Centraliser near-ring representations

Abstract

Let V be a group, written additively but not necessarily abelian, and let S be a semigroup of endomorphisms of V. The set C( S; V)={ f: V→ V| fσ=σ f for all σ∈ S and f(0)=0} forms a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centraliser near-ring determined by S and V. Centraliser near-rings are very general, for if N is any zero-symmetric near-ring with 1 then there exists a group V and a semigroup S of endomorphisms of V such that N≃ C( S; V).