Rough sets by reflexive relations and their algebras

Abstract

We consider various types of algebras defined on the completion DM(RS) of the partially ordered set of rough sets induced by a reflexive relation. We restrict ourselves to the cases in which the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe reflexive relations for which DM(RS) has the same properties as in the case of an equivalence relation: it forms a completely distributive and spatial regular double Stone algebra.

Figures (4)

• Figure 1: $\mathrm{RS}$ is isomorphic to $\mathbf{2} \times \mathbf{3}$.
• Figure 2: The Hasse diagrams of $\wp(U)^\blacktriangle$ and $\mathrm{RS} = \mathrm{DM(RS)}$.
• Figure 3: The Hasse diagrams of $\wp(U)^\blacktriangle$ and $\mathrm{RS} = \mathrm{DM(RS)}$.
• Figure 4: The Hasse diagram of $\mathrm{RS}$.

Theorems & Definitions (46)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Example 3.1
• Lemma 3.2
• proof
• Proposition 3.3