Segments in a planar nearring Original Research Article

Suppose (N,+,·) is a finite (left) planar nearring. Define the (line) segment in N with endpoints a and b by a,b=a,b→∩b,a→=[N(b−a)+a]∩[N(a−b)+b].We prove that under many circumstances we havea,b=c,d ⇔ {a,b}={c,d},which is the same phenomenon as in the Euclidean plane.