A New Methodology on Rough Lattice Using Granular Concepts

Abstract

Rough set theory has a vital role in the mathematical field of knowledge representation problems. Hence, a Rough algebraic structure is defined by Pawlak. Mathematics and Computer Science have many applications in the field of Lattice. The principle of the ordered set has been analyzed in logic programming for crypto-protocols. Iwinski extended an approach towards the lattice set with the rough set theory whereas an algebraic structure based on a rough lattice depends on indiscernibility relation which was established by Chakraborty. Granular means piecewise knowledge, grouping with similar elements. The universe set was partitioned by an indiscernibility relation to form a Granular. This structure was framed to describe the Rough set theory and to study its corresponding Rough approximation space. Analysis of the reduction of granular from the information table is based on object-oriented. An ordered pair of distributive lattices emphasize the congruence class to define its projection. This projection of distributive lattice is analyzed by a lemma defining that the largest and the smallest elements are trivial ordered sets of an index. A Rough approximation space was examined to incorporate with the upper approximation and analysis with various possibilities. The Cartesian product of the distributive lattice was investigated. A Lattice homomorphism was examined with an equivalence relation and its conditions. Hence the approximation space exists in its union and intersection in the upper approximation. The lower approximation in different subsets of the distributive lattice was studied. The generalized lower and upper approximations were established to verify some of the results and their properties.