Presentations of faithful d.g. near-rings

This is the authors' abstract. We don't add key points for this paper.

Many basic definitions and results in the theory of near-rings can be found in G. Pilz ( 4 ). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum ( 2 ), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S , which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by ( R, S ). Then ( R , +) = Gp < S ; > where is a set of defining relations in S . Let ( T, U ) be a d.g. near-ring. Then a d.g. homomorphism θ from ( R, S ) to ( T, U ) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U . If ( G , +) is a group, let T 0 ( G ) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G . Then ( E ( G ), End G ) is a d.g. near-ring. A d.g. near-ring ( R, S ) is faithful if there exists a d.g. monomorphism θ:( R, S ) → ( E ( G ), End G ) for some group G .