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The Scientific World Journal
Volume 2014, Article ID 785932, 6 pages
http://dx.doi.org/10.1155/2014/785932
Research Article

A Lattice-Theoretic Approach to Multigranulation Approximation Space

College of Science, Xi’an Shiyou University, Xi’an 710065, China

Received 16 June 2014; Accepted 1 August 2014; Published 27 August 2014

Academic Editor: Yunqiang Yin

Copyright © 2014 Xiaoli He and Yanhong She. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 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 when , and if and only if . The obtained results help us gain more insights into the mathematical structure of multigranulation approximation spaces.