$On $0$-semisimplicity of linear hulls of generators for semigroups generated by idempotents$

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
Hauptverfasser:	Bondarenko, Vitaliy M., Tertychna, Olena M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Lugansk National Taras Shevchenko University 2018
Schlagworte:	semigroup matrix representations defining relations $0$-semisimple matrix 16G 20M30
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

id	oai:ojs.admjournal.luguniv.edu.ua:article-718
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spelling	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
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
work_keys_str_mv	AT bondarenkovitaliym on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents AT tertychnaolenam on0semisimplicityoflinearhullsofgeneratorsforsemigroupsgeneratedbyidempotents
first_indexed	2025-07-17T10:32:51Z
last_indexed	2025-07-17T10:32:51Z
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On \(0\)-semisimplicity of linear hulls of generators for semigroups generated by idempotents

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