A Note on Rough Set Algebra and Core Regular Double Stone Algebras

Abstract

Rough Set Theory (RST), first introduced by Pawlak in 1982, is an approach for dealing with information systems where knowledge is uncertain or incomplete.\cite{Pawlak} It is of fundamental importance in many subfields of artificial intelligence and cognitive science.\cite{RSTppf} Given a universe $U$ with an equivalence relation $\theta$, the pair $\langle U,\theta\rangle$ is referred to as an information system and we denote its collection of rough sets $R_\theta$. In our main Theorem we show $R_\theta$ with $|\theta_u| > 1\ \forall\ u \in U$ to be isomorphic to core regular double Stone algebras, CRDSA, that are complete and atomic, and that the crisp, or definable, sets form a complete atomistic Boolean algebra. These guarantees of infimum/supremeum for arbitrary subsets and formulations in terms of fundamental elements are likely useful if dealing with equivalence relations with an infinite number of partitions, such as projective Hilbert spaces. We further derive that every CRDSA is isomorphic to a subalgebra of a principal rough set algebra, $R_\theta$, for some approximation space $\langle U,\theta \rangle$. In our main Corollary we show explicitly how to embed $R_\theta$ into the CRDSA and first demonstrate by extending the culminating finite example of \cite{RCRDSA}. As our capstone, we consider the projective Hilbert space of complex numbers, $\mathbb{C}$ and show, among other things, the power set of the set of pure states is a complete, atomistic Boolean algebra. In closing, we suggest other Quantum relevant applications that may be useful, such as Hilbert spaces of operators