N Fold Hemiring

This article covers the theoretical proof''s of 1 Let A be a non-empty set and θ_1θ_2 〖θ〗_3......θ_(n+1) be binary operations on A . Then A=〖(Aθ〗_1θ_2 〖θ〗_3......θ_(n+1)) is said to be n fold Hemiring if 〖(Aθ〗_1) is an abelian group 〖 (Aθ〗_2) is Monoid 〖 (Aθ〗_3) is Monoid .......〖 (Aθ〗_(n+1)) is Monoid θ_2 is distributive over θ_1 θ_3 is distributive over θ_1 ...... θ_(n+1 )is distributive over θ_1 . 2 If A is a n-fold Hemiring with zero element 0 Then for all a b c ϵ A 1) aQi0 = 0Qia = O ∀ i = 23-- n+1. 2) aQi(-b) = (-a)Qib = - (aQib) ∀ i =23...... 3) (-a) Qi (-b) = aQib ∀ i = 2131....... n+1 4) aQi (bQ1(-c)) = (aQib) Q1(aQi (-c)) ∀ i = 23...... n+1 5) (-1) Qi a = (-a) ∀ i = 23....... n+1. 6) (-1) Qi (-1) = 1 ∀ I = 234...... 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 Q1Q2----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