# Algebraic Hyperstructures of Vague Soft Sets Associated with Hyperrings and Hyperideals

Authors: Ganeshsree Selvachandran; Abdul Razak Salleh
Journal: The Scientific World Journal
 (2015)

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## Abstract

We apply the classical theory of hyperrings to vague soft sets to derive the concepts of vague soft hyperrings, vague soft hyperideals, and vague soft hyperring homomorphism. The properties and structural characteristics of these concepts are also studied and discussed. Furthermore, the relationship between the concepts introduced here and the corresponding concepts in classical hyperring theory and soft hyperring theory is studied and investigated.


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## Body

## 1. Introduction

Hyperstructure theory which is a generalization of the notion of classical algebraic structures was introduced by Marty in 1934 (see [1]). Since its inception, hyperstructure theory has seen tremendous development. Corsini (see [2]) introduced the concept of a hyperring and gave the definition of the most general form of hyperring, besides being responsible for introducing the notion of hyperring homomorphism. In 1994, Vougiouklis (see [3]) introduced another type of hyperrings called the Hν-ring which is a generalization of the notion of hyperring introduced by [2]. Vougiouklis (see [3]) also introduced the notion of Hν-subring and left and right Hν-ideal of a Hν-ring.The theory of hyperstructures has also been actively applied to various mathematical theories such as the theory of fuzzy sets introduced by Zadeh in 1965 (see [4]), intuitionistic fuzzy theory introduced by Atanassov in 1986 (see [5]), and soft set theory introduced by Molodtsov in 1999 (see [6]), to produce several important concepts and theories. The study of fuzzy algebra began with the introduction of the notion of a fuzzy subgroup of a group by Rosenfeld (see [7]). Consequently, this led to the study of fuzzy hyperalgebra beginning with the introduction of the notion of a fuzzy subhyperring of a hyperring (resp., fuzzy Hν-subrings) of a hyperring (resp., Hν-ring) and fuzzy hyperideals (resp., fuzzy Hν-ideals) of hyperrings (resp., Hν-rings). The notion of fuzzy hyperideals introduced by Davvaz (see [8]) is a generalization of the concept of Liu’s (see [9]) definition of a fuzzy ideal of a ring.Soft set theory is a general mathematical tool which was designed to deal with uncertainties and is widely acknowledged as an effective alternative to the classical mathematical tools that were used to deal with uncertainties and imprecision. Prior to the introduction of soft set theory, mathematical theories such as the theory of fuzzy sets, rough sets, and probability theory were used as tools to deal with uncertainties, imprecision, and vagueness. However, soft set theory has been proven to be a superior alternative to these classical theories as it is free from the inherent difficulties that affect these classical mathematical tools. Soft set theory has been successfully applied in various areas of mathematics such as in classical algebra and hyperalgebra, fuzzy algebra, and fuzzy hyperalgebra to develop various algebraic structures. However, an inherent weakness of soft set theory is that it is difficult to be used to represent the vagueness of problem parameters, particularly in problem-solving and decision making contexts. To overcome this disadvantage, various generalizations of soft sets such as fuzzy soft sets, vague soft sets, and intuitionistic fuzzy soft sets were introduced as better alternatives to the concept of soft sets. In the context of this paper, the notion of vague soft sets is used as a base to develop the initial theory of vague soft hyperrings.In this paper, we develop the initial theory of vague soft hyperrings in Rosenfeld’s sense as a continuation to the theory of vague soft hypergroups introduced in [10]. Subsequently some of the fundamental properties and structural characteristics of these concepts are studied and discussed. Lastly, we prove that there exists a one-to-one correspondence between some of these concepts and their corresponding concepts in soft hyperring theory as well as classical hyperring theory.


