Rough Sets Determined by Quasiorders
Jouni J\"arvinen, S\'andor Radeleczki, Laura Veres
TL;DR
This paper investigates the structure of rough sets determined by quasiorders, proving they form a complete, distributive lattice with various complementation operations and characterizing when it is a Stone lattice.
Contribution
It generalizes previous results by characterizing the lattice structure of rough sets under quasiorders, extending known cases where the relation is an equivalence.
Findings
The lattice of rough sets is complete and completely distributive.
Three types of complementation operations are defined on this lattice.
Conditions for the lattice to be a Stone lattice are characterized.
Abstract
In this paper, the ordered set of rough sets determined by a quasiorder relation is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykala and J. A. Pomykala (1988) and M. Gehrke and E. Walker (1992) in case is an equivalence.
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