An Application of Linear Automata to Near Rings

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) is abelian, (b) N has an identity 1, (c) There is some such that is generated by {1,d}; 2) Let  , N  : h   d d N  0 N  S S be a GSAepimorphism. Then there exists a near-ring epimorphism h from   N S to   N  S with       h qn  h q h n for all and q Q    n N  S ; 3) Let   , , , , Q A B F G  A be a GA. Then (a)       : , a a a a Q A G Q A : Q N , , , A B F Q     A A is accessible, (b) Q = 0N(A), (c)   ~ , , , , B F ~ Q : ~  / Q A A/ ~ with       , ~ , : F q a F  q a     and       , : , a G q a  ~ G q is reduced, (d) is minimal. a A / ~