Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets
In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-c...
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MDPI AG
2024-12-01
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| author | Pablo Hernández-Varela Francisco Javier Talavera Susana Cubillo Carmen Torres-Blanc Jorge Elorza |
| author_facet | Pablo Hernández-Varela Francisco Javier Talavera Susana Cubillo Carmen Torres-Blanc Jorge Elorza |
| author_sort | Pablo Hernández-Varela |
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| description | In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-conorms are of particular significance, as they are applied to model intersection and union, respectively. Furthermore, the existence of a range of these operators allows for the selection of the t-norm or t-conorm that offers the optimal performance, in accordance with the specific context of the system. In this paper, we obtain new t-norms and t-conorms on some important subfamilies of the set of functions from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. The structure of these families provides a more solid algebraic foundation for the applications. In particular, we define these new operators on the subsets of the functions that are convex, normal, and normal and convex, as well as the functions taking only the values 0 or 1 and the subset of functions whose support is a finite union of closed intervals. These t-norms and t-conorms are generalized to the type-2 fuzzy set framework. |
| format | Article |
| id | doaj-art-0d9c906d0e6d4f4297d1e6abe1d55cfc |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-0d9c906d0e6d4f4297d1e6abe1d55cfc2025-01-24T13:22:11ZengMDPI AGAxioms2075-16802024-12-011412710.3390/axioms14010027Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy SetsPablo Hernández-Varela0Francisco Javier Talavera1Susana Cubillo2Carmen Torres-Blanc3Jorge Elorza4Departamento de Ciencias Exactas, Facultad de Ingeniería, Arquitectura y Diseño, Universidad San Sebastián, Bellavista 7, Santiago 8420524, ChileDepartamento de Física y Matemática Aplicada, Facultad de Ciencias, Universidad de Navarra, C. Irunlarrea 1, 31008 Pamplona, SpainDepartamento de Matemática Aplicada, Universidad Politécnica de Madrid, Boadilla del Monte, 28660 Madrid, SpainDepartamento de Matemática Aplicada, Universidad Politécnica de Madrid, Boadilla del Monte, 28660 Madrid, SpainDepartamento de Física y Matemática Aplicada, Facultad de Ciencias, Universidad de Navarra, C. Irunlarrea 1, 31008 Pamplona, SpainIn certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-conorms are of particular significance, as they are applied to model intersection and union, respectively. Furthermore, the existence of a range of these operators allows for the selection of the t-norm or t-conorm that offers the optimal performance, in accordance with the specific context of the system. In this paper, we obtain new t-norms and t-conorms on some important subfamilies of the set of functions from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. The structure of these families provides a more solid algebraic foundation for the applications. In particular, we define these new operators on the subsets of the functions that are convex, normal, and normal and convex, as well as the functions taking only the values 0 or 1 and the subset of functions whose support is a finite union of closed intervals. These t-norms and t-conorms are generalized to the type-2 fuzzy set framework.https://www.mdpi.com/2075-1680/14/1/27function from [0,1] to [0,1]normal functionconvex functiontype-2 fuzzy setinterval type-2 fuzzy sett-norm |
| spellingShingle | Pablo Hernández-Varela Francisco Javier Talavera Susana Cubillo Carmen Torres-Blanc Jorge Elorza Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets Axioms function from [0,1] to [0,1] normal function convex function type-2 fuzzy set interval type-2 fuzzy set t-norm |
| title | Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets |
| title_full | Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets |
| title_fullStr | Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets |
| title_full_unstemmed | Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets |
| title_short | Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets |
| title_sort | definition of triangular norms and triangular conorms on subfamilies of type 2 fuzzy sets |
| topic | function from [0,1] to [0,1] normal function convex function type-2 fuzzy set interval type-2 fuzzy set t-norm |
| url | https://www.mdpi.com/2075-1680/14/1/27 |
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