On lower and upper intension order relations by different cover concepts

  • Authors:

  • M. H. Ghanim;H. I. Mustafa;S. Abd El Aziz

  • Affiliations:

  • Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt;Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt;Modern Academy for Engineering and Technology, Maadi, Egypt

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2011

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Abstract

In this paper, the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set. These types of ordering relations are distinct from the well-known ordering relation based on set inclusion. For these ordering relations, we consider the algebraic structures that arise in various types of covers. We show that in the case of a representative cover, the algebraic structure resulting from the lower intension inclusion is a double Stone algebra, while in the case of a reduced cover, it is a Boolean algebra. In addition, the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a Boolean algebra, and in the case of a reduced cover, the two Boolean algebraic structures from lower and upper intension inclusions are isomorphic.