Geodesic
On $0$-semisimplicity of linear hulls of generators for semigroups generated by idempotents
Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 168-173.

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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
Mots-clés : 0-semisimple matrix. 
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     title = {On $0$-semisimplicity of linear hulls of  generators for semigroups generated by idempotents},
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Vitaliy M. Bondarenko; O. M. Tertychna. On $0$-semisimplicity of linear hulls of  generators for semigroups generated by idempotents. Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 168-173. http://geodesic.mathdoc.fr/item/ADM_2012_14_2_a2/

[1] V. M. Bondarenko, O. M. Tertychna, “On infiniteness of type of infinite semigroups generated by idempotents with partial null multiplication”, Trans. Inst. of Math. of NAS of Ukraine, 3 (2006), 23–44 (in Russian) | MR | Zbl

[2] V. M. Bondarenko, O. M. Tertychna, “On tame semigroups generated by idempotents with partial null multiplication”, Algebra Discrete Math., 2008, no. 4, 15–22 | MR | Zbl

[3] O. M. Tertychna, Matrix representations of semigroups generated by idempotents with partial null multiplication, Thesis for a candidate's degree by speciality 01.01.06 – algebra and number theory, Kyiv National Taras Shevchenko University, 2009, 167 pp. (In Ukrainian)