The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Abstract

Following the idea of $L$ -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on $L$ -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on $L$ -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on $L$ -fuzzy relations. Furthermore, if $L$ satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on $L$ -fuzzy generalized neighborhood systems; when $L$ -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on $L$ -fuzzy relations are complete lattice isomorphism.