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

摘要

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, Φ, ψ) where L and M are cl-monoids (in particular, L = M = [0,1]) and ψ : L → M and Φ : M → 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 roughness of rough approximation. Basic properties of such measures are studied. A realization of measures of rough approximation in terms of L-fuzzy topologies is presented.