Congruence-Simple Acts Over Completely Simple Semigroups

We prove that an act X over a completely simple semigroup S=MG,I,Λ,P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\mathcal{M}\left(G,I,\Lambda ,P\right)$$\end{document} 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 = {z1, z2}, where z1 and z2 are zeros; (4) X ≅ R/ρ, where R is a minimal right ideal of the semigroup S and ρ is a maximal proper congruence of the right ideal R, which is considered as an act over S. We describe these congruences.