Completeness for concrete near-rings

In this paper a completeness criterion for near-rings over a finite group is derived using techniques from clone theory. The relationship between near-rings and clones containing the group operations of the underlying group shows that the unary parts of such clones correspond precisely to near-rings containing the identity function. Rosenbergʹs characterization of maximal clones is then applied to describe maximal near-rings containing the identity map, while maximal near-rings not containing the identity are described using typical near-ring methods. This finally provides us with a completeness criterion. We apply this criterion to show that if the order of Γ is large then with high probability the set containing a single bijection is complete.