How round is a circle? Constructions of double and circular planar near-rings. (Q1403507)

Let \(V\) a ring with identity and \(S\) a commutative group faithfully acting on \(V\). Let \(T\) be the group of units of \(V\). Let \(\delta\colon T\to S\) be an \(S\)-homogeneus group homomorphism. Then, under natural suitable conditions, \((V,+,*)\) and \((V,+,\odot)\) are planar nearrings where \(w\odot v=w\delta(v)\) and \(w*v=w(v\delta(v)^{-1})\) if \(v\in T\), \(0\) otherwise. Moreover, in interesting cases, \(\odot\) and \(*\) distribute from the right over each other. But the core of the paper is given by many simple and interesting examples of such situations, with geometric interpretations and applications (mainly concerning the Euclidean plane). In conclusion more investigations are suggested and it is noted that nearrings provide a very convenient setting to algebraically describe geometric shapes.