Many-level fuzzy rough approximation spaces induced by many-level fuzzy preorders and the related ditopological structures

Abstract

We present a many-level version for the Pawlak -Dubois&Prade theory of rough approximation of fuzzy sets. Basing on the many-level upper and lower fuzzy rough approximation operators, we define the measure of rough approximation that in a certain sense characterizes the quality of the obtained approximation. Further, the fuzzy rough approximation operators give rise to two alternative topological-type structures considered in the paper. Keywords: Many-level fuzzy rough approximation operators, measure of fuzzy rough approximation, LM -fuzzy (di)topologies, Mlevel L-fuzzy (di)-topologies.2 Many-level L-fuzzy relations 2.1 Basic definitions Let L = (L, ≤ L , ∧ L , ∨ L , * ) be an integral commutative complete lattice monoid (in particular, L = [0, 1] and * a lower semi-continuous t-norm), see, e.g. [8], and let M = (M, ≤ M , ∧ M , ∨ M ) be a complete infinitely distributive lattice. Definition 2.1 An M -level L-fuzzy relation on a set 11th

Rough Semiring-Valued Fuzzy Sets with Application

“…Another possible use of this universal method of (R, R * )-fuzzy sets offers the possibility of creating new types of fuzzy sets, defined using suitable examples of dual pairs of semirings. We also gave examples of this procedure, when we mentioned the possibility of introducing fuzzy sets with values in Abelian l-groups of application of (R, R * )-fuzzy sets in many-level fuzzy sets and relations introduced by Šostak [24]. It can be assumed that further research in this direction will show many other examples of new types of fuzzy sets.…”

“…Therefore, instead of the fuzzy relation Q ∶ X × X → L, there is the fuzzy relation Q ′ ∶ X × X × Ω → L, where Ω is the set of observation points. This approach was first discussed by A. Šostak in [24], where he introduced the concept of many-level L-fuzzy relations. We show that these structures can be transformed into (R 5 , R * 5 )-fuzzy relations.…”

Cut Systems with Relational Morphisms for Semiring-Valued Fuzzy Structures

“…This category represents a generalization of the classical category Set of sets, with the difference that the more general L-valued similarity relation is used instead of the classical identity. Among recent applications of L-sets theory is, for example, the theory of multilevel fuzzy sets, introduced by Šostak [10,11].…”