L-fuzzifying rough sets based on KM-fuzzy pseudo metrics

In this paper, we construct an L-fuzzifying lower and an L-fuzzifying upper approximation operator (L denotes a completely distributive De Morgan algebra) based on a KM-fuzzy pseudo metric and study their properties. Then we discuss some equivalent characterizations of the corresponding axiom of $KM$-fuzzy pseudo metrics via its induced L-fuzzifying rough set, respectively. Moreover, we explore the relationships between the crisp rough set induced by a KM-fuzzy pseudo metric M and the crisp rough set induced by the corresponding crisp pseudo metrics d_{a}^{M}. Besides, we investigate the relationships between the L-fuzzifying rough set induced by a famliy of crisp pseudo metrics \mathcal{D} and the L-fuzzifying rough set induced by the corresponding KM-fuzzy pseudo metric {M}^{\mathcal{D}}. Finally, we get the conclusions that \underline{R}^{\mathcal{D}}\leq {\underline{R}^{M}}^{\mathcal{D}} and {\overline{R}^{M}}^{\mathcal{D}}\leq \overline{R}^{\mathcal{D}}.