Geodesic
Congruence-simple acts over completely simple semigroups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2023) no. 4, pp. 133-142

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We prove that an act $X$ over a completely simple semigroup $S=\mathcal M (G,I,\Lambda,P)$ is congruence-simple (i.e., it has no nontrivial congruences) if and only if one of the following conditions holds: (1) $|X|=1$; (2) $|X|=2$ and $|XS|=1$; (3) $X=\{z_1,z_2\}$, where $z_1$ and $z_2$ are zeros; (4) $X\cong R/\rho$, where $R$ is a minimal right ideal of the semigroup $S$ and $\rho$ is a maximal proper congruence of the right ideal $R$, which is considered as an act over $S$. We describe these congruences. 
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     author = {I. B. Kozhukhov and K. A. Kolesnikova},
     title = {Congruence-simple acts over completely simple semigroups},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     year = {2023},
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I. B. Kozhukhov; K. A. Kolesnikova. Congruence-simple acts over completely simple semigroups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2023) no. 4, pp. 133-142. http://geodesic.mathdoc.fr/item/FPM_2023_24_4_a7/