Homomorphism Of Tripolar Fuzzy Soft Γ−Semiring

AHLAM FALLATAHTaibah University

Department of MathematicsMadina

SAUDI ARABIA JORDAN JORDAN Afallatah@taibahu.edu.sa

MOURAD OQLA MASSA’DEHAl-Balqa Applied university

Department of Applied scienceAjloun

mourad.oqla@bau.edu.jo

ABD ULAZEEZ ALKOURIAjloun National UniversityDepartment of Mathematics

Ajloun

alkouriabdulazez@gmail.com

Abstract: Given the notion of tripolar fuzzy soft sets, the concepts of a tripolar fuzzy soft Γ−Semirings, a tripolarfuzzy soft Γ−Semiring homomorphism and a tripolar fuzzy soft ideal in Γ−Semirings are discussed, and relatedproperties and corollaries are investigated. On the other hand, in this paper, we also define the image and pre-imageof tripolar fuzzy soft Γ−Semirings. Some properties and results involving these concepts are stated and proved. .

Key Words Soft set, fuzzy Soft set, tripolar fuzzy soft set, tripolar fuzzy soft Γ−Semiring, tripolar fuzzy soft ideal,tripolar fuzzy soft Γ−Semiring homomorphism.

Received: March 1, 2020. Revised: May 1, 2020. Accepted: May 11, 2020. Published: May 28, 2020. Mathematics Subject Classification: 06B10, 16y30, 20N25, 06Y60.

May 2, 2020

1 IntroductionIn 1934 Vandiver [1] introduced the concept of semir-ing, a semiring concepet is the best algebraic structurebecause its comman generalization of distribitive lat-tices, rings and an universal algebra with two binaryoperations addition and multiplication such that oneof them distributive over the other. Semiring used forsolving problems in applied mathematics, informationsciences and in the areas of theortical computer sci-ence as well as in optimazation theory, coding theory,graph theory and formal languages.

In 1964 Nobusawa [2] gave Γ−ring notation asa generalization of ring, after that Sen [3] disscusedΓ−semigroup concept. While, Γ−semiring conceptwas given by Muirali Krishna [4] as a generalizesto the concept of Γ−ring and semiring. A fuzzy settheory was discussed and introduced by Zadeh [5] in1965 as the most appropriate theory for dealing withuncertainty. Rosenfeld [6] in 1971 studied fuzzy sub-group concepts. The idea of fuzzy subgroup and itsapplication on theory and properties studied by someresearchers, see [7, 8, 9, 10, 11, 12]. In 1999, the con-cept of soft set theory was introduced by Molodtsov[13] as a new mathematical tool for dealing with un-certainties. Furthermore, in 2001 Maji et al [14] in-troduced fuzzy soft sets as extended of soft set theory.Ghosh et al and Murali [15, 16] introduced and stud-ied fuzzy soft ring, fuzzy soft ideals and fuzzy soft k-ideals over a Γ−Semiring. ozturk et al [17] discussedsoft Γ−rings and fuzzy subnear rings. In 1999, At-tanassov [18] gave the idea of intuitionistic fuzzy set.

Massa’deh et al [19, 20, 21, 22, 23] extended the in-tuitionistic fuzzy set notation to Γ−Semiring and itsideals, Ku-ideals, subrings, M-subgroups and homo-morphisms. On the other hand, Maji et al [24] studiedand introduced the intuitionistic fuzzy soft set con-cept. Then Yaqoob et al [25] studied the concep ongroups induced by (t, s)-norm.

In 1998, Zhang [26] introduced bipolar fuzzysets concepts as a generlazation of fuzzy sets. Lee[27] in 2000 used this concept and applied it to al-gebraic structures. Also Massa’deh [28, 29, 30] in-troduced the concepts of bipolar Q-fuzzy H-idealsover Γ−hemiring, anti bipolarQ−fuzzy normal semi-group and bipolar for any cosets, isomorphisms andΓ−hemiring. Bipolar fuzzy soft set concept intro-duced in 2013 by Akram [31] where he studied thisconcept on subalgebras.

The ideas of tripolar fuzzy set was introduced byMurali Krishna Rao [32] in 2018 where he discussedthis concept on interior ideal of Γ−semigroup. Also,Murali et al discussed this concept on interior idealof Γ−semiring and on soft interior ideal over semir-ing [33]. In this paper we introduced and discuss theconcept of tripolar fuzzy soft Γ−semiring homomor-phism and some of its theorems and properties of ho-momorphic image of tripolar fuzzy soft Γ−semiring.

