A BOUND ON THE NUMBER OF UNARY POLYNOMIAL FUNCTIONS AND A DECIDABILITY RESULT FOR NEAR-RINGS

Abstract

Given a finite zero-symmetric near-ring with identity N, we ask whether there is a group G such that N is isomorphic to the inner automorphism near-ring <I(G);+,◦>, or whether N is a compatible near-ring. We will show that there are algorithms that decide these questions. To this end, we study polynomial functions on subdirectly irreducible expanded groups. We prove that the size of a finite subdirectly irreducible expanded group is bounded from above by a function of the number of its zero-preserving unary polynomial functions.