An Application of Linear Automata to Near Rings

Songfa You, Yijun Feng, Ming Cao, Yaping Wei

School of Mathematics and Computer Science, Hubei University, Wuhan, China.

DOI: 10.4236/am.2012.311223                

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 N₀ 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.

Keywords

Linear Automata; Accessible; GSA-Homomorphism; Near-Ring

Conflicts of Interest

The authors declare no conflicts of interest.

References

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