A1 Refereed original research article in a scientific journal
Rough sets determined by tolerances
Authors: Jarvinen J, Radeleczki S
Publisher: ELSEVIER SCIENCE INC
Publication year: 2014
Journal: International Journal of Approximate Reasoning
Journal name in source: INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
Journal acronym: INT J APPROX REASON
Volume: 55
Issue: 6
First page : 1419
Last page: 1438
Number of pages: 20
ISSN: 0888-613X
DOI: https://doi.org/10.1016/j.ijar.2013.12.005(external)
Abstract
We show that for any tolerance R on U, the ordered sets of lower and upper rough approximations determined by R form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if R is induced by an irredundant covering of U, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set RS of rough sets determined by a tolerance R on U is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that R is a tolerance induced by an irredundant covering of U if and only if RS is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on RS. We present necessary and sufficient conditions which guarantee that for a tolerance R on U, the ordered set RSx is a lattice for all X subset of U, where R-x denotes the restriction of R to the set X and RSx is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are DedekindMacNeille completions of RS. (C) 2013 Elsevier Inc. All rights reserved.
We show that for any tolerance R on U, the ordered sets of lower and upper rough approximations determined by R form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if R is induced by an irredundant covering of U, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set RS of rough sets determined by a tolerance R on U is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that R is a tolerance induced by an irredundant covering of U if and only if RS is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on RS. We present necessary and sufficient conditions which guarantee that for a tolerance R on U, the ordered set RSx is a lattice for all X subset of U, where R-x denotes the restriction of R to the set X and RSx is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are DedekindMacNeille completions of RS. (C) 2013 Elsevier Inc. All rights reserved.
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