Centralizer near-rings determined by unions of groups
Let \(P=\{G_{\alpha}\); \(\alpha\in A\}\) be a set of disjoint groups, \(X=\cup_{\alpha \in A}G_{\alpha}\). Let S be a monoid of functions on X such that \(\sigma\in S\) induces homomorphisms from each \(G_{\alpha}\) to some \(G_{\beta}\). Define \(M_ S(X,P)=\{f: X\to X\); \(f(G_{\alpha})\subseteq G_{\alpha}\) for all \(\alpha\in A\), \(f\sigma =\sigma f\) for all \(\sigma\in S\}\). This is a near-ring under function composition and pointwise addition, generalizing the idea of centralizer near-ring, see e.g. \textit{C. J. Maxson} and \textit{K. C. Smith} [Commun. Algebra 8, 211-230 (1980; Zbl 0425.16028)]. The case when S is a group of automorphisms is studied here. First the authors show that every near-ring is of this type, with S a monoid. If S is a group, a direct product of groups of automorphisms of the \(G_{\alpha}'s\), then \(M_ S(X,P)\) is a direct product of centralizer near-rings. Otherwise P splits into equivalence classes in which the individual groups are linked by automorphisms in S. This leads to a decomposition of \(M_ S(X,P)\) as a direct product of centralizer near-rings. There follows a series of results which characterize in terms of the triple (S,X,P) when \(M_ S(X,P)\) is a near-field, 2-semisimple, 2-primitive or simple, although in some cases there are restrictions preventing a complete solution.
direct product of centralizer near-rings
scientific article; zbMATH DE number 4010716
scientific article