Axiomatic approaches to rough approximation operators via ideal on a complete completely distributive lattice

AbstractA lattice is said to be essentially metrizable if it is an essential extension of a countable lattice. The main result of this paper is that for a completely distributive lattice the following conditions are equivalent: (1) the interval topology on L is metrizable, (2) L is essentially metrizable, (3) L has a doubly ordergenerating sublattice, (4) L is an essential extension of a countable chain.

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A Lattice-Theoretic Approach to Multigranulation Approximation Space

The Scientific World JOURNAL ◽

10.1155/2014/785932 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Xiaoli He ◽

Yanhong She

Keyword(s):

Distributive Lattice ◽

Mathematical Structure ◽

Galois Connection ◽

Theoretic Approach ◽

Approximation Space ◽

Approximation Spaces ◽

Approximation Operators ◽

Rough Approximation ◽

Definable Sets

In this paper, we mainly investigate the equivalence between multigranulation approximation space and single-granulation approximation space from the lattice-theoretic viewpoint. It is proved that multigranulation approximation space is equivalent to single-granulation approximation space if and only if the pair of multigranulation rough approximation operators(Σi=1nRi¯,Σi=1nRi_)forms an order-preserving Galois connection, if and only if the collection of lower (resp., upper) definable sets forms an (resp., union) intersection structure, if and only if the collection of multigranulation upper (lower) definable sets forms a distributive lattice whenn=2, and if and only if∀X⊆U, Σi=1nRi_(X)=∩i=1nRi_(X). The obtained results help us gain more insights into the mathematical structure of multigranulation approximation spaces.

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Aximatics for Rough Set Model Based on Vague Relations

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.282-283.283 ◽

2011 ◽

Vol 282-283 ◽

pp. 283-286

Author(s):

Hai Dong Zhang ◽

Yan Ping He

Keyword(s):

Rough Set ◽

Rough Sets ◽

General Framework ◽

Axiomatic Approach ◽

Constructive Approach ◽

Approximation Operators ◽

Rough Approximation ◽

Axiomatic Approaches ◽

Set Approximation

This paper presents a general framework for the study of rough set approximation operators in vague environment in which both constructive and axiomatic approaches are used. In constructive approach, by means of a vague relation defined by us, a new pair of vague rough approximation operators is ﬁrst deﬁned. Also some properties about the approximation operators are then discussed. In axiomatic approach, an operator-oriented characterization of vague rough sets is proposed, that is, vague rough approximation operators are defined by axioms.

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Essential completions of distributive lattices

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700002471 ◽

1985 ◽

Vol 32 (3) ◽

pp. 361-374 ◽

Cited By ~ 1

Author(s):

Gerhard Gierz ◽

Albert R. Stralka

Keyword(s):

Distributive Lattice ◽

Salient Feature ◽

Distributive Lattices ◽

Compact Topological Space ◽

Completely Distributive ◽

Completion Process ◽

The One ◽

One Point Compactification ◽

Essential Completion ◽

Completely Distributive Lattice

The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the some way as the one-point compactification of locally compact topological space does in its setting.

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