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.

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

Cite this paper

S. You, Y. Feng, M. Cao and Y. Wei, "An Application of Linear Automata to Near Rings,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1614-1618. doi: 10.4236/am.2012.311223.

S. You, Y. Feng, M. Cao and Y. Wei, "An Application of Linear Automata to Near Rings,"

References

[1] S. Eilenberg, “Automata, Language, and Machines,” Academic Press, New York, 1974.

[2] G. Pilz, “Near Rings,” North-Holland, Amsterdam, 1977.

[3] S. F. You, M. Cao and Y. J. Feng, “Semiautomata and Near Rings,” Quantitative Logic and Soft Computing, Vol. 5, 2012, pp. 428-431.

[4] S. F. You, H. Y. Zhao, Y. J. Feng and M. Cao, “An Application of Eulerian Graph to PI on Mn(C),” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 809-811.

[5] S. F. You, “An Application of Eulerian Graph to Polynomial Identity,” IEEE Proceedings of the 2011 International Conference on Computational Intelligence and Software Engineering (CiSE 2011), Wuhan, 9-11 December 2011.

[6] S. F. You, et al., “Eulerian Graph and Polynomial Identities on Matrix Rings,” Advances in Mathematics, Vol. 32, No. 4, 2003, pp. 425-428.

[7] S. F. You, “The Primitivity of Extended Centroid Extension on Prime GPI-Rings,” Advances in Mathematics, Vol. 29, No. 4, 2000, pp. 331-336.

[1] S. Eilenberg, “Automata, Language, and Machines,” Academic Press, New York, 1974.

[2] G. Pilz, “Near Rings,” North-Holland, Amsterdam, 1977.

[3] S. F. You, M. Cao and Y. J. Feng, “Semiautomata and Near Rings,” Quantitative Logic and Soft Computing, Vol. 5, 2012, pp. 428-431.

[4] S. F. You, H. Y. Zhao, Y. J. Feng and M. Cao, “An Application of Eulerian Graph to PI on Mn(C),” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 809-811.

[5] S. F. You, “An Application of Eulerian Graph to Polynomial Identity,” IEEE Proceedings of the 2011 International Conference on Computational Intelligence and Software Engineering (CiSE 2011), Wuhan, 9-11 December 2011.

[6] S. F. You, et al., “Eulerian Graph and Polynomial Identities on Matrix Rings,” Advances in Mathematics, Vol. 32, No. 4, 2003, pp. 425-428.

[7] S. F. You, “The Primitivity of Extended Centroid Extension on Prime GPI-Rings,” Advances in Mathematics, Vol. 29, No. 4, 2000, pp. 331-336.