Journals Information
Mathematics and Statistics Vol. 13(5), pp. 329 - 336
DOI: 10.13189/ms.2025.130508
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A New Approach towards Rough Lattice Using Rough Relation with a Condition
B. Srirekha 1, Shakeela Sathish 2,*, B. Muthu Deepika 1, S. Sangeetha 1
1 Department of Mathematics, SRM Institute of Science and Technology-Ramapuram Campus, India
2 Department of Computing Technologies, School of Computing, SRM Institute of Science and Technology, Karrankulathur, Tamilnadu, India
ABSTRACT
The concept of rough relations plays a significant role in rough set theory, introduced by Zdzis艂aw Pawlak in 1981. This study employs the rough membership function as a key tool to represent and analyze rough relations over a universe of discourse. A specific condition is applied to these relations and examined through ordered pairs, allowing systematic evaluation of approximations and their behavior. The work investigates the algebraic properties of rough relations within a lattice-theoretic framework, particularly on distributive lattices equipped with a complementary operation. This structure provides a clear interpretation of the approximation process and the relationship between elements. The existence and behavior of upper approximations are illustrated through examples, with emphasis on how granularity refines approximation boundaries and improves the classification of indiscernible objects. Key theoretical results demonstrate that in a rough lattice, if two elements are ordered, then their meet and join operations preserve non-negativity under the rough membership function, reflecting algebraic consistency. This property extends to complementary elements, ensuring that their mutual relationships also maintain non-negative rough membership values. Additionally, when comparing two rough lattices over the same universe, inclusion between their approximation sets leads to a monotonic increase in rough membership values, indicating order preservation. In distributive lattices, specific inequalities involving elements and their complements reinforce internal consistency between the algebraic and rough structures. Within the upper approximation space, element ordering is symmetrically reflected through complementation, supporting the duality principle. The study also confirms the transitivity of the rough membership function: if one element is roughly related to a second, and that second to a third, then the first is also roughly related to the third鈥攈ighlighting logical coherence. Overall, these findings advance the theoretical understanding of rough lattice structures and underscore their importance in modeling uncertain and incomplete information, with applications in logic programming, data mining, and formal concept analysis.
KEYWORDS
Indiscernibility Relation, Rough Membership Function, Rough Lattice, Distributive Lattice, Rough Relation
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] B. Srirekha , Shakeela Sathish , B. Muthu Deepika , S. Sangeetha , "A New Approach towards Rough Lattice Using Rough Relation with a Condition," Mathematics and Statistics, Vol. 13, No. 5, pp. 329 - 336, 2025. DOI: 10.13189/ms.2025.130508.
(b). APA Format:
B. Srirekha , Shakeela Sathish , B. Muthu Deepika , S. Sangeetha (2025). A New Approach towards Rough Lattice Using Rough Relation with a Condition. Mathematics and Statistics, 13(5), 329 - 336. DOI: 10.13189/ms.2025.130508.