Some Conditions On Ternary Semi Rings
Journal Title: International Journal of Engineering and Science Invention - Year 2018, Vol 7, Issue 10
Abstract
Here, we introduce the notions of simple w.r.t addition, infinite, center of a ternary semi ring and proved (1) If p, q and r are elements of a ternary semi ring T while n, m and s are nonnegative integers then : (i) [ ] [ ] 0 ( ) n n i n i i l q l q , (ii) l [n] q [m] r [s] = 1 0 ( ) n i i s n i m i i l q r l q r . (2) The laws (i) U is simple, (ii) f = fur + fu1+ ur1 + f ∀ f, u, r ∈ U, (iii) f = urf + uf1 + rf1 + f ∀ f, u, r ∈ U, (iv) fur = fur + fsutr ∀ f, u, r, s, t ∈ U are equivalent. (3) For a ternary semi ring U : (i) U is simple and ternary multiplicatively idempotent, (ii) (j + u)(j + r)(j + s) =j + urs ∀ j, u, r, s ∈ U, (ii) If p, q ∈ U then f + u = f ⟺ fuu = uuf = ufu = u are equivalent. (4) For each element of a simple ternary semi ring U, if J(a) = {0} ⋃ {r ∈ U / r + a = 1} then (i) J(a) is a ternary sub semi ring of U for all a ∈ U, (ii) J(a) ⋂ J(b) ⋂ J(c) = J(abc) ∀ a, b, c ∈ U. (5) If U is a ternary semi ring and f, r ∈ I + (U), then (1) (H(l),+) is a commutative ternary semi group, (2) J(l) = {l + s / s ∈ H(l)} is a ternary subgroup of H(l), (3) J(r)J( r " )J( r " ) ⊆ J( rr r " ).
Authors and Affiliations
G. Srinivasa Rao, D. Madhusudhana Rao, P. Sivaprasad
Keywords
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