Metric-based [formula omitted]-fuzzy rough sets: Approximation operators and definable sets

Abstract

Binary relations, coverings and neighborhood systems/operators are useful tools to study rough set theory. In this paper, we use the notion of ⊕-hemimetric, a weak version of the standard metric in topology and analysis, as the basic structure to study L-fuzzy rough set theory, where L is a complete residuated lattice. We define a pair of L-fuzzy upper and lower approximation operators and then investigate their properties and relations. It is shown that both operators are monotone with respect to the L-fuzzy order of fuzzy inclusion relation between L-fuzzy subsets. The L-fuzzy upper approximation operator has more nice properties than the lower one, and if L is regular and the hemimetric is symmetric, then they are dual to each other. We then study the upper and lower definable sets in this model. The family of upper definable sets forms an Alexandrov stratified L-topology while that of lower definable ones does not necessarily. If L is regular (even if the hemimetric is not symmetric), the upper definability coincides with the lower definability. We finally present an application of metric-based L-fuzzy set theory to fuzzy clustering for weighted graphs.