Near-rings of compatible functions

Abstract

SummaryIn this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance,M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). IfNis a near-ring thenN0: = {n∈N/n0 = 0}. A group (Γ, +) is anN-group (we writeNΓ) if a “product”nyis defined with (n+n‛)γ =nγ +n‛γ and (nn‛)γ =n(n‛γ). Ideals of near-rings andN-groups are kernels of (N-) homomorphisms. If Γ is a vector-space,Maff(Γ) is the near-ring of all affine transformations on Γ.Nis 2-primitive onNΓ ifNΓ is non-trivial, faithful and without properN-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.