International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-9 Issue-1S5, December, 2019 306 Published By: Blue Eyes Intelligence Engineering & Sciences Publication Retrieval Number: A11101291S52019/2019©BEIESP DOI: 10.35940/ijeat.A1110.1291S52019 Partial Addition and Ternary Product Based  -So- Semirings-2 Bhagyalakshmi Kothuru, V. Amarendra Babu Abstract: Here we are introducing thenotions i-system, idempotent, centre of a ternary  -SO semiring, Nilpotent are introduced and it is proved that some equivalent conditions. Further it is also proved that (i) if C be a ternary  - SO semiring, m is a “strongly regular element”, then ∃, ∈Г also n∈C ∋m = mnm,n = nmn (ii) If “I be an Ideal of A strongly regular ternary  - SOsemiring R then I is strongly regular and any ideal J of I is an ideal of R” and many more properties were proved. Mathematical subject classification: 16Y60. Keywords:Idempotent, i-system, strongly regular, m-system, n- system. I. INTRODUCTION Sen in 1981 introduced the concepts of  - semigroup as generalization of semi group. In 1934 H.S Vandiver develops the theory of semi ring. In 1995 the notion of  - semiringswas introduced by M.MuralikrishnaRao. In this paper we are introducing some classical notions of ternary  - SO semiring. II. MATHEMATICAL BACKGROUND: Some of the main definitions and results are as follows. Throughout this paper C is a CTSS means “complete ternary Γ- SO semiring” and “ternary  - SO semiring” is denoted by TГ-S-SR. Definition2.1:Let R is a TГ-S-SR along with ≠S⊆R then S is known as “m-system” if for any s, t, u∈ S implies that RΓRΓsΓRΓRΓtΓRΓRΓuΓRΓR⊆S. Definition2.2: Let R be a TГ-S-SR along with≠S⊆R then S is known as “n-system” if any l∈S implies that RΓRΓlΓRΓRΓlΓRΓRΓlΓRΓR⊆S. Revised Manuscript Received on December 15, 2019. Bhagyalakshmi Kothuru, Research Scholar, Department of Mathematics, Acharya Nagarjuna University,KKR&KSR Institute of Technology &Sciences Vinjanampadu, Guntur, E-mail:mblakshmi12@gmail.com V. Amarendra Babu, Department of Mathematics, AcharyaNagarjuna University. Vinjanampadu, Guntur, India. E-mail: amarendrab4@gmail.com Notation2.3: For ∈, CTSSC define (x] = {d∈C: d≤x} and for a subset X of a CTSSC. (x] = (] ∈ . Remark: 2.4:“Let F be a PO-ternary semiring”and L⊆F, M⊆F, N⊆F, then (i) L⊆(L] (ii) ((L]]=(L] (iii) (L]Γ(M]Γ(N]⊆(LΓMΓN] (iv) L⊆M⇒L⊆(M] (v) L⊆M⇒(L]⊆(M] (vi) (L∩M]=(L]∩(M] (vii) (L∪M]=(L]∪(M] Definition 2.5: F is any proper ideal of a TГ-S-SRM is known as “prime”for any ideals R, S, T of M, RΓSΓT ⊆ F⟹ R ⊆ F or S ⊆ F or T ⊆ F. Definition2.6: F is any proper ideal of a TГ-S-SRM is known as “semiprime” for any ideals R of M, RΓRΓR ⊆ F⟹ R ⊆ F. And for more preliminaries the references [12] [13] [14] [15][16][17] and [18]. III. EXPERIMENTS AND RESULTS DESCRIPTION Definition3.1: Let M be a TГ-S-SRmoreover≠S⊆Mthen S is known as an “i-system” ∀ r, t, u∈ S,<  >∩<  >∩<  >∩≠∅ Example: 3.2: Let S= {0,  1 ,  2 ,  3 ,  4 ′  5 },  ={, } define Σ on S as    =   if   =0   ≠ , and for some  not defined , elsewhere Here S is “ternary SO –monoid”.