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Approximate Description of Indefinable Granules Based on Classical and Three-Way Concept Lattices

Author

Listed:
  • Hongwei Wang

    (Faculty of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China)

  • Huilai Zhi

    (Faculty of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China)

  • Yinan Li

    (Big Data Institute, Central South University, Changsha 410075, China)

Abstract

Granule description is a fundamental problem in granular computing. However, how to describe indefinable granules is still an open, interesting, and important problem. The main objective of this paper is to give a preliminary solution to this problem. Before proceeding, the framework of approximate description is introduced. That is, any indefinable granule is characterized by an ordered pair of formulas, which form an interval set, where the first formula is the β -prior approximate optimal description and the second formula is the α -prior approximate optimal description. More concretely, given an indefinable granule, by exploring the description of its lower approximate granule, its β -prior approximate optimal description is obtained. Likewise, by consulting the description of its upper approximate granule, its α -prior approximate optimal description can also be derived. Following this idea, the descriptions of indefinable granules are investigated. Firstly, ∧-approximate descriptions of indefinable granules are investigated based on the classical concept lattice, and ( ∧ , ∨ ) -approximate descriptions of indefinable granules are given via object pictorial diagrams. And then, it is revealed from some examples that the classical concept lattice is no longer effective and negative attributes must be taken into consideration. Therefore, a three-way concept lattice is adopted instead of the classical concept lattice to study ( ∧ , ¬ ) -approximate descriptions and ( ∧ , ∨ , ¬ ) -approximate descriptions of indefinable granules. Finally, some discussions are presented to show the differences and similarities between our study and existing ones.

Suggested Citation

  • Hongwei Wang & Huilai Zhi & Yinan Li, 2025. "Approximate Description of Indefinable Granules Based on Classical and Three-Way Concept Lattices," Mathematics, MDPI, vol. 13(4), pages 1-27, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:672-:d:1594059
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    References listed on IDEAS

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    1. Pedrycz, Witold, 2014. "Allocation of information granularity in optimization and decision-making models: Towards building the foundations of Granular Computing," European Journal of Operational Research, Elsevier, vol. 232(1), pages 137-145.
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