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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2012, Volume 14, Issue 2, Pages 168–173 (Mi adm91)

RESEARCH ARTICLE

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

Vitaliy M. Bondarenkoa, O. M. Tertychnab

a Institute of Mathematics, NAS, Kyiv, Ukraine
b Vadim Hetman Kyiv National Economic University, Kiev, Ukraine

Abstract: 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.

Keywords: semigroup, matrix representations, defining relations, 0-semisimple matrix.

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Bibliographic databases:
MSC: 16G, 20M30
Revised: 09.01.2013
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Citation: Vitaliy M. Bondarenko, O. M. Tertychna, “On $0$-semisimplicity of linear hulls of generators for semigroups generated by idempotents”, Algebra Discrete Math., 14:2 (2012), 168–173

Citation in format AMSBIB
\Bibitem{BonTer12} \by Vitaliy~M.~Bondarenko, O.~M.~Tertychna \paper On $0$-semisimplicity of linear hulls of generators for semigroups generated by idempotents \jour Algebra Discrete Math. \yr 2012 \vol 14 \issue 2 \pages 168--173 \mathnet{http://mi.mathnet.ru/adm91} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3099967} \zmath{https://zbmath.org/?q=an:1288.20090}