The max-plus algebra of exponent matrices of tiled orders
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| Format: | Preprint |
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2017
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| _version_ | 1866917817684590592 |
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| author | Dokuchaev, Mikhailo Kirichenko, Vladimir V. Kudryavtseva, Ganna Plakhotnyk, Makar |
| author_facet | Dokuchaev, Mikhailo Kirichenko, Vladimir V. Kudryavtseva, Ganna Plakhotnyk, Makar |
| contents | An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1703_08349 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The max-plus algebra of exponent matrices of tiled orders Dokuchaev, Mikhailo Kirichenko, Vladimir V. Kudryavtseva, Ganna Plakhotnyk, Makar Rings and Algebras 15A80, 16H99, 16Z99 An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$ |
| title | The max-plus algebra of exponent matrices of tiled orders |
| topic | Rings and Algebras 15A80, 16H99, 16Z99 |
| url | https://arxiv.org/abs/1703.08349 |