Kernels of covered groups with operators (Q1101818)

A set \({\mathcal C}=\{C_ i\); \(i\in I\}\), of proper non-trivial subgroups of a group G is a geometric cover if \(G=\cup {\mathcal C}\) and \(C_ i\nleq C_ j\) for \(i\neq j\). A monoid S of endomorphisms of G is a semigroup of operators for (G,\({\mathcal C})\) if for each \(\sigma\in S\), \(C_ i\in {\mathcal C}\) there exists some \(C_ j\in {\mathcal C}\) such that \(\sigma (C_ i)\leq C_ j\). Then (G,\({\mathcal C},S)\) is a generalized translation space with operators. All structures are finite in this paper, which studies the near-rings \(M_ S(C,{\mathcal C})\) consisting of all mappings \(G\to G\), which respect each \(C_ i\in {\mathcal C}\) and whose actions commute with the elements of S. The authors obtain necessary and sufficient conditions for \(M_ S(G,{\mathcal C})\) to be semisimple or simple when S is a group of automorphisms. The case in which S is a cyclic semigroup is also considered and reduced to the case when the endomorphism is nilpotent. Again necessary and sufficient conditions for the simplicity of \(M_ S(G,{\mathcal C})\) are established. There are several examples and some further results along these lines. This paper generalizes the work of \textit{C. J. Maxson} and \textit{A. Oswald} [Arch. Math. 48, 353-368 (1987; Zbl 0598.16038)].