On Tolerance-Based Rough Set Operators and Their Covering Generalizations

We investigate approximation operators determined by arbitrary families of granules which cover the space of objects and without posing any conditions on the nature of granules. We recall from literature some rough sets approximation operators: classical equivalential rough sets operators by Z. Pawlak and their generalizations for tolerance relations: tolerance approximation operators by A. Skowron and J. Stepaniuk and tolerance-granular approximation operators suggested by Z. Pawlak. Then we proposed granular generalization of tolerance approaches to tolerance-based rough sets by means of arbitrary covering of the object space proposed by Y.Y. Yao Then we propose a version of tolerance-granular operators suggested by Z. Pawlak with biting procedure proposed in our paper. We prove basic properties of recalled tolerance rough sets operators. We discuss which pair of operators possesses the property of mutual definability. Then we show generalizations of tolerance rough sets approximation operators which possess mutual definability property by granular covering operators from.