Single axioms for lower fuzzy rough approximation operators determined by fuzzy implications

This paper further investigates axiomatic characterizations of lower fuzzy rough approximation operators determined by a semicontinuous fuzzy implication I. The I-lower fuzzy rough approximation operators on a general fuzzy relation can be characterized by only one axiom. The axiomatic characterizations of I-lower fuzzy rough approximation operators are studied in two approaches, which characterize abstract I-lower fuzzy rough approximation operators with ordinary fuzzy operations on fuzzy sets and product operation, respectively. Considering the properties of fuzzy implications, different axioms are proposed for I-lower fuzzy rough approximation operators, which are determined by an S-implication, an R-implication and a fuzzy implication satisfying left neutrality property and regular property, respectively. When the fuzzy relation is special, such as serial, reflexive, symmetric and so on, I-lower fuzzy rough approximation operators can also be characterized by a single axiom.