## Notions from rough set theory in a generalized dependency relation context.(English)Zbl 1446.03087

Summary: In this paper, we introduce a notion of generalized dependency relation between subsets of an arbitrary (not necessarily finite) set $$\Omega$$ starting with the classical Armstrong’s rule. More specifically, we fix a given set system $$\mathcal{F}$$ on $$\Omega$$ and call any transitive binary relation $$\leftarrow$$ on the power set $$\mathcal{P}(\Omega)$$ such that = 0.5 cm
$$\bullet$$
$$B \subseteq A \in \mathcal{F}$$ implies $$B \leftarrow A$$;
$$\bullet$$
$$B \leftarrow A$$ if and only if $$b \leftarrow A \; \forall b \in B$$;
a $$[\mathcal{F}]$$-dependency relation on $$\Omega$$. We use the generality of such a notion to investigate some common analogies between rough set theory on information tables, formal context analysis, Scott’s information systems and possibility theory. More specifically, taking as inspirational models some classical notions derived by rough set theory on attribute set of an information table, we first generalize and study such notions for any $$[\mathcal{F}]$$-dependency relation. Next, we interpret such general results relatively to natural dependency relations derived by rough set theory on objects of an information table, formal context analysis, Scott’s information systems and possibility theory. Finally, we study the generation of $$[\mathcal{F}]$$-dependency relations by starting from a fixed set $$\mathcal{D}$$ of subset ordered pairs of $$\Omega$$.

### MSC:

 03E72 Theory of fuzzy sets, etc. 68T37 Reasoning under uncertainty in the context of artificial intelligence 68T30 Knowledge representation
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