N Fold Hemiring

This article covers the theoretical proof's of 1 Let A be a non-empty set and ¿_1,¿_2 ¿,¿¿_3,......,¿_(n+1) be binary operations on A . Then A=¿(A,¿¿_1,¿_2 ¿,¿¿_3,......,¿_(n+1)) is said to be n fold Hemiring if ¿(A,¿¿_1) is an abelian group ¿ (A,¿¿_2) is Monoid , ¿ (A,¿¿_3) is Monoid , .......¿ (A,¿¿_(n+1)) is Monoid , ¿_2 is distributive over ¿_1 , ¿_3 is distributive over ¿_1 , ......, ¿_(n+1 )is distributive over ¿_1 . 2 If A is a n-fold Hemiring with zero element 0 Then for all a ,b ,c ¿ A 1) aQi0 = 0Qia = O, ¿ i = 2,3,----, n+1. 2) aQi(-b) = (-a)Qib = - (aQib), ¿ i =2,3,...... 3) (-a) Qi (-b) = aQib , ¿ i = 2131......., n+1 4) aQi (bQ1(-c)) = (aQib) Q1(aQi (-c)) , ¿ i = 2,3,......, n+1 5) (-1) Qi a = (-a) , ¿ i = 2,3,......., n+1. 6) (-1) Qi (-1) = 1 , ¿ I = 2,3,4,......, n+1. 3 A finite n fold integral domain is a n-fold field . 4 The set of units in a commutative n-fold Hemiring is a abelian group with respect to Q2 ,-------, Qn+1 . 5 Any nonempty subset S of a n-fold Hemiring A = (A1 Q1, Q2, Q3,---------,Qn+1) Is called sub n-fold Hemiring ; if S = (S, Q1,Q2,--------,Qn+1) is a n-fold Hemiring . 6 A nonempty subset S of a n-fold Hemiring A is a sub n fold Hemiring of A iff