Relocation Ternary Γ-Semihyperrings Pp

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) fundamental 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 semiring.

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] introduced 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 hyperoperation 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 ∈ Γ and a, b, c, d, f ∈ 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.

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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 = {z :[x1] + [x2] ≤ z < [x1] + [x2] + 1}, α(x1, x2, x3) = {z : α[x1][x2][x3] ≤ z < α[x1][x2][x3] + 1},

for every x1, x2, x3 ∈ R and α ∈ 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 {x, y} ⊆ u, where u ∈ U(R,L). Let γ be the transitive closure of γ. If for every α ∈ Γ there exists an element eα ∈ R such that eα +x = x, α(eα, eα, x) = α(eα, x, eα) = α(x, eα, eα) = x, for every x ∈ R, then eα is called an α-unit element and R is called ternary Γ-semihyperring with unit. { } Theorem∪ 2.3. Let R be a ternary Γ-semiring, A r∈R be a collection of non-empty disjoint sets ∈ ∈ and S = r∈R Ar. For every x, y, z S and γ Γ, we define ⊕ x y = Ar1+r2 , α(x, y, z) = Aα(r1,r2,r3), ∈ ∈ ∈ ∈ 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) ∈ ρI ⇐⇒ x1 + x = x2 + y.

We say that (φ, f) is a homomorphism between R1 and R2 as Γ1- and Γ2- ternary semihyperrings, respectively, when φ : R1 −→ R2, and f :Γ1 −→ Γ2 are maps such that for every a, b, c ∈ R1 and α1 ∈ Γ1, we have

φ(a + b) = φ(a) + φ(b), φ(α1(a, b, c)) = f(α1)(φ(a), φ(b), φ(c)).

Proposition 2.4. Let R1 and R2 be multiplicative Γ1- and Γ2- semihyperrings, respectively, (ψ, f):(R1, Γ1) −→ (R2, Γ2) be a epimorphism and I be an ideal of R1. Then, there exists a homomorphism (ψ/ϱI , f/ϱI ):[R1 : ϱI ] −→ [R2 : ϱψ(I)], such that ◦ e e ◦ (ψ/ϱI , f/ϱI ) (πR1 , πΓ1 ) = (πR2 , πΓ2 ) (ψ, f). A ternary Γ- semihyperring R is called relocation ternary Γ-semihyperring, if for every α, β ∈ Γ and a, b, c, d, e, f ∈ R, α(β{(a, b, c), d, f) = β(α(a, b, c), d, f). Let R be a relocation} ternary Γ- ∏n ∗ ∗ b b semihyperring and F (R) = (γ (xi), γ (yi), αbi): αbi ∈ Γ, xi, yi ∈ R, n ∈ N , where Γ = {αb : i=1 α ∈ Γ}. We deﬁne a relation ρ on F (R) as follows:   ∏n ∏m  ∗ ∗ ∗ ∗ b  (γ (ai), γ (bi), αbi), (γ (cj), γ (dj), βj) ∈ ρ ⇐⇒ i=1 j=1 ∑n ∑m ∗ ∗ ∗ b ∗ ∗ ∗ αbi(γ (ai), γ (bi), γ (x)) = βj(γ (cj), γ (dj), γ (x)), i=1 j=1 ∗ ∗ for every γ (x) ∈ [R : γ ]. This relation( is congruence on )F (R). The congruence class containing ∏n ∏n ∗ ∗ ∗ ∗ (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 ( )   ∏n ∏m ∗ ∗  ∗ ∗ b  ρ (γ (ai), γ (bi), αbi) ρ (γ (cj), γ (dj), βj) i=1 j=1

116

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43rd Annual Iranian Mathematics Conference, 27-30 August 2012 University of Tabriz

Talk Relocation ternary Γ-semihyperrings pp. 115-118   ∏  ∗ ∗ ∗ ∗ b  = 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 RFS. 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 { ( ) ( ) } ∏n ∏n ∗ ∗ ∗ ∗ QRl = ρ (αbi, γ (xi), γ (yi)) ∈ [F (R): ρ]l : ρ (αbi, γ (xi), (yi)) [R : γ ] ⊆ Q , i=1 i=1 { ∑ } ∗ ∗ b ∗ ∗ ∗ ∗ AlR = γ (a) ∈ [R : γ ]: ρ (Γ, γ (x), γ (a)) ⊆ A, for every γ (x) ∈ [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:  ( )   ∏n ∏m  ∗ ∗  ∗ ∗ b  ∗ ρ (γ (ai), γ (bi), αbi) , ρ (γ (cj), γ (dj), βj) ∈ Ω ⇐⇒ i=1 j=1   ∑n ∑m  ∗ ∗ ∗ b ∗ ∗ ∗  αbi(γ (ai), γ (bi), γ (x)), βj(γ (cj), γ (dj), γ (x)) ∈ Ω, i=1 j=1 for every γ∗(x) ∈ [R : γ∗].

Theorem 2.5. Let R1 and R2 be relocation ternary Γ1- and Γ2- semihyperrings, respectively, and there exists an epimorphism (φ, f):(R1, Γ1) −→ (R2, Γ2). Then, ∗ ∼ [[F (R1): ρ1]:(kerφ) ] = [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 \ (φ, f):(R1, Γ1) −→ (R2, Γ2) is an epimorphism, then there is a homomorphism (φ, f):[F (R1): ρ1] −→ [F (R2): ρ2] such that following diagram is commutative:

(φ,f) (R1, Γ1) −− −→ (R2, Γ2) ↓ ↓ (φ,fd) [F (R1): ρ1] −− −→ [F (R2): ρ2] −→ −→ where (ΠR1 , φ1):(R1, Γ1) [F (R1): ρ1] and (ΠR2 , φ2):(R2, Γ2) [F (R2): ρ2], are canonical maps.

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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. [4] S. O. Dehkordi and B. Davvaz, A strong regular relation on Γ- semihyperrings, Journal of Sciences, Islamic Republic of Iran, 22(3) (2011), 257-266. [5] F. Marty, Sur une generalization de la notion de groupe, in Proceedings of the 8th Congress des Mathematiciens Scandinaves,Stockholm, Sweden, (1934), 45-49.

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