Triangular norms on finite lattices
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911737015435264 |
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| author | He, Peng Wang, Xue-Ping |
| author_facet | He, Peng Wang, Xue-Ping |
| contents | This article intends to characterize triangular norms on a finite lattice. We first give a method for generating a triangular norm on an atomistic lattice by the values of atoms. Then we prove that every triangular norm on a non-Boolean atomistic lattice is not left-continuous and $T_M$ is the uniquely left-continuous triangular norm on an atomistic Boolean lattice. Furthermore, we show that each atomistic Boolean lattice can be represented by a family of triangular norms on an atomistic lattice with the same number of atoms. Finally, we construct a triangular norm on a finite lattice by restricting a triangular norm on an extended atomistic lattice of the finite lattice to the finite lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triangular norms on finite lattices He, Peng Wang, Xue-Ping Combinatorics This article intends to characterize triangular norms on a finite lattice. We first give a method for generating a triangular norm on an atomistic lattice by the values of atoms. Then we prove that every triangular norm on a non-Boolean atomistic lattice is not left-continuous and $T_M$ is the uniquely left-continuous triangular norm on an atomistic Boolean lattice. Furthermore, we show that each atomistic Boolean lattice can be represented by a family of triangular norms on an atomistic lattice with the same number of atoms. Finally, we construct a triangular norm on a finite lattice by restricting a triangular norm on an extended atomistic lattice of the finite lattice to the finite lattice. |
| title | Triangular norms on finite lattices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.04944 |