2 Preliminaries

Definition 1. [4] If S is a set together with two asso-ciative operations called addition + and multiplica-

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tion · then will be called a semiring if the followingconditions hold:1. + is a commutative operation.2. ∃ 0 ∈ S such that s + 0 = s and s · 0 = 0 · s =0 ∀s ∈ S.3. Distribute low hold from left and right.

Definition 2. [4] If (S,+) and (Γ,+) are commuta-tive semigroups. Then S is said to be Γ−semiring, ifthere exists a mapping S × Γ × S → S written as(s1, α, s2) as s1αs2 such that it satisfies the followingconditions:1.s1α(s2 + s3) = s1αs2 + s1αs32.(s1 + s2)αs3 = s1αs3 + s2αs33. s1(α+ β)s2 = s1αs2 + s1βs2.4. s1α(s2βs3) = (s1αs2)βs3, ∀s1, s2, s3 ∈ S andα, β ∈ Γ.

Definition 3. [4] If M is a Γ−semiring and I bea non empty subset of M. Then I is said to bea Γ−subsemiring of M if I is a sub-semigroup of(M,+) and IΓI ⊆ I.

Definition 4. [4] IfM is a Γ−semiring and I is a nonempty subset of M. Then:

1. I is called a right ideal of M if:(i) I is closed under addition.(ii) IΓM ⊆ I.

2. I is called a left ideal of M if:(i) I is closed under addition.(ii) MΓI ⊆ I.

3. I is called an ideal of M , if it is both a right andleft ideal.

Definition 5. [18] An intuitionistic fuzzy set of a nonempty set A is an object of the form δ = (δµ, δλ) ={(a, δµ(a), δλ(a)); a ∈ A}, such that δµ : A →[0, 1], δλ : A → [0, 1] are membership functions,δµ, δλ are respectively and 0 ≤ δµ(a) + δµ(a) ≤1, ∀a ∈ A.

Definition 6. [26] A bipolar fuzzy set γ of anon empty set A is an object of the form γ ={(a, γµ(a), γλ(a)); a ∈ A} such that γµ : A → [0, 1]and γλ : A → [−1, 0]. γµ(a) represents satisfactiondegree of a to the property corresponding to fuzzy setγ and γλ(a) represents satisfaction degree of a to theimplicit counter property of fuzzy set γ.

Definition 7. [13] If U is an intial universe set, Eis the of parameters set, X ⊂ E. If P (U) representto the power set of U. Then a pair (φ,X) is said tobe a soft set over U such that φ is a map given byφ : X → P (u).

Definition 8. [14] If U is an intial universe set, E isa parameters set and X ⊆ E. A pair (φ,X) is saidto be fuzzy soft over U , such that φ is a map given byφ : X → IU where IU denotes the collection of allfuzzy subset of U.

Definition 9. [4] If R1 and R2 are two Γ−semirings,a function Ψ : R1 → R2 is called a homomor-phism Γ−semiring if Ψ(x + y) = Ψ(x) + Ψ(y) andΨ(xαy) = Ψ(x)αΨ(y), ∀x, y ∈ R1, α ∈ Γ.

Definition 10. [4] If R1 and R2 are two sets and Ψ :R1 → R2 is any function. A bipolar fuzzy subset δof R1 is called a Ψ−invariant if Ψ(a) = Ψ(b) ⇒δ(a) = δ(b).

Definition 11. [30] If ψ : R1 → R2 is a map and δ =(δ+, δ−) and γ = (γ+, γ−) are bipolar fuzzy subsetinR1 andR2 respectively. Then the image ψ(δ) of δ isthe bipolar fuzzy subset ψ(δ) = ((ψ(δ))+, (ψ(δ))−)of R2 defined by:

(ψ(δ))+(a) =

{max{δ+(a); a ∈ ψ−(a); ifψ−(a) 6= φ}0; otherwise

(ψ(δ))−(a) =

{max{δ−(a); a ∈ ψ−(a); ifψ−(a) 6= φ}0; otherwise

and the pre-image ψ−1(γ) of γ under ψ isthe bipolar fuzzy subset of R1 defined by fora ∈ R1, (ψ

−1(γ))+(a) = γ+(ψ(a)) and(ψ−1(γ))−(a) = γ−(ψ(a)).

