A Note on Rough Set Algebra and Core Regular Double Stone Algebras
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Given an approximation space ⟨ U,θ⟩, assume that E is the indexing set for the equivalence classes of θ and let R_θ denote the collection of rough sets of the form ⟨X,X⟩ as a regular double Stone algebra and what I. Dunstch referred to as a Katrinak algebra.[7],[8] We give an alternate proof from the one given in [1] of the fact that if |θ_u| > 1∀ u ∈ U then R_θ is a core regular double Stone algebra. Further let C_3 denote the 3 element chain as a core regular double Stone algebra and TP_U denote the collection of ternary partitions over the set U. In our Main Theorem we show R_θ with |θ_u| > 1 ∀ u ∈ U to be isomorphic to TP_E and C_3^E and that the three CRDSA's are complete and atomic. In our Main Corollary we show explicitly how we can embed such R_θ in TP_U, C_3^U, respectively, ϕ∘α_r:R_θ↪ TP_U↪ C_3^U, and hence identify it with its specific images. Following in the footsteps of Theorem 3. and Corollary 2.4 of [7], we show C_3^J ≅ R_θ for ⟨ U,θ⟩ the approximation space given by U = J ×{0,1}, θ = {(j0),(j1)} : j ∈ J} and every CRDSA is isomorphic to a subalgebra of a principal rough set algebra, R_θ, for some approximation space ⟨ U,θ⟩. Finally, we demonstrate this and our Main Theorem by expanding an example from [1]. We feel this could be very useful when dealing with a specific R_θ in an application. Further, we know a little more about the subalgebras of TP_U and C_3^U in general as they must exist for every E that is an indexing set for the equivalence classes of any equivalence relation θ on U satisfying |θ_u| > 1 ∀ u ∈ U.
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