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Published in:
2011 | OriginalPaper | Chapter
On Lattice–Based Fuzzy Rough Sets
Author : Anna Maria Radzikowska
Published in: 35 Years of Fuzzy Set Theory
Publisher: Springer Berlin Heidelberg
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Rough sets were originally proposed by Pawlak as a formal tool for analyzing and processing incomplete information represented in data tables. Later on, fuzzy generalizations of rough sets were introduced and investigated to be able to deal with imprecision. In this paper we present
L
-fuzzy rough sets as a further generalization of rough sets. As an underlying algebraic structure we take an
extended
residuated
lattice
, that is a residuated lattice endowed with a De Morgan negation. The signature of these structures gives algebraic counterparts of main fuzzy logical connectives. Properties of
L
-fuzzy rough sets are presented. We show that under some conditions families of all lower (resp. upper)
L
-fuzzy rough sets are complete (distributive) lattices. It is also pointed out that in some specific cases lower and upper fuzzy rough approximation operators are
L
-fuzzy topological operators of interior and closure, respectively.