A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Rough sets determined by tolerances
Tekijät: Jarvinen J, Radeleczki S
Kustantaja: ELSEVIER SCIENCE INC
Julkaisuvuosi: 2014
Journal: International Journal of Approximate Reasoning
Tietokannassa oleva lehden nimi: INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
Lehden akronyymi: INT J APPROX REASON
Vuosikerta: 55
Numero: 6
Aloitussivu: 1419
Lopetussivu: 1438
Sivujen määrä: 20
ISSN: 0888-613X
DOI: https://doi.org/10.1016/j.ijar.2013.12.005
Tiivistelmä
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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