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). If N is a near-ring then N0: = {n ∈ N/n0 = 0}. A group (Γ, +) is an N-group (we write NΓ) if a “product” ny is defined with (n + n‛)γ = nγ + n‛γ and (nn‛)γ = n(n‛γ). Ideals of near-rings and N-groups are kernels of (N-) homomorphisms. If Γ is a vector-space, Maff (Γ) is the near-ring of all affine transformations on Γ. N is 2-primitive on NΓ if NΓ is non-trivial, faithful and without proper N-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.