## 2. Preliminaries

In this section, some well-known and useful definitions pertaining to the development of the theory of vague soft hyperrings introduced here will be presented. These definitions and concepts will be used throughout this chapter.Definition 1 (see [6]).
A pair(F,A) is called asoft set over U, where F is a mapping given by F:A→P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε∈A, F(ε) may be considered as the set of ε-elements of the soft set F,A or as the ε-approximate elements of the soft set.Definition 2 (see [6]).
For a soft set(F,A), the set SuppF,A=x∈A:Fx≠∅ is called thesupport of the soft set (F,A). Thus anull soft set is a soft set with an empty support and a soft set (F,A) is said to be nonnull if SuppF,A≠∅.Definition 3 (see [11]).
LetX be a space of points (objects) with a generic element of X denoted by x. Avague setV in X is characterized by a truth membership function tV:X→[0,1] and a false membership function fV:X→[0,1]. The value tV(x) is a lower bound on the grade of membership of x derived from the evidence for x and fV(x) is a lower bound on the negation of x derived from the evidence against x. The values tV(x) and fV(x) both associate a real number in the interval [0,1] with each point in X, where tVx+fV(x)≤1. This approach bounds the grade of membership of x to a subinterval [tVx,1-fVx] of 0,1. Hence a vague set is a form of fuzzy set, albeit a more accurate form of fuzzy set.Definition 4 (see [12]).
A pair(F^,A) is called avague soft set over U where F^ is a mapping given by F^:A→V(U) and V(U) is the power set of vague sets over U. In other words, a vague soft set over U is a parameterized family of vague sets of the universe U. Every set F^(e), for all e∈A, from this family may be considered as the set of e-approximate elements of the vague soft set (F^,A). Hence the vague soft set (F^,A) can be viewed as consisting of a collection of approximations of the following form:(1)F^,A=F^axi:i=1,2,3,…=tF^eixi,1-fF^eixixi:i=1,2,3,…,where F^axi is a subset of F^,A for all e∈A and for all x∈U.Example 5 (see [12]).
Consider a vague soft setF^,E, where U is a set of six houses under consideration of a decision maker to purchase which is denoted by U=h1,h2,h3,h4,h5,h6 and E is a parameter set, where E=e1,e2,e3,e4,e5 = {expensive, beautiful, wooden, cheap, inthegreensurroundings}. The vague soft set F^,E describes the “attractiveness of the houses” to this decision maker.
Suppose that(2)
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.Definition 6 (see [10]).
Let(F^,A) be a vague soft set over X. Thesupport of (F^,A) denoted by Supp(F^,A) is defined as(4)
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It is to be noted that anull vague soft set is a vague soft set where both the truth and false membership functions are equal to zero. Therefore, a vague soft set (F^,A) is said to be nonnull if Supp(F^,A)≠∅.Definition 7 (see [12]).
Let(F^,A) and (G^,B) be vague soft sets over U. The set (F^,A) is called avague soft subset of G^,B if A⊆B and, for every a∈A, F^(a) and G^(a) are the same approximation. In this case, (G^,B) is called thevague soft superset of (F^,A) and this relationship is denoted by F^,A⊆G^,B.Definition 8 (see [12]).
IfF^,A and G^,B are two vague soft sets over U, theintersection of F^,A and G^,B denoted as “F^,A∩~G^,B” is defined by F^,A∩~G^,B=H^,A×B, where(5)tH^cx=tF^cx,c∈A-B,tG^cx,c∈B-A,min⁡tF^cx,tG^cx,c∈B∩A,1-fH^cx=1-fF^cx,c∈A-B,1-fG^cx,c∈B-A,min⁡1-fF^cx,1-fG^cx,c∈B∩A,for all c∈SuppH^,C and for all x∈U.Definition 9 (see [12]).
LetF^,A and G^,B be vague soft sets over U. Then “F^,AANDG^,B” denoted by F^,A∧~G^,B is defined as F^,A∧~G^,B=H^,A×B, where(6)tH^α,βx=min⁡tF^αx,tG^βx,1-fH^α,βx=min⁡1-fF^αx,1-fG^βx,for every α,β∈A×B and x∈U, where tH^(α,β) and fH^(α,β) are the truth membership function and false membership function of H^(α,β), respectively. This relationship can be written as H^(α,β)x=F^αx∩G^βx, where ∩ represents the intersection operation between two vague soft sets as defined in the case of c∈B∩A in Definition 8.Definition 10 (see [10]).
LetF^,A be a vague soft set over U. Then, for all α,β∈0,1, where α≤β, the α,β-cut or thevague soft(α,β)-cut of F^,A is a subset of U which is as defined below:(7)
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Ifα=β, then it is called thevague soft  (α,α)-cut of F^,A or the α-level set of F^,A, denoted by (F^,A)(α,α), is a subset of U which is as defined below:(8)
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Let(F^,A) be a vague soft set over X and let G be a nonnull subset of X. Then (F^,A)G is called avague soft characteristic set over G in [0,1] and the lower bound and upper bound of F^aG are as defined below:(9)tF^aGx=1-fF^aGx=s,ifx∈G,w,otherwise,where F^aG is a subset of (F^,A)G, x∈X,s,w∈[0,1], and s>w.This paper pertains to the application of classical hyperring theory to the concept of vague soft sets. Furthermore, the relationships that exist between the concepts introduced in this paper and the corresponding concepts in classical hyperring theory are also studied and discussed. The definitions of several important concepts pertaining to the theory of hyperstructures are presented below.Definition 12 (see [1]).
AhypergroupH,∘ is a set H equipped with an associative hyperoperation ∘:H×H→P(H) which satisfies the reproduction axiom given by x∘H=H∘x=H for all x in H.Definition 13 (see [13]).
A hyperstructureH,∘ is called an Hv-group if the following axioms hold:(i)
x∘y∘z∩(x∘y)∘z≠∅ for all x,y,z∈H,(ii)
x∘H=H∘x=H for all x in H.If H,∘ only satisfies the first axiom, then it is called a Hv-semigroup.Definition 14 (see [1]).
A subsetK of H is called asubhypergroup if K,∘ is a hypergroup.Definition 15 (see [3]).
AHν-ring is a multivalued system (R,+,∘) which satisfies the following axioms:(i)
(R,+) is a Hν-group,