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<p>Fuzzy sets and rough sets are valid tools for dealing with uncertain and inconsistent problems due to their complementary characteristics. Compared to fuzzy sets, axiomatic fuzzy sets are a new understanding of fuzzy concepts from a global perspective. The difference between fuzzy sets and axiomatic fuzzy sets motivates us to study the approximation of rough approximation operators under axiomatic fuzzy sets from a theoretical point of view. And, overlap functions are not necessarily associative aggregation functions differ from triangular norms (t-norms, in short). Motivated by these two factors, this paper introduces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)</EquationSource> </InlineEquation>-fuzzy rough sets (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)</EquationSource> </InlineEquation>-FRSs, in short), which are based on overlap functions. First, two operators are defined by overlap functions and residual implications induced by the overlap functions, respectively. Then, the connection between <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)</EquationSource> </InlineEquation>-FRSs and fuzzy relations are considered. Finally, the relationship between fuzzy sets and axiomatic fuzzy sets is analogous to the relationship between mappings and continuous functions. Therefore, it is interesting to study the topological properties of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)</EquationSource> </InlineEquation>-FRSs. Also, a brief discussion of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)</EquationSource> </InlineEquation>-FRSs and the established rough sets.</p>

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Axiomatic fuzzy set-based \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\)-fuzzy rough sets and their potential applications in machine learning

  • 期刊论文

  • 发表日期：

    2026年2月28日

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  • Siyu Xu,
  • Xiaodong Pan,
  • Longyu He,
  • Chao Fu,
  • Keyun Qin,
  • Yexing Dan

摘要

Fuzzy sets and rough sets are valid tools for dealing with uncertain and inconsistent problems due to their complementary characteristics. Compared to fuzzy sets, axiomatic fuzzy sets are a new understanding of fuzzy concepts from a global perspective. The difference between fuzzy sets and axiomatic fuzzy sets motivates us to study the approximation of rough approximation operators under axiomatic fuzzy sets from a theoretical point of view. And, overlap functions are not necessarily associative aggregation functions differ from triangular norms (t-norms, in short). Motivated by these two factors, this paper introduces \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\) -fuzzy rough sets ( \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\) -FRSs, in short), which are based on overlap functions. First, two operators are defined by overlap functions and residual implications induced by the overlap functions, respectively. Then, the connection between \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\) -FRSs and fuzzy relations are considered. Finally, the relationship between fuzzy sets and axiomatic fuzzy sets is analogous to the relationship between mappings and continuous functions. Therefore, it is interesting to study the topological properties of \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\) -FRSs. Also, a brief discussion of the \((\mathcal {I}_{\mathcal {O}}, \mathcal {O})\) -FRSs and the established rough sets.