An Application of Linear Automata to Near Rings

Abstract

In this paper , we have established an intimate connection between near-nings and linear automata，and obtain the following results: 1) For a near-ring N there exists a linear GSA S with N ≌ N(S) iff (a) (N, +) is abelian, (b) N has an identity 1, (c) There is some d ∈ N d such that N 0 is generated by {1,d}; 2) Let h: S → S’ be a GSA- epimorphism. Then there exists a near-ring epimorphism from N(S) to N(S’) with h(qn) = h(q)h(n) for all q ∈ Q and n ∈ N(S); 3) Let A = (Q,A,B,F,G) be a GA. Then (a) A a :=(Q(N(A)) =: Q a ,A,B,F/Q a × A) is accessible, (b) Q = 0N(A), (c) A/~:= (Q/~,A,B,F ~ ), Q ~ ) with F ~ ([q], a):= [F(q,a)] and G ~ ([q], a):= G(q,a) is reduced, (d) A a /~ is minimal.