TOPOLOGICAL SIMILARITY OF L-RELATIONS

$L$-fuzzy rough sets are extensions of the classical rough sets by relaxing the equivalence relations to $L$-relations. The topological structures induced by $L$-fuzzy rough sets have opened up the way for applications of topological facts and methods in granular computing. In this paper, we firstly prove that each arbitrary $L$-relation can generate an Alexandrov $L$-topology. Based on this fact, we introduce the topological similarity of $L$-relations, denote it by T-similarity, and we give intuitive characterization of T-similarity. Then we introduce the variations of a given $L$-relation and investigate the relationship among them. Moreover, we prove that each $L$-relation is uniquely topological similar to an $L$-preorder. Finally, we investigate the related algebraic structures of different sets of $L$-relations on the universe.