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A Geometry of ApproximationFrames (Part II)

A Geometry of Approximation: Frames (Part II) Chapter 10 10.1 Frame – Rough Set Systems and Chain-Based Lattices The equivalence between semi-simple Nelson algebras and three-valued L  ukasiewicz algebras was stated in [Monteiro, 1967]. The transformation stated in Example 6.5.1 at point 5 is supported by the following results from [Epstein & Horn, 1974a]): Lemma 10.1.1. If A = A, ∨, ∧, ¬,0= e ≤ ... ≤ e =1 is a 0 n−1 pseudo-complemented P -lattice, then A has a chain base f ≤ f ≤ 0 0 1 ... ≤ f s.t. f is the smallest dense element of A. n−1 1 Lemma 10.1.2. If A = A, ∨, ∧, ¬, =⇒,0= e ≤ ... ≤ e =1 is 0 n−1 a P -lattice and A, ∨, ∧, ¬, =⇒, 0, 1 is a Heyting algebra, then there exists a chain base 0= f ≤ f ≤ ... ≤ f =1 s.t. A, ∨, ∧, ¬, →, 0 1 n−1 0= f ≤ f ≤ ... ≤ f =1 is a P -lattice. 0 1 n−1 1 In order to get the required chain base one can refer to Lemma 10.1.1, taking as f the first dense element of A, and inductively as f http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationFrames (Part II)

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
Springer Journals — Jan 1, 2008
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79 pages

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Publisher
Springer Netherlands
Copyright
© Springer Netherlands 2008
ISBN
978-1-4020-8621-2
Pages
281 –359
DOI
10.1007/978-1-4020-8622-9_10
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 10 10.1 Frame – Rough Set Systems and Chain-Based Lattices The equivalence between semi-simple Nelson algebras and three-valued L  ukasiewicz algebras was stated in [Monteiro, 1967]. The transformation stated in Example 6.5.1 at point 5 is supported by the following results from [Epstein & Horn, 1974a]): Lemma 10.1.1. If A = A, ∨, ∧, ¬,0= e ≤ ... ≤ e =1 is a 0 n−1 pseudo-complemented P -lattice, then A has a chain base f ≤ f ≤ 0 0 1 ... ≤ f s.t. f is the smallest dense element of A. n−1 1 Lemma 10.1.2. If A = A, ∨, ∧, ¬, =⇒,0= e ≤ ... ≤ e =1 is 0 n−1 a P -lattice and A, ∨, ∧, ¬, =⇒, 0, 1 is a Heyting algebra, then there exists a chain base 0= f ≤ f ≤ ... ≤ f =1 s.t. A, ∨, ∧, ¬, →, 0 1 n−1 0= f ≤ f ≤ ... ≤ f =1 is a P -lattice. 0 1 n−1 1 In order to get the required chain base one can refer to Lemma 10.1.1, taking as f the first dense element of A, and inductively as f

Published: Jan 1, 2008

Keywords: Boolean Algebra; Atomic Formula; Kripke Model; Heyting Algebra; Stone Algebra

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