Type-2 and Type-n Rough Sets: A Hierarchical Generalization of Pawlak Approximations with some Applications"

Takaaki Fujita, Independent Researcher, Tokyo, JapanFollow
Ajoy Kanti Das, Associate Professor, Department of Mathematics, Tripura University, Agartala-799022, Tripura, IndiaFollow
Suman Das, Assistant Professor Grade II (Mathematics), Department of Education, National Institute of Technology Calicut, Kozhikode-673601, Kerala, IndiaFollow
Sankar Prasad Mondal, Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata-741249, West Bengal, IndiaFollow

ORCID

Takaaki Fujita: https://orcid.org/0000-0002-9380-386X

Ajoy Kanti Das: https://orcid.org/0000-0002-9326-1677

Suman Das: https://orcid.org/0000-0001-5682-9334

Sankar Prasad Mondal: https://orcid.org/0000-0003-4690-2598

Keywords

Rough set, Type-2 rough set, Type-n rough set

Article Type

Original Article

Abstract

Uncertainty is ubiquitous in real-world systems and has motivated the development of various generalized set-theoretic frameworks, including rough sets, fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets. In particular, rough set theory represents uncertainty by approximating a target subset of a universe through two crisp sets, namely the lower and upper approximations, which are induced by an equivalence, or indiscernibility, relation.
Motivated by the well-established hierarchy of Type-2 and, more generally, Type-(n) fuzzy sets, this paper proposes and studies analogous higher-order extensions of rough sets. First, we introduce Type-2 rough sets as a two-level parameterized framework in which each primary context is assigned a family of Pawlak approximation pairs determined by secondary parameters. We then extend this construction recursively to define Type-n rough sets. The proposed hierarchical rough-set frameworks provide a principled mechanism for modeling context-dependent granulations and multi-level uncertainty within a unified approximation-based formalism.

How to Cite

Fujita, Takaaki; Das, Ajoy Kanti; Das, Suman; and Mondal, Sankar Prasad (2026) "Type-2 and Type-n Rough Sets: A Hierarchical Generalization of Pawlak Approximations with some Applications"," Sustainable Machine Intelligence Journal: Vol. 14: Iss. 2, Article 2.
DOI: https://doi.org/10.63689/3005-3617.1087