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A Geometry of ApproximationBasic Logico-Algebraic Structures

A Geometry of Approximation: Basic Logico-Algebraic Structures Chapter 6 Basic Logico-Algebraic Structures In order to appreciate the polymorphism of Rough Set Systems the essential ideas and notions of the logico-algebraic structures we shall deal with will be introduced. In Mathematical toolkit 16.3 the reader will find the basic princi- ples of bounded lattices. Moreover, all the algebraic structures needed are not only bounded lattices, but finite distributive bounded lattices, that is, finite structures D = A, ∨, ∧, 0, 1 such that the following distributive properties hold: a ∧ (b ∨ c)=(a ∧ b) ∨ (a ∧ c) (6.0.1) a ∨ (b ∧ c)=(a ∨ b) ∧ (a ∨ c) (6.0.2) Remarks. The restriction to finite structures is not a limitation when we have to deal with practically given Rough Set Systems. This consideration lies behind our choice to focus on finite algebras. However, in general the results which will follow do not require finiteness. Anyway, we shall indicate when the finiteness assumption is essential to prove a result. Among bounded distributive lattices Heyting algebras play a pro- minent role. 193 194 6 Basic Logico-Algebraic Structures 6.1 Heyting Algebras Heyting algebras aim at modeling Intuitionistic Logic. In contrast to Classical Logic which maintains that given a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationBasic Logico-Algebraic Structures

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
A Geometry of Approximation — Jan 1, 2008
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Publisher
Springer Netherlands
Copyright
© Springer Netherlands 2008
ISBN
978-1-4020-8621-2
Pages
193 –210
DOI
10.1007/978-1-4020-8622-9_6
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 6 Basic Logico-Algebraic Structures In order to appreciate the polymorphism of Rough Set Systems the essential ideas and notions of the logico-algebraic structures we shall deal with will be introduced. In Mathematical toolkit 16.3 the reader will find the basic princi- ples of bounded lattices. Moreover, all the algebraic structures needed are not only bounded lattices, but finite distributive bounded lattices, that is, finite structures D = A, ∨, ∧, 0, 1 such that the following distributive properties hold: a ∧ (b ∨ c)=(a ∧ b) ∨ (a ∧ c) (6.0.1) a ∨ (b ∧ c)=(a ∨ b) ∧ (a ∨ c) (6.0.2) Remarks. The restriction to finite structures is not a limitation when we have to deal with practically given Rough Set Systems. This consideration lies behind our choice to focus on finite algebras. However, in general the results which will follow do not require finiteness. Anyway, we shall indicate when the finiteness assumption is essential to prove a result. Among bounded distributive lattices Heyting algebras play a pro- minent role. 193 194 6 Basic Logico-Algebraic Structures 6.1 Heyting Algebras Heyting algebras aim at modeling Intuitionistic Logic. In contrast to Classical Logic which maintains that given a

Published: Jan 1, 2008

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