LM-valued equalities, LM-rough approximation operators and ML-graded ditopologies

We introduce a certain many-valued generalization of the concept of an $L$-valued equality called an $L^M$-valued equality. Properties of $L^M$-valued equalities are studied and a construction of an $L^M$-valued equality from a pseudo-metric is presented. $L^M$-valued equalities are applied to introduce upper and lower $L^M$-rough approximation operators, which are essentially many-valued generalizations of Z. Pawlak's rough approximation operators and of their fuzzy counterparts. We study properties of these operators and their mutual interrelations. In its turn, $L^M$-rough approximation operators are used to induce topological-type structures, called here $ML$-graded ditopologies.