By Rossini M.

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**Sample text**

Because fields of sets are subalgebras of the power set algebras, the Stone Representation Theorem can be rephrased in the following way: Every abstract Boolean algebra is embeddable into some power set algebra or into some complete field of sets. Theorem 2 The family of definable sets of every equivalence approximation space is a complete and atomic field of sets, and so it is a complete and atomic Boolean algebra. Proof. Let (U, R) be an equivalence approximation space. The family of definable sets Def(GR(U)) by its definition (definable sets are unions of granules) is closed for arbitrary unions of sets.

We define Pt :℘(U ) →℘(U ), P t :℘(U ) →℘(U ) and BN P :℘(U ) →℘(U ) such that, for any X⊆U: Pt ( X ) := {t( x) : t( x) ⊆ X }, P t ( X ) := {t( x) : t( x) ∩ X ≠ ∅}, BN P ( X ) := P t ( X ) \ Pt ( X ). We call Pt(X) and Pt(X) as P-lower and Pupper approximations of X respectively, while BN P ( X ) is called P-boundary of X. Notice that a set is definable if it is a union of knowledge granules. From Proposition 4 it follows that if t is transitive, t hen Pt ( X ) = {t( x) : t( x) ⊆ X } = t∗ ( X ) a nd P t ( X ) = {t( x) : t( x) ∩ X ≠ ∅} = t∗ ( X ) .

In such representation theorems, the concrete structures have to be derivated from information systems. ,: Banerjee & Chakraborty, 1996; Cattaneo, 1997, 1998; Comer, 1991, 1995; Demri & Orłowska, 1998; Järvinen, 1999, 2006; Orłowska, 1993b; Pagliani, 1996, 1998a, 1998b; Vakarelov, 1989, 1991, 1998; Wasilewska, 1997; Wasilewska & Banerjee, 1995; Wasilewska & Vigneron, 1995). The notion of informational representability was proposed in Orłowska (1993b). In this section, we recall informational representation theorems for Boolean and Stone algebras.

### 2D-Discontinuity Detection from Scattered Data by Rossini M.

by Christopher

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