Functional equations and topological \(N\)-groups (Q1390821)

There is up to isomorphism exactly one topological nearring \(N\) with additive group the euclidean plane, which has an identity and is not zero-symmetric [the author, Acta Sci. Math. 62, No. 1-2, 115-125 (1996; Zbl 0862.16029)]. The aim is to determine all the topological \(N\)-groups \(G\), where \(G=\mathbb{R}^n\) the \(n\)-dimensional real vector space. A topological \(N\)-group is a group \(G\) with a continuous action \(N\times G\to G\) such that \((a+b)x=ax+bx\) and \((ab)x=a(bx)\) for all \(a,b\in N\), \(x\in G\). To achieve his aim the author first solves a rather complicated functional equation involving two functions on \(G\) and various scalars. He also gives examples for solutions. This is then applied to the main problem. Finally, continuous \(N\)-group homomorphisms for the \(N\)-groups under consideration are discussed.