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Roughness in quantales

Authors
Journal
Information Sciences
0020-0255
Publisher
Elsevier
Publication Date
Volume
220
Identifiers
DOI: 10.1016/j.ins.2012.07.042
Keywords
  • Rough Set
  • Quantale
  • Rough Ideal
  • Rough Prime Ideal
  • Rough Prime Radical Theorem
  • Quantale Homomorphism
Disciplines
  • Logic
  • Mathematics

Abstract

Abstract Rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems. In this paper, rough approximations are introduced into quantales, a kind of partially ordered algebraic structure with an associative binary multiplication. The notions of (upper, lower) rough (prime, semi-prime, primary) ideals of quantales are proposed and verified to be the extended notions of usual (prime, semi-prime, primary) ideals of quantales respectively. Global order properties of all lower (upper) rough ideals of a quantale are also investigated. The rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales. Some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales.

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