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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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | 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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oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-04-04T10:03:23Z |
| collection |
OJS |
| language |
English |
| topic |
semigroup matrix representations defining relations \(0\)-semisimple matrix 16G 20M30 |
| 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 |
| topic_facet |
semigroup matrix representations defining relations \(0\)-semisimple matrix 16G 20M30 |
| format |
Article |
| author |
Bondarenko, Vitaliy M. Tertychna, Olena M. |
| author_facet |
Bondarenko, Vitaliy M. Tertychna, Olena M. |
| author_sort |
Bondarenko, Vitaliy M. |
| title |
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_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_sort |
on \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/718 |
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AT bondarenkovitaliym on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents AT tertychnaolenam on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents |
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2025-07-17T10:32:51Z |
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2025-07-17T10:32:51Z |
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