Archive of SID 43rd Annual Iranian Mathematics Conference, 27-30 August 2012 University of Tabriz Talk Relocation ternary Γ-semihyperrings pp. 115-118 Relocation ternary Γ-semihyperrings S. Ostadhadi−Dehkordi M. Heidari Hormozgan University Bu-Ali Sina University Abstract Our main purpose of this study is to introduce the notions of right (left) fundamen- tal semirings and we show that there is a covariant functor between the category of relocation ternary Γ-semihyperring and semirings. In a ternary Γ-semihyperring, addition is a hyperoperation and multiplication is a ternary operation. Indeed, the notion of ternary Γ-semihyperrings is a generalization of ternary Γ-semirings, a generalization of Γ-semihyperrings and a generalization of ternary semirings. Keywords: Relocation ternary Γ-semihyperring, right(left) fundamental semir- ing. Mathematics Subject Classification: 20N20, 16Y99. 1 Introduction Algebraic hyperstructures which are based on the notion of hyperoperation were introduced by Marty [5] and studied extensively by many mathematicians. Recently, Davvaz et. al. [3, 4] in- troduced the notion Γ- semihyperrings. A recent book on hyperstructures [1], points out on their applications in fuzzy and rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Ternary Γ-semihyperrings are one of the generalized structures of ternary Γ-semirings, Γ- semihyperrings and ternary semirings. Indeed, in a ternary Γ-semihyperring, addition is a hyper- operation and multiplication is a ternary operation. In this study, we show that there is a covariant functor between the category of relocation ternary Γ-semihyperrings and the category of semirings. 2 Main Result Definition 2.1. [2] Let R and Γ be non-empty sets such that R be a additive commutative semi- groups. Then, R is said to be a ternary Γ- semihyperring if for each α; β 2 Γ and a; b; c; d; f 2 R, there exist a ternary operations α : R3 −! R and β : R3 −! R such that satisfying the following conditions: 1. α(a + b; c; d) = α(a; c; d) + α(b; c; d); 2. α(a; b + c; d) = α(a; b; d) + α(a; c; d); 3. α(a; b; c + d) = α(a; b; c) + α(a; b; d); 4. α(β(a; b; c); d; f) = β(a; α(b; c; d); f) = α(a; b; β(c; d; f)). In this definition if α and β are ternary hyperoperations, then R is called general ternary Γ- semihyperring and R is called multiplicative ternary Γ-semihyperring when + and α; β are operation and ternary hyperoperations, respectively. 115 www.SID.ir Archive of SID 43rd Annual Iranian Mathematics Conference, 27-30 August 2012 University of Tabriz Talk Relocation ternary Γ-semihyperrings pp. 115-118 Example 2.2. Let R be a real numbers. Then, R is a general ternary Z-semihyperring with respect to the following hyperaddition and ternary hyperoperation: x1 ⊕ x2 = fz :[x1] + [x2] ≤ z < [x1] + [x2] + 1g; α(x1; x2; x3) = fz : α[x1][x2][x3] ≤ z < α[x1][x2][x3] + 1g; for every x1; x2; x3 2 R and α 2 Z. Let us denote the set U(R;L) as a finite sum of finite products of elements of R. Suppose that xγy, ∗ if and only if fx; yg ⊆ u, where u 2 U(R;L). Let γ be the transitive closure of γ. If for every α 2 Γ there exists an element eα 2 R such that eα +x = x; α(eα; eα; x) = α(eα; x; eα) = α(x; eα; eα) = x; for every x 2 R, then eα is called an α-unit element and R is called ternary Γ-semihyperring with unit. f g TheoremS 2.3. Let R be a ternary Γ-semiring, A r2R be a collection of non-empty disjoint sets 2 2 and S = r2R Ar. For every x; y; z S and γ Γ, we define ⊕ x y = Ar1+r2 ; α(x; y; z) = Aα(r1;r2;r3); 2 2 2 2 where x Ar1 ; y Ar2 and z Ar3 , for some α Γ. Then, S is a general ternary Γ-semihyperring. ∗ ∼ Moreover, if R = Γ(R; R; R), then [S : γ ] = R: Let I be an ideal of multiplicative ternary Γ-semihyperring R. We define (x1; x2) 2 ρI () x1 + x = x2 + y: We say that ('; f) is a homomorphism between R1 and R2 as Γ1- and Γ2- ternary semihyperrings, respectively, when ' : R1 −! R2, and f :Γ1 −! Γ2 are maps such that for every a; b; c 2 R1 and α1 2 Γ1, we have '(a + b) = '(a) + '(b);'(α1(a; b; c)) = f(α1)('(a);'(b);'(c)): Proposition 2.4. Let R1 and R2 be multiplicative Γ1- and Γ2- semihyperrings, respectively, ( ; f):(R1; Γ1) −! (R2; Γ2) be