Fuzzy Extension of Rough Sets Theory

Abstract

In this paper we consider many approximation spaces for rough theories, all of which are concrete realizations of an abstract structure stronger than the one introduced in [2] and defined in the following way:

$$ \mathfrak{A}: = \langle \Sigma ,\mathcal{D}(\Sigma ), \leqslant ,0,1\rangle $$

where:

• 1. 〈Σ, ∧. ∨, 0, 1〉 is a complete lattice with respect to the partial order relation ≤, bounded by the least element 0 $(\forall x \in \Sigma ,0 \leqslant x)$ and the greatest element 1 $(\forall x \in \Sigma ,x \leqslant 1)$; elements from Σ are interpreted as concepts, data, etc., and are said to be approximable elements;

• 2. $\mathcal{D}(\Sigma )$ is a sublattice of Σ whose elements are called definable;

and satisfying the following axioms:

• (Ax1): For any approximable element x ∈ Σ, there exists (at least) one element i(x) such that:

  $$ i(x) \in \mathcal{D}(\Sigma ) $$

  ((1.1a))
  $$ i(x) \leqslant x $$

  ((1.1b))
  $$ \forall \alpha \in \mathcal{D}(\Sigma ), (\alpha \leqslant x \Rightarrow \alpha \leqslant i(x)) $$

  ((1.1c))

• (Ax2): For any approximable element x ∈ Σ, there exists (at least) one element o(x) such that:

  $$ o(x) \in \mathcal{D}(\Sigma ) $$

  ((1.2a))
  $$ x \leqslant o(x) $$

  ((1.2b))
  $$ \forall \gamma \in \mathcal{D}(\Sigma ), (x \leqslant \gamma \Rightarrow o(x) \leqslant \gamma ) $$

  ((1.2c))

  i.e., i(x) [resp., o(x)] is the best approximation of the “vague” element x from the bottom [resp., top] by definable elements.