Algebraic aspects of generalized approximation spaces

  • Authors:

  • Lingyun Yang;Luoshan Xu

  • Affiliations:

  • Department of Mathematics, Yangzhou University, Yangzhou 225002, China and Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China;Department of Mathematics, Yangzhou University, Yangzhou 225002, China

  • Venue:
  • International Journal of Approximate Reasoning
  • Year:
  • 2009

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Abstract

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.