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

Some remarks on topological structures in the context of fuzzy relational mathematical morphology

“…• The pairs (ε R , δ R −1 ) and (ε R −1 , δ R ) were used here to develop a certain "topological viewpoint" on the operators of erosion and dilation. However, probably even more interesting especially from the point of possible application will be to apply these pairs for description of certain "remote" fuzzy rough approximation models (cf [22], [8], [10], [33]). In particular, it could be useful in the study of big volumes of transformed data.…”

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

“…Proof We omit the proofs of properties (1u) -(4u) and (1l) -(4l) since they can be easily obtained by modifying the proofs of the corresponding properties of fuzzy rough approximation operators in the proof of Proposition 4 in [12] for the many-level case. Properties (5u) and (5l) are clear from the definitions.…”

“…The proof of the following theorem can be done by level-wise modification in the proofs of Proposition 3 and Theorem 1 in [12].…”

“…Let (X, L R , U R ) be an M -level L-fuzzy L-rough approximation space and let α ∈ M be fixed. Properties (1), (5) and (6) of Theorem 5.3 characterize the relation's L R : L X ×M → L restriction to the set L X ×{α} as a stratified L-fuzzy topology on the set X [10], [9]. In its turn, properties (2), (7) and (8) characterize the relation's U R : L X × M → L restriction to the set L X × {α} as a stratified L-fuzzy co-topology on a set X.…”

“…Definition 5.6 An M -level L-fuzzy topology on a set X is a mapping L R : L X ×M → L satisfying properties (1), (5) and (6) of Theorem 5.3. Respectively, an Mlevel L-fuzzy co-topology on a set X is a mapping U R : L X × M → L satisfying properties (2), (7) and (8) Now from Theorem 5.5 we get the following:…”