On the Class of Subsets of Residuated lattice which induces a Congruence Relation

In this manuscript,  we study the class of special subsets connected with a  subset in a residuated lattice and investigate some related properties. We describe the union of elements of this class. Using the intersection of all special subsets connected with a subset, we give a necessary and sufficient condition for a subset to be a filter.  Finally, by defining some operations, we endow this class with a residuated lattice structure and prove that it is isomorphic to the set of all congruence classes with respect to a filter.

Keywords:

Residuated lattice , Filter , Complement-closed , Congruence Relation