a epimorphism and I be an ideal of R1. Then, there exists a homomorphism ( =%I ; f=%I ):[R1 : %I ] −! [R2 : % (I)]; such that ◦ e e ◦ ( =%I ; f=%I ) (πR1 ; πΓ1 ) = (πR2 ; πΓ2 ) ( ; f): A ternary Γ- semihyperring R is called relocation ternary Γ-semihyperring, if for every α; β 2 Γ and a; b; c; d; e; f 2 R, α(β(a; b; c); d; f) = β(α(a; b; c); d; f). Let R be a relocation) ternary Γ- Yn ∗ ∗ b b semihyperring and F (R) = (γ (xi); γ (yi); αbi): αbi 2 Γ; xi; yi 2 R; n 2 N ; where Γ = fαb : i=1 α 2 Γg. We define a relation ρ on F (R) as follows: 0 1 Yn Ym @ ∗ ∗ ∗ ∗ b A (γ (ai); γ (bi); αbi); (γ (cj); γ (dj); βj) 2 ρ () i=1 j=1 Xn Xm ∗ ∗ ∗ b ∗ ∗ ∗ αbi(γ (ai); γ (bi); γ (x)) = βj(γ (cj); γ (dj); γ (x)); i=1 j=1 ∗ ∗ for every γ (x) 2 [R : γ ]. This relation is congruence on !F (R). The congruence class containing Yn Yn ∗ ∗ ∗ ∗ (γ (ai); γ (bi); αbi) denoted by ρ (γ (ai); γ (bi); αbi) . Hence, [F (R): ρ]r forms a semiring i=1 i=1 with multiplication defined by ! 0 1 Yn Ym ∗ ∗ @ ∗ ∗ b A ρ (γ (ai); γ (bi); αbi) ρ (γ (cj); γ (dj); βj) i=1 j=1 116 www.SID.ir Archive of SID 43rd Annual Iranian Mathematics Conference, 27-30 August 2012 University of Tabriz Talk Relocation ternary Γ-semihyperrings pp. 115-118 0 1 Y @ ∗ ∗ ∗ ∗ b A = ρ (αbi(γ (ai); γ (bi); γ (cj)); γ (dj); βj) : i;j We can see that [F (R): ρ]r forms a semiring and is called right fundamental semiring and we show that with RF S. Dually we define left fundamental semiring of ternary Γ-semihyperring and denoted by [F (R): ρ]l and we show that with LF S. b Let R be a relocation ternary Γ-semihyperring, A ⊆ [F (R): ρ]r, ∆ ⊆ Γ and Q ⊆ [R : γ∗]. We define ( ! ! ) Yn Yn ∗ ∗ ∗ ∗ QRl = ρ (αbi; γ (xi); γ (yi)) 2 [F (R): ρ]l : ρ (αbi; γ (xi); (yi)) [R : γ ] ⊆ Q ; i=1 i=1 n P o ∗ ∗ b ∗ ∗ ∗ ∗ AlR = γ (a) 2 [R : γ ]: ρ (Γ; γ (x); γ (a)) ⊆ A; for every γ (x) 2 [R : γ ] : In the same way, we can define ArR and QRr. Let R be a relocation ternary Γ-semihyperring and Ω be a binary relation on [R : γ∗]. We ∗ define a relation Ω on [F (R): ρ]r, as follows: 0 ! 0 11 Yn Ym @ ∗ ∗ @ ∗ ∗ b AA ∗ ρ (γ (ai); γ (bi); αbi) ; ρ (γ (cj); γ (dj); βj) 2 Ω () i=1 j=1 0 1 Xn Xm @ ∗ ∗ ∗ b ∗ ∗ ∗ A αbi(γ (ai); γ (bi); γ (x)); βj(γ (cj); γ (dj); γ (x)) 2 Ω; i=1 j=1 for every γ∗(x) 2 [R : γ∗]. Theorem 2.5. Let R1 and R2 be relocation ternary Γ1- and Γ2- semihyperrings, respectively, and there exists an epimorphism ('; f):(R1; Γ1) −! (R2; Γ2). Then, ∗ ∼ [[F (R1): ρ1]:(ker') ] = [F (R2): ρ2]: Proposition 2.6. Let R be a relocation ternary Γ-semihyperring with zero and I be an ideal of ∗ [F (R): ρ]r. Then, IrR is a right ideal (left ideal) of [R : γ ]. Proposition 2.7. Let R be a relocation ternary Γ-semihyperring with unit, I be a right ideal of ∗ [R : γ ] and J be a right ideal of [F (R): ρ]r. Then, IRrR = I and JrRr = J. Theorem 2.8. Let R be a relocation ternary Γ-semihyperring with unit. Then, [R : γ∗] is additively cancellative if and only if so [F (R): ρ]r. Theorem 2.9. Let T GSH be the category of all relocation ternary Γ-semihyperring and SR be the category of semirings. Then, there exists a covariant functor between T GSH and SR. Theorem 2.10. Let R1 and R2 be relocation ternary Γ1- and Γ2- semihyperrings, respectively, if \ ('; f):(R1; Γ1) −! (R2; Γ2) is an epimorphism, then there is a homomorphism ('; f):[F (R1): ρ1] −! [F (R2): ρ2] such that following diagram is commutative: (';f) (R1; Γ1) −− −! (R2; Γ2) # # (';fd) [F (R1): ρ1] −− −! [F (R2): ρ2] −! −! where (ΠR1 ;'1):(R1; Γ1) [F (R1): ρ1] and (ΠR2 ;'2):(R2; Γ2) [F (R2): ρ2]; are canonical maps. 117 www.SID.ir Archive of SID 43rd Annual Iranian Mathematics Conference, 27-30 August 2012 University of Tabriz Talk Relocation ternary Γ-semihyperrings pp. 115-118 References [1] B. Davvaz and V. Leoreanu-Fotea, Hyperring theory and applications, International Academic Press, USA, 2007. [2] S. O. Dehkordi and B. Davvaz, Ternary Γ-semihyperring: ideals, regular ideals, submitted for publication. [3] S. O. Dehkordi and B. Davvaz, Γ-semihyperrings: Approximations and rough ideals, accepted for publication in Bulletin of the Malaysian Mathematical Sciences Society.