Definition 12. [33] If Y is a universe set, a tripo-lar fuzzy set γ in Y is an object having the formγ = {(a, λγ(a), µγ(a), δγ); a ∈ Y and 0 ≤ λγ(a) +µγ(a) ≤ 1} such that, λγ : Y → [0, 1], µγ : Y →[0, 1], δγ : Y → [−1, 0]; 0 ≤ λγ(a) + µγ(a) ≤ 1. Thedegree of membership λγ(a) characterize the extentthat a satisfies the property corresponding to tripolarfuzzy set γ, µγ(a) characterize the extent that a satis-fies to the not property corresponding to tripolar fuzzyset γ and δγ(a) characterize the extent that a satisfiesto the implicit counter property of tripolar fuzzy set γ.

Remark 13. γ = (λγ , µγ , δγ) has been used forγ = {(a, λγ(a), µγ(a), δγ); a ∈ Y and 0 ≤ λγ(a) +µγ(a) ≤ 1}.

Definition 14. [16] Assume that R is a Γ−semiring,E is a set of parameter and X ⊆ E. If φ is a mappinggiven by φ : X → ρ(R) such that ρ(R) is the powerset of R. Then (φ,X) is called a soft Γ−semiringover R if and only if for each x ∈ X,φ(x) isΓ−subsemiring of R. This means that:

1. a, b ∈ R⇒ a+ b ∈ φ(a).

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2. a, b ∈ R,α ∈ Γ⇒ aαb ∈ φ(a).

Definition 15. If S is a Γ−semiring a tripo-lar fuzzy soft (φ,X) over S is said to be tripo-lar fuzzy soft Γ−semiring over S if φ(x) ={λφ(x)(s), µφ(x)(s), δφ(x)(s); s ∈ S, x ∈ X} suchthat λφ(x)(s) : S → [0, 1], µφ(x)(s) : S →[0, 1], δφ(x)(s) : S → [−1, 0], 0 ≤ λφ(x)(s) +µφ(x)(s) ≤ 1, ∀s ∈ S satisfying the following ax-ioms:

1. λφ(x)(s1 + s2) ≥ min{λφ(x)(s1), λφ(x)(s2)}

2. µφ(x)(s1 + s2) ≤ max{µφ(x)(s1), µφ(x)(s2)}

3. δφ(x)(s1 + s2) ≤ max{δφ(x)(s1), δφ(x)(s2)}

4. λφ(x)(s1αs2) ≥ min{λφ(x)(s1), λφ(x)(s2)}

5. µφ(x)(s1αs2) ≤ max{µφ(x)(s1), µφ(x)(s2)}

6. δφ(x)(s1αs2) ≤ max{δφ(x)(s1), δφ(x)(s2)},∀s1, s2 ∈ S, x ∈ X and α ∈ Γ.

Definition 16. [4] If S is a Γ−semiring, E is a pa-rameter set and X ⊆ E. If φ is a mapping given byφ : X → ρ(S). Then (φ,X) is said to be a soft right(left) ideal over S if and only if for each x ∈ X,φ(x)is a right (left) ideal of S. This means that:

1. s1, s2 ∈ φ(X) then s1 + s2 ∈ φ(X)

2. s1, s2 ∈ φ(X), α ∈ Γ, s ∈ S then s1αs(sαs1) ∈φ(X).

Definition 17. A tripolar fuzzy soft set (φ,X) overΓ−semiring S is said to be a tripolar fuzzy soft right(left) ideal over S if

1. λφ(x)(s1 + s2) ≥ min{λφ(x)(s1), λφ(x)(s2)}

2. µφ(x)(s1 + s2) ≤ max{µφ(x)(s1), µφ(x)(s2)}

3. δφ(x)(s1 + s2) ≤ max{δφ(x)(s1), δφ(x)(s2)}

4. λφ(x)(s1αs2) ≥ λφ(x)(s1)(λφ(x)(s2))

5. µφ(x)(s1αs2) ≤ µφ(x)(s1)(µφ(x)(s2))

6. δφ(x)(s1αs2) ≤ δφ(x)(s1)(δφ(x)(s2)),∀s1, s2 ∈S, x ∈ X and α ∈ Γ.

