On rough set and fuzzy sublattice

Rough sets based on fuzzy ideals in distributive lattices

“…Let µ be a fuzzy sublattice of L. Then µ is a fuzzy ideal of L, if µ(x ∨ y) = µ(x) ∧ µ(y) for all x, y ∈ L. Proposition 2.4. [20] Let µ be a fuzzy sublattice of a lattice L. Then µ is a fuzzy ideal of L if and only if x ≤ y implies that µ(x) ≥ µ(y), for all x, y ∈ L. Proposition 2.5. [21] Let µ be a fuzzy set of a lattice L. Then µ is a fuzzy ideal of L if and only if any one of the following sets of conditions is satis ed: for all x, y ∈ L, (1) µ( ) = and µ(x ∨ y) = µ(x) ∧ µ(y); (2) µ( ) = , µ(x ∨ y) ≥ µ(x) ∧ µ(y) and µ(x ∧ y) ≥ µ(x) ∨ µ(y).…”

Generalized Rough Fuzzy Ideals in Quantales

“…Late years have seen its wide applications in algebraic systems, knowledge discovery, data mining, expert systems, pattern recognition, granular computing, graph theory, machine learning, partially ordered sets, and so forth [3][4][5][6][7][8][9][10][11][12][13][14][15]. It is noted that the significant concepts in the classical theory of rough set are the lower and upper approximations obtained from equivalence relation on a universal set.…”

The Order Structure of the Soft Lattice

“…For example, Estaji et al [3] introduced the notion of  -upper and  -lower approximations of a fuzzy subset of a lattice. The study of constructing the lattice structures of soft sets are begun by Qin et al [9] and the study of lattice structure have become one of hot spots in related fields [10].…”