of approximate systems: variable-range categories. Proceedings of ICTA2011, Cambridge Univ. Publ. (2012) 265–284.) Such variable-range (quasi-)approximate systems can be realized as special families of L-fuzzy rough sets indexed by elements of a complete lattice.

Fuzzifying topology induced by a strong fuzzy metric

A construction of a fuzzifying topology induced by a strong fuzzy metric is presented. Properties of this fuzzifying topology, in particular, its convergence structure are studied. Our special interest is in the study of the relations between products of fuzzy metrics and the products of the induced fuzzifying topologies.

M-valued Measure of Roughness for Approximation of L-fuzzy Sets and Its Topological Interpretation

We develop a scheme allowing to measure the “quality” of rough approximation of fuzzy sets. This scheme is based on what we call “an approximation quadruple” \((L,M,\varphi ,\psi )\) where L and M are cl-monoids (in particular, \(L=M=[0,1]\)) and \(\psi : L \rightarrow M\) and \(\varphi : M \rightarrow L\) are satisfying certain conditions mappings (in particular, they can be the identity mappings). In the result of realization of this scheme we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with a reflexive transitive M-fuzzy relation R. In case the relation R is also symmetric, these measures coincide and we call their value by the measure of rou…

Ideal-valued topological structures

With L a complete lattice and M a continuous lattice, this paper demonstrates an adjunction between M -valued L-topological spaces (i.e. (L,M )-topological spaces) and Idl(M )-valued L-topological spaces where Idl(M ) is the complete lattice of all ideals of M . It is shown that the right adjoint functor provides a procedure of generating (L,M )-topologies from antitone families of (L,M )-topologies. This procedure is then applied to give an internal characterization of joins in the complete lattice of all (L,M )-topologies on a given set.

Axiomatic Foundations Of Fixed-Basis Fuzzy Topology

This paper gives the first comprehensive account on various systems of axioms of fixed-basis, L-fuzzy topological spaces and their corresponding convergence theory. In general we do not pursue the historical development, but it is our primary aim to present the state of the art of this field. We focus on the following problems:

A construction of a fuzzy topology from a strong fuzzy metric

<p>After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper  (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems,  6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology ${\mathcal T}:2^X \to [0,1]$ …

On the Measure of Many-Level Fuzzy Rough Approximation for L-Fuzzy Sets

We introduce a many-level version of L-fuzzy rough approximation operators and define measures of approximation obtained by such operators. In a certain sense, theses measures characterize the quality of the resulting approximation. We study properties of such measures and give a topological interpretation of the obtained results.

On inductive dimensions for fuzzy topological spaces

An approach to the dimension theory for fuzzy topological spaces is being developed. The appropriate context for this theory is not the category CFT of Chang fuzzy topological spaces or some of its modifications, but the category Hut introduced in the paper (this category is a slight extension of the category H of Hutton fuzzy topological spaces Hutton (1980). The frames of this category allow us to make exposition simple and uniform, and on the other hand to make it applicable in quite a general setting.

Fuzzy $$\varphi $$ -pseudometrics and Fuzzy $$\varphi $$ -pseudometric Spaces

By replacing the axiom \(m(x,x,t) = 1\) for all \(x\in X, t>0\) in the definition of a fuzzy pseudometric in the sense of George-Veeramani with a weaker axiom \(m(x,x,t) = \varphi (t)\) for all \(x\in X, t>0\) where \(\varphi : {\mathbb R}^+ \rightarrow (0,1]\) is a non-decreasing function, we come to the concept of a fuzzy \(\varphi \)-pseudometric space. Basic properties of fuzzy \(\varphi \)-pseudometric spaces and their mappings are studied. We show also an application of fuzzy \(\varphi \)-pseudometrics in the words combinatorics.

Extremal problems of approximation theory in fuzzy context

Abstract The problem of approximation of a fuzzy subset of a normed space is considered. We study the error of approximation, which in this case is characterized by an L -fuzzy number. In order to do this we define the supremum of an L -fuzzy set of real numbers as well as the supremum and the infimum of a crisp set of L -fuzzy numbers. The introduced concepts allow us to investigate the best approximation and the optimal linear approximation. In particular, we consider approximation of a fuzzy subset in the space L p m of differentiable functions in the L q -metric. We prove the fuzzy counterparts of duality theorems, which in crisp case allows effectively to solve extremal problems of the…

On approximate-type systems generated by L-relations

The aim of this work is to study approximate-type systems induced by L-relations in the framework of the general theory of M-approximate systems introduced in [42] and its generalizations. Special attention is payed to the structural properties of lattices of such systems and to the study of connections between categories of such systems and the corresponding categories of sets endowed with L-relations.

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

Abstract 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 Csaszar syntopogenous structures, Katsaras–Petalas fuzzy syntopogenous structures as well as fuzzy syntopogeneous structures introduced in the previous work of the second named author (A. Sostak, 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 spac…

On spline methods of approximation under L-fuzzy information

This work is closely related to our previous papers on algorithms of approximation under L-fuzzy information. In the classical theory of approximation central algorithms were worked out on the basis of usual, that is crisp splines. We describe central methods for solution of linear problems with balanced L-fuzzy information and develop the concept of L-fuzzy splines.