Definition 18. If S is a Γ−semiring,E is a parameterset and X ⊆ E. A tripolar fuzzy soft set (φ,X) overS is said to be a tripolar fuzzy soft ideal if the follwingaxioms are hold:

1. λφ(x)(s1 + s2) ≥ min{λφ(x)(s1), λφ(x)(s2)}

2. µφ(x)(s1 + s2) ≤ max{µφ(x)(s1), µφ(x)(s2)}

3. δφ(x)(s1 + s2) ≤ max{δφ(x)(s1), δφ(x)(s2)}

4. λφ(x)(s1αs2) ≥ max{λφ(x)(s1), λφ(x)(s2)}

5. µφ(x)(s1αs2) ≤ min{µφ(x)(s1), µφ(x)(s2)}

6. δφ(x)(s1αs2) ≤ min{δφ(x)(s1), δφ(x)(s2)},∀s1, s2 ∈ S, x ∈ X and α ∈ Γ.

3 Homomorphism in tripolar fuzzysoft Γ−semiring

The homomorphism concept over tripolar fuzzy softΓ−semiring is introduced and studied their propertiesin this section.

Definition 19. If (φ1, X) and (φ2, Y ) are tripolarfuzzy soft set over Γ−semirings R1 and R2 respec-tiely. Let ψ1 : R1 → R2 and ψ2 : X → Y are twofunctions such that X and Y are parameter sets forthe crisp sets R1 and R2 respectiely. Then (ψ1, ψ2) issaid to be a tripolar fuzzy soft function fromR1 toR2.

Definition 20. If (φ1, X) and (φ2, Y ) are tripolarfuzzy soft set over Γ−semiringsR1 andR2 respectielyand (ψ1, ψ2) are tripolar fuzzy soft functions fromR1 to R2. Then (ψ1, ψ2) is called tripolar fuzzy softΓ−semiring homomorphism if satisfying the follow-ing axioms:

1. ψ1 is a Γ−semiring homomorphism from R1

onto R2.

2. ψ2 is a mapping from X onto Y.

3. ψ1(λφ1(x)) = φ2ψ2(x), ψ1(µφ1(x)) = φ2ψ2(x)

and ψ1(δφ1(x)) = φ2ψ2(x), ∀x ∈ X.

Remark 21. If there exist a tripolar fuzzy softΓ−semiring homomorphism between (φ1, X) and(φ2, Y ). Then we say that (φ1, X) is soft homomor-phic to (φ2, Y ).

Definition 22. If (ψ1, ψ2) is a tripolar fuzzy soft func-tion from R1 to R2. The pre-image of (φ2, Y ) underthe tripolar fuzzy soft function (ψ1, ψ2), denoted by(ψ1, ψ2)

−1((φ2, Y ) defined by (ψ1, ψ2)−1(φ2, Y ) =

(ψ−11 (φ2), ψ

−12 (Y )) is a tripolar fuzzy soft set.

Theorem 23. If (φ,X) is a tripolar fuzzy softΓ−semiring over R2, ψ : R1 → R2 is monomor-phism and for each x ∈ X, define (ψφ)x(r) =φx(ψ(r)),∀r ∈ R, then (ψφ,X) is a tripolar fuzzysoft Γ−semiring over R2.

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Proof. Let r1, r2 ∈ R, x ∈ X and α ∈ Γ. Then:1.(ψφ)x(r1 + r2) = φx(ψ(r1 + r2))= λφ(x)(ψ(r1) + ψ(r2))

≥ min{λφ(x)(ψ(r1)), λφ(x)(ψ(r2))}= min{(ψφ)x(r1), (ψφ)x(r2)}.2.(ψφ)x(r1 + r2) = φx(ψ(r1 + r2))= µφ(x)(ψ(r1) + ψ(r2))

≤ max{µφ(x)(ψ(r1)), µφ(x)(ψ(r2))}= max{(ψφ)x(r1), (ψφ)x(r2)}.3.(ψφ)x(r1 + r2) = φx(ψ(r1 + r2))= δφ(x)(ψ(r1) + ψ(r2))

≤ max{δφ(x)(ψ(r1)), δφ(x)(ψ(r2))}= max{(ψφ)x(r1), (ψφ)x(r2)}.4.(ψφ)x(r1αr2) = φx(ψ(r1ℵr2))= λφ(x)(ψ(r1)αψ(r2))

≥ min{λφ(x)(ψ(r1)), λφ(x)(ψ(r2))}= min{(ψφ)x(r1), (ψφ)x(r2)}.5.(ψφ)x(r1αr2) = φx(ψ(r1αr2))= µφ(x)(ψ(r1)αψ(r2))

≤ max{µφ(x)(ψ(r1)), µφ(x)(ψ(r2))}= max{(ψφ)x(r1), (ψφ)x(r2)}.6.(ψφ)x(r1αr2)= φx(ψ(r1αr2))= δφ(x)(ψ(r1)αψ(r2))

≤ max{δφ(x)(ψ(r1)), δφ(x)(ψ(r2))}= max{(ψφ)x(r1), (ψφ)x(r2)}.

