Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912311999987712 |
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| author | Briec, Walter |
| author_facet | Briec, Walter |
| contents | This article analyzes the geometric properties of an idempotent, non-associative algebraic structure that extends the Max-Times semiring. This algebraic structure is useful for studying systems of Max-Times and Max-Plus equations, employing an appropriate notion of a non-associative determinant. We consider a connected ultrametric distance and demonstrate that it implies, among other properties, an analogue of the Pythagorean relation. To this end, we introduce a suitable notion of a right angle between two vectors and investigate a trigonometric concept associated with the Chebyshev unit ball. Following this approach, we explore the potential implications of these properties in the complex plane.
We provide an algebraic definition of a line passing through two points, which corresponds to the Painlevé-Peano-Kuratowski limit of a sequence of generalized lines. We establish that this definition leads to distinctive geometric properties; in particular, two distinct parallel lines may share an infinite number of points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure Briec, Walter Rings and Algebras Classical Analysis and ODEs 06D50, 32F17 This article analyzes the geometric properties of an idempotent, non-associative algebraic structure that extends the Max-Times semiring. This algebraic structure is useful for studying systems of Max-Times and Max-Plus equations, employing an appropriate notion of a non-associative determinant. We consider a connected ultrametric distance and demonstrate that it implies, among other properties, an analogue of the Pythagorean relation. To this end, we introduce a suitable notion of a right angle between two vectors and investigate a trigonometric concept associated with the Chebyshev unit ball. Following this approach, we explore the potential implications of these properties in the complex plane. We provide an algebraic definition of a line passing through two points, which corresponds to the Painlevé-Peano-Kuratowski limit of a sequence of generalized lines. We establish that this definition leads to distinctive geometric properties; in particular, two distinct parallel lines may share an infinite number of points. |
| title | Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure |
| topic | Rings and Algebras Classical Analysis and ODEs 06D50, 32F17 |
| url | https://arxiv.org/abs/2401.07317 |