On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents
Let \(I\) be a finite set (without \(0\)) and \(J\) a subset of \(I\times I\) without diagonal elements. Let \(S(I,J)\) denotes the semigroup generated by \(e_0=0\) and \(e_i\), \(i\in I\), with the following relations: \(e_i^2=e_i\) for any \(i\in I\), \(e_ie_j=0\) for any \((i,j)\in J\). In thi...
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/718 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543265340260352 |
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| author | Bondarenko, Vitaliy M. Tertychna, Olena M. |
| author_facet | Bondarenko, Vitaliy M. Tertychna, Olena M. |
| author_sort | Bondarenko, Vitaliy M. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T10:03:23Z |
| description | Let \(I\) be a finite set (without \(0\)) and \(J\) a subset of \(I\times I\) without diagonal elements. Let \(S(I,J)\) denotes the semigroup generated by \(e_0=0\) and \(e_i\), \(i\in I\), with the following relations: \(e_i^2=e_i\) for any \(i\in I\), \(e_ie_j=0\) for any \((i,j)\in J\). In this paper we prove that, for any finite semigroup \(S=S(I,J)\) and any its matrix representation \(M\) over a field \(k\), each matrix of the form \(\sum_{i \in I}\alpha_i M(e_i)\) with \(\alpha_i\in k\) is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra. |
| first_indexed | 2025-12-02T15:32:17Z |
| format | Article |
| id | admjournalluguniveduua-article-718 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:32:17Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-7182018-04-04T10:03:23Z On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents Bondarenko, Vitaliy M. Tertychna, Olena M. semigroup, matrix representations, defining relations, \(0\)-semisimple matrix 16G, 20M30 Let \(I\) be a finite set (without \(0\)) and \(J\) a subset of \(I\times I\) without diagonal elements. Let \(S(I,J)\) denotes the semigroup generated by \(e_0=0\) and \(e_i\), \(i\in I\), with the following relations: \(e_i^2=e_i\) for any \(i\in I\), \(e_ie_j=0\) for any \((i,j)\in J\). In this paper we prove that, for any finite semigroup \(S=S(I,J)\) and any its matrix representation \(M\) over a field \(k\), each matrix of the form \(\sum_{i \in I}\alpha_i M(e_i)\) with \(\alpha_i\in k\) is similar to the direct sum of some invertible and zero matrices. We also formulate this fact in terms of elements of the semigroup algebra. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/718 Algebra and Discrete Mathematics; Vol 14, No 2 (2012) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/718/250 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | semigroup matrix representations defining relations \(0\)-semisimple matrix 16G 20M30 Bondarenko, Vitaliy M. Tertychna, Olena M. On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title | On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_full | On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_fullStr | On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_full_unstemmed | On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_short | On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| title_sort | on \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| topic | semigroup matrix representations defining relations \(0\)-semisimple matrix 16G 20M30 |
| topic_facet | semigroup matrix representations defining relations \(0\)-semisimple matrix 16G 20M30 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/718 |
| work_keys_str_mv | AT bondarenkovitaliym on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents AT tertychnaolenam on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents |