Abstract
It is known ([15]) that the propositional aspect of rough set theory is adequately captured by the modal system S5. A Kripke model gives the approximation space (A,R) in which well formed formulas are interpreted as rough sets. Banejee and Chakraborty ([1]) introduced a new binary connective in S5, the intended interpretation of which was the notion of rough equality, defined by Pawlak in 1982. They called the resulting Lindenbaum-Tarski like algebra a rough algebra. We show here that their rough algebra is a particular case of a quasi-Boolean algebra (as introduced in [4]). It also leads to a definition of the new classes of algebras, called topological quasi-Boolean algebras2 and topological rough algebras. We introduce, following Rasiowa and Białynicki-Birula’s representation theorem for the quasi-Boolean algebras ([4], [20]), a notion of quasi field of sets and generalize it to a new notion of a topological quasi field of sets. We use it to give the representation theorems for the topological quasi-Boolean algebras and topological rough algebras, and hence to provide a mathematical characterization of the rough algebra.
This paper was initiated in November 1993 during the author’s discussions with M. Banejee who also visited the Institute of Computer Science, Warsaw University of Technology, Warsaw, Poland. The research was supported by a Fulbright grant no. 93-68818 (1993-1994).
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Wasilewska, A. (1997). Topological Rough Algebras. In: Rough Sets and Data Mining. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1461-5_21
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DOI: https://doi.org/10.1007/978-1-4613-1461-5_21
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