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.

Object-oriented concept approach on rough granular lattice with discernibility object granular

“…Hence, we denote 𝐼𝑆 = ⟨ 𝑂, 𝐴, 𝑉, 𝑓 ⟩. [1,6] Let 𝑆 ⊆ 𝐴, then we can define the relation as a indiscernibility relation as…”

“…A relation between the classical formal context and the incomplete fuzzy formal context was introduced by Binghan [5]. Srirekha [6] analyzed the distributive lattice and its projection towards the granular concepts. Also, an equivalence relation has been examined using the homomorphism condition and was generalized its approximation space.…”

“…Further, he established the object-oriented graded fuzzy concept lattice with an example. Srirekha [17] introduced an object ranking concept in the SE-ISI context and studied its relationship between the four kinds of attribute reduction.…”

An Innovative Method for Attribute Reduction: Weighted Attribute Concepts for Probabilistic Analysis of Decision Attributes

“…Definition 2.1.1: Let us consider the formal context (O, At, R) where O and At are the finite non-empty set of objects and attributes respectively, R is a binary relation which is a subset of a cartesian product of object and attributes i.e., (R⊆O×At) where the elements are represented as (g, m)∈R, ∀g∈O, m∈At [8,11,12].…”

“…They also examined the interpretation of lattice theory and set theory. Srirekha et al [11] investigated the projection of a distributive lattice by defining the trivial ordered set and its properties. The lattice homomorphism was analyzed based on an equivalence relation, and its condition was verified.…”