Binary Relation, Basis Algebra, Approximation Operator Form and Its Property in L-Fuzzy Rough Sets

Abstract

The rough set theory usually is used in data-mining. At this time, we will suppose the approximation operator to satisfy some wonderful properties. It need us to think about the basis algebra, the binary relation and the approximation operator’s form in the rough set model. L-fuzzy rough approximation operator is a general fuzzy rough approximation operator. Comparing with the others rough approximation operator, it is more clearly for the L-fuzzy rough approximation operator that not only the binary relation and the basis algebra, but also the form of approximation operator influence its properties. To inhibit the relations between binary relation on the universe, basis algebra, the approximation operators’ properties and its form in the L-fuzzy rough set, we introduce some basis algebra (residuated lattice, MTL algebra and IMTL algebra) and some binary relations(the reflexive, symmetric, transitive, L-similar, L-indistinguish-able fuzzy binary relation) into three types L-fuzzy rough approximation operators and proof some L-fuzzy rough approximation operators’ properties as its properties. As usual, we discuss the approxima-tion operator in the constructive approach and axiomatic approach. This paper focus on the basis algebra and the form of the approximation operat-ors. As the preliminaries, we use some spaces to dis-cuss the properties of the non-classic logic algebra.