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2016 | OriginalPaper | Buchkapitel

Covering Rough Sets and Formal Topology – A Uniform Approach Through Intensional and Extensional Constructors

verfasst von : Piero Pagliani

Erschienen in: Transactions on Rough Sets XX

Verlag: Springer Berlin Heidelberg

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Abstract

Approximation operations induced by coverings are reinterpreted through a set of four “constructors” defined by simple logical formulas. The very logical definitions of the constructors make it possible to readily understand the properties of such operators and their meanings.

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Vorheriges Kapitel Algebraic Semantics of Proto-Transitive Rough Sets

Nächstes Kapitel Multiple-Source Approximation Systems, Evolving Information Systems and Corresponding Logics: A Study in Rough Set Theory

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The members of U will be usually denoted by g after the German term Gegenstand which means an object before interpretation, while M is after Merkmal, which means “property”. In Formal Topology, M is thought of as a set of abstract neighborhoods. This interpretation will be used later on in the paper.

R(A) and \(R^\smile (B)\) are also called the the left Peirce product of R and A, and, respectively, the right Peirce product of R (left Peirce product of \(R^\smile \)) and B.

Often a lower adjoint is called “left adjoint” and an upper adjoint “right adjoint”. We avoid the terms “right” and “left” because they could make confusion with the position of the arguments of the operations of binary relations. For the general notion of adjoint functors see for instance [3]. For Galois connections induced by binary relations a classic reference is [19]. For the present use in Rough Set Theory see [23] or [25].

It is worth noticing that there are constructive logics between Intuitionistic and Classical logics such that the opposite of the above implication holds if the premise is a negated formula (see [14]).

In these works \({\mathcal N}(\mathbf{P})\) is denoted as \({\mathcal N}(U)\) and instead of \({\mathcal Z}\) the entire powerset \(\wp (U)\) is considered. The present is a slight generalization.

For the notions of a “neighborhood” and a “pretopology”, see [5, 16].

If R is not serial and \(R(g)=\emptyset \), then \(\emptyset \) does not belong to \({\mathcal N}_g\). Otherwise stated, \(\emptyset \) is different from \(\{\emptyset \}\). 0 does not hold if there exists \(g\in G\) such that \(\langle g,\emptyset \rangle \in R\).

In [22] \({\mathcal N}_*(U)\) is denoted as \({\mathcal N}_{F(R)}(U)\), and \({\mathcal N}^*_x\) as \({\mathcal N}^R_x\).

A wider reference about covering-based approximation operators and the scientific literature about the topic can be found in Sect. 5 of [33].

The original definition of \((uC)_7\) is \((lC)_1(X)\cup (\bigcup \{n(x):x\in X\cap -(lC)_1(X)\})\).

Actually, from Facts 3.(iii), Corollary 1 and Lemma 11.(5) and (6) one trivially derives that in any SRS P with R a preorder: \(\mathbf{S}_{\langle i\rangle }(\mathbf{P})=\mathbf{S}_{[e]\langle i\rangle }(\mathbf{P})=\mathbf{S}_{[e]}(\mathbf{P})= \mathbf{S}_{\langle i\rangle [e]}(\mathbf{P})\); \(\mathbf{S}_{\langle e\rangle }(\mathbf{P})=\mathbf{S}_{[i]\langle e\rangle }(\mathbf{P})=\mathbf{S}_{[i]}(\mathbf{P})= \mathbf{S}_{\langle e\rangle [i]}(\mathbf{P})\).

In general, from Facts 3, if R is a preorder then the set of fixpoints of the constructors \(\langle \cdot \rangle \) and \([\cdot ]\) coincides with the sets of fixpoint of their derived operators \(\langle \cdot \rangle [\cdot ]\) and \([\cdot ]\langle \cdot \rangle \) (where the directions, intension or intension, alternate). Since the sets of fixpoints of the derived operators form distributive lattices, the same happens for the constructors.

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Titel: Covering Rough Sets and Formal Topology – A Uniform Approach Through Intensional and Extensional Constructors
verfasst von: Piero Pagliani
Verlag: Springer Berlin Heidelberg
Buch: Transactions on Rough Sets XX
Print ISBN: 978-3-662-53610-0

Electronic ISBN: 978-3-662-53611-7

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-662-53611-7_4

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