Therefore (ψφ)x is a tripolar fuzzyΓ−subsemiring of S. Thus (ψf,X) is a tripolarfuzzy soft Γ−semiring over R2.

Theorem 24. If (γ,X) is a tripolar fuzzy soft semir-ing over Γ−semiring R, if ψ is an endomomorphismof R and defined (γψ)x = γxψ for each x ∈ X. Then(γψ,X) is a tripolar fuzzy soft Γ−semiring over R.

Proof. Let r1, r2 ∈ R, x ∈ X and α ∈ Γ. Then:1.(λψ)x(r1 + r2)= λγ(x)(ψ(r1 + r2))

= λγ(x)(ψ(r1) + ψ(r2))

≥ min{λγ(x)(ψ(r1)), λγ(x)(ψ(r2))}= min{(λψ)x(r1), (λψ)x(r2)}.2.(µψ)x(r1 + r2)= µγ(x)(ψ(r1 + r2))

= µγ(x)(ψ(r1) + ψ(r2))

≤ max{µγ(x)(ψ(r1)), µγ(x)(ψ(r2))}= max{(µψ)x(r1), (µψ)x(r2)}.3.(δψ)x(r1 + r2)= δγ(x)(ψ(r1 + r2))

= δγ(x)(ψ(r1) + ψ(r2))

≤ max{δγ(x)(ψ(r1)), δγ(x)(ψ(r2))}= max{(δψ)x(r1), (δψ)x(r2)}.

4.(λψ)x(r1αr2)= λγ(x)(ψ(r1ℵr2))= λγ(x)(ψ(r1)αψ(r2))

≥ min{λγ(x)(ψ(r1)), λγ(x)(ψ(r2))}= min{(λψ)x(r1), (λψ)x(r2)}.5.(µψ)x(r1αr2)= µγ(x)(ψ(r1αr2))

= µγ(x)(ψ(r1)αψ(r2))

≤ max{µγ(x)(ψ(r1)), µγ(x)(ψ(r2))}= max{(µψ)x(r1), (µψ)x(r2)}.6.(δψ)x(r1αr2)= δγ(x)(ψ(r1αr2))

= δγ(x)(ψ(r1)αψ(r2))

≤ max{δγ(x)(ψ(r1)), δγ(x)(ψ(r2))}= max{(δψ)x(r1), (δψ)x(r2)}.

Thus (γψ)x is a tripolar fuzzy Γ−subsemiring ofR. Then (γψ,X) is a tripolar fuzzy soft Γ−semiringover R

Theorem 25. If ψ : R1 → R2 is an eipomomorphismof Γ−semiring and (γ,X) is a tripolar fuzzy soft rightideal over R2. If for each x ∈ X, ζx = ψ−1(γx) then(ζ,X) is a tripolar fuzzy soft right ideal over R1.

Proof. If x ∈ X and α ∈ Γ. Then γx is a tripolarfuzzy soft right ideal over R2. If r1, r2 ∈ R1 andα ∈ Γ, then:

1.ψ−1(λx)(r1 + r2) = λγ(x)(ψ(r1 + r2))

= λγ(x)(ψ(r1) + ψ(r2))

≥ min{λγ(x)(ψ(r1)), λγ(x)(ψ(r2))}= min{ψ−1(λx)(r1), ψ

−1(λx)(r2)}.2.ψ−1(µx)(r1 + r2) = µγ(x)(ψ(r1 + r2))

= µγ(x)(ψ(r1) + ψ(r2))

≤ max{µγ(x)(ψ(r1)), µγ(x)(ψ(r2))}= max{ψ−1(µx)(r1), ψ

−1(µx)(r2)}.3.ψ−1(δx)(r1 + r2) = δγ(x)(ψ(r1 + r2))

= δγ(x)(ψ(r1) + ψ(r2))

≤ max{δγ(x)(ψ(r1)), δγ(x)(ψ(r2))}= max{ψ−1(δx)(r1), ψ

−1(δx)(r2)}.4.ψ−1(λx)(r1αr2) = λx(ψ(r1αr2))

= λx(ψ(r1)αψ(r2))≥ λx(ψ(r1))= ψ−1(λx)(r1).

5.ψ−1(µx)(r1αr2) = µx(ψ(r1αr2))= µx(ψ(r1)αψ(r2))≤ µx(ψ(r1))= ψ−1(µx)(r1).

6.ψ−1(δx)(r1αr2) = δx(ψ(r1αr2))= δx(ψ(r1)αψ(r2))≤ δx(ψ(r1))= ψ−1(δx)(r1).

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Therefore ζx = ψ−1(γx) is a tripolar fuzzy rightideal of R1. Thus (ζ,X) is a tripolar fuzzy soft rightideal over R1.

Theorem 25 is true for tripolar fuzzy soft leftideal.

Proposition 26. If R1 and R2 are Γ−semirings, ψ :R1 → R2 is a Γ−semiring homomorphism and φ is aψ−invariant bipolar fuzzy subset of R1, if b = ψ(a)then ψ(φ)(b) = φ(a); a ∈ R1.

Proof. strightforword.

Theorem 27. If (γ,X) is a tripolar fuzzy soft rightideal over Γ−semiring R1 and ψ is a homomor-phism from R1 onto R2. For each x ∈ X, γx is aψ−invariant bipolar fuzzy right ideal of R1, if ζx =ψ(γx) then (ζ,X) is a tripolar fuzzy soft right idealover R2.

Proof. Let r1, r2 ∈ R2, x ∈ X and α ∈ Γ. Then thereexists r3, r4 ∈ R1 such that ψ(r2) = r1, ψ(r4) =r2, r1 + r2 = ψ(r3 + r4) and r1αr2 = ψ(r3αr4). γxis ψ−invariant. Thus by propostion 26, we have:

1.λζ(x)(r1 + r2) = ψ(λγ(x))(r1 + r2)

= λγ(x)(r3 + r4)

≥ min{λγ(x)(r3), λγ(x)(r4)}= min{ψ(λγ(x)(r1), ψ(λγ(x)(r2)}= min{λζ(x)(r1), λζ(x)(r2)}

2.µζ(x)(r1 + r2) = ψ(µγ(x))(r1 + r2)

= µγ(x)(r3 + r4)

≤ max{µγ(x)(r3), µγ(x)(r4)}= max{ψ(µγ(x))(r1), ψ(µγ(x)(r2))}= max{µζ(x)(r1), µζ(x)(r2)}

3.δζ(x)(r1 + r2) = ψ(δγ(x))(r1 + r2)

= δγ(x)(r3 + r4)

≤ max{δγ(x)(r3), δγ(x)(r4)}= max{ψ(δγ(x)(r1)), ψ(δγ(x)(r2))}= max{δζ(x)(r1), δζ(x)(r2)}

4.λζ(x)(r1αr2) = ψ(λγ(x))(r1αr2)

= λγ(x)(ψ(r3αr2))

= λγ(x)(ψ(r3)αψ(r4))

≥ λγ(x)(ψ(r3))

= ψ(λγ(x)(r1))

= λζ(x)(r1)

5.µζ(x)(r1αr2) = ψ(µγ(x))(r1αr2)

= µγ(x)(ψ(r3αr2))

= µγ(x)(ψ(r3)αψ(r4))

≤ µγ(x)(ψ(r3))

= ψ(µγ(x)(r1))

= µζ(x)(r1)

6.δζ(x)(r1αr2) = ψ(δγ(x))(r1αr2)

= δγ(x)(ψ(r3αr2))

= δγ(x)(ψ(r3)αψ(r4))

≤ δγ(x)(ψ(r3))

= ψ(δγ(x)(r1))

= δζ(x)(r1)

then ζx is a tripolar fuzzy ideal of R2. Hence(ζ,X) is a tripolar fuzzy soft right ideal over R2.

Theorem 28. If (γ1, X1) and (γ2, X2) are two bipo-lar fuzzy soft Γ−semirings over R1 and R2 respec-tively, and (φ, ψ) is a tripolar fuzzy soft Γ−semiringhomomorphism from (γ1, X1) onto (γ2, X2). Then(φ(γ1), X2) is a tripolar fuzzy soft Γ−semiring overR2.

Proof. By definition 20, φ is a Γ−semiring homomor-phism from R1 onto R2 and ψ is a mapping from X1

onto X2 for each y ∈ X2 there exist x ∈ X; such thatψ(x) = y. Define (φ(γ1))y = φ(γ1x). If r1, r2 ∈ R2

and α ∈ Γ, then there exist r3, r4 ∈ R1 such thatφ(r3) = r1, φ(r4) = r2 and φ(r3 + r4) = r1 + r2 andφ(r3αr4) = r1αr2. Thus we have:

1.(φ(λγ1)ψ(x)(r1 + r2) = φ(λγ1(x))(r1 + r2)

= λγ1(x)(r3 + r4)≥ min{λγ1(x)(r3), λγ1(x)(r4)}= min{φ(λγ1(x))(r1), φ(λγ1(x))(r2)}= min{φ(λγ1)ψ(x)(r1), φ(λγ1)ψ(x)(r2)}2.(φ(µγ1)ψ(x)(r1 + r2) = φ(µγ1(x))(r1 + r2)

= µγ1(x)(r3 + r4)≤ max{µγ1(x)(r3), µγ1(x)(r4)}= max{φ(µγ1(x))(r1), φ(µγ1(x))(r2)}= max{φ(µγ1)ψ(x)(r1), φ(µγ1)ψ(x)(r2)}3.(φ(δγ1)ψ(x)(r1 + r2) = φ(δγ1(x))(r1 + r2)

= δγ1(x)(r3 + r4)≤ max{δγ1(x)(r3), δγ1(x)(r4)}= max{φ(δγ1(x))(r1), φ(δγ1(x))(r2)}= max{φ(δγ1)ψ(x)(r1), φ(δγ1)ψ(x)(r2)}4.(φ(λγ1))ψ(x)(r1αr2) = φ(λγ1(x))(r1αr2)

= λγ1(x)(r3αr4)

≥ min{λγ1(x)(r3), λγ1(x)(r4)}= min{φ(λγ1(x))(r1), φ(λγ1(x))(r2)}= min{φ(λγ1)ψ(x)(r1), φ(λγ1)ψ(x)(r2)}5.(φ(µγ1))ψ(x)(r1αr2) = φ(µγ1(x))(r1αr2)

= µγ1(x)(r3αr4)

≤ max{µγ1(x)(r3), µγ1(x)(r4)}= max{φ(µγ1(x))(r1), φ(µγ1(x))(r2)}= max{φ(µγ1)ψ(x)(r1), φ(µγ1)ψ(x)(r2)}

WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.23

Ahlam Fallatah, Mourad Oqla Massa’deh, Abd Ulazeez Alkouri

E-ISSN: 2224-2880 243 Volume 19, 2020

6.(φ(δγ1))ψ(x)(r1αr2) = φ(δγ1(x))(r1αr2)

= δγ1(x)(r3αr4)

≤ max{δγ1(x)(r3), δγ1(x)(r4)}= max{φ(δγ1(x))(r1), φ(δγ1(x))(r2)}= max{φ(δγ1)ψ(x)(r1), φ(δγ1)ψ(x)(r2)}.

Then φ(γ1)y is a tripolar fuzzy Γ−subsemiringof R2. Hence (φ(γ1), X2) is a tripolar fuzzy softΓ−semiring over R2.

Theorem 29. If R1, R2 are two Γ−semirings, φ :R1 → R2 is a Γ−semiring homomorphism,(γ1, X1), (γ2, X2) are tripolar fuzzy soft Γ−semiringsover R1 and (γ1, X1) is a tripolar fuzzy softΓ−subsemiring of (γ2, X2). Then (φ(γ1), X1) and(φ(γ2), X2) are tripolar fuzzy soft Γ−subsemiringsover R2 and (φ(γ1), X1) is a tripolar fuzzy softΓ−subsemiring of (φ(γ2), X2).

Proof. Since (φ(γ1))x = φ(γ1(x)) is a tripolar fuzzyΓ−subsemiring of R2 for all x ∈ X1 and (φ(γ2))y =φ(γ2(y)) is a tripolar fuzzy Γ−subsemiring of R2

for all y ∈ X2. Hence (φ(γ1), X1) and (φ(γ2), X2)are tripolar fuzzy soft Γ−semiring over R2. Since(γ1, X1) is a tripolar fuzzy soft Γ−subsemiring of(γ2, X2). And γ1(x) is a tripolar fuzzy subsemir-ing of γ2(x). Hence φ(γ1(x)) is a tripolar fuzzyΓ−subsemiring of φ(γ2(x)) for all x ∈ X1. Therefore(φ(γ1), X1) is a tripolar fuzzy soft Γ−subsemiring of(φ(γ2), X2).

Theorem 30. If (γ1, X) and (γ2, Y ) are tripolar fuzzysoft Γ−semirings over R1 and R2 respectively and(φ, ψ) is a tripolar fuzzy soft homomorphism from(γ1, X) onto (γ2, Y ) then the pre-image of (γ2, Y ) un-der tripolar fuzzy soft Γ−semiring homomorphism isa tripolar fuzzy soft Γ−subsemiring of (γ1, X) overR1.

Proof. By Definition 22 (φ, ψ)−1(γ2, Y ) =(φ−1(γ2), ψ

−1(Y )). Define (φ−1(γ2))x(r1) =γ2ψ(x)(φ(r1)) for all r1 ∈ R1 and x ∈ ψ−1(Y ). Takev, w ∈ R1 and α ∈ Γ. Then

1.(φ−1(λγ2))x(v + w) = λγ2ψ(x)(φ(v + w))

= λγ2ψ(x)(φ(v) + φ(w))

≥ min{λγ2ψ(x)(φ(v)), λγ2ψ(x)

(φ(w))}= min{φ−1(λγ2)x(v), φ−1(λγ2)x(w)}2.(φ−1(µγ2))x(v + w) = µγ2ψ(x)

(φ(v + w))

= µγ2ψ(x)(φ(v) + φ(w))

≤ max{µγ2ψ(x)(φ(v)), µγ2ψ(x)

(φ(w))}= max{φ−1(µγ2)x(v), φ−1(µγ2)x(w)}3.(φ−1(δγ2))x(v + w) = δγ2ψ(x)

(φ(v + w))

= δγ2ψ(x)(φ(v) + φ(w))

≤ max{δγ2ψ(x)(φ(v)), δγ2ψ(x)

(φ(w))}= max{φ−1(δγ2)x(v), φ−1(δγ2)x(w)}

4.(φ−1(λγ2))x(vαw) = λγ2ψ(x)(φ(vαw))

= λγ2ψ(x)(φ(v)αφ(w))

≥ min{λγ2ψ(x)φ(v), λγ2ψ(x)

φ(w)}= min{φ−1(λγ2)x(v), φ−1(λγ2)x(w)}5.(φ−1(µγ2))x(vαw) = µγ2ψ(x)

(φ(vαw))

= µγ2ψ(x)(φ(v)αφ(w))

≤ max{µγ2ψ(x)φ(v), µγ2ψ(x)

φ(w)}= max{φ−1(µγ2)x(v), φ−1(µγ2)x(w)}6.(φ−1(δγ2))x(vαw) = δγ2ψ(x)

(φ(vαw))

= δγ2ψ(x)(φ(v)αφ(w))

≤ max{δγ2ψ(x)φ(v), δγ2ψ(x)

φ(w)}= max{φ−1(δγ2)x(v), φ−1(δγ2)x(w)}

Thus (φ−1(γ2))x is a tripolar fuzzyΓ−subsemiring of R1 for all x ∈ φ−1(Y ). There-fore ((φ−1(γ2)), (ψ

−1(Y )) is a tripolar fuzzy softΓ−subsemiring of (γ1, X) over R1.

4 ConclusionIn this paper, we studied the concept of tripolar fuzzysoft Γ− semiring homomorphism and discussed someproperties of homomorphic image and pre image oftripolar fuzzy soft Γ−semiring. These concepts arebasic supporting structures for development the the-ory of soft set. This work can be extended to the prop-erties of different notions of kernel of tripolar fuzzysoft Γ− semiring homomorphism, tripolar fuzzy softfilters over Γ− semirings and tripolar fuzzy soft primeand maximal ideals.

Acknowledgements: Authors are thankful to the ref-eree for there valuable suggestions.

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E-ISSN: 2224-2880 244 Volume 19, 2020

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WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.23

Ahlam Fallatah, Mourad Oqla Massa’deh, Abd Ulazeez Alkouri

E-ISSN: 2224-2880 246 Volume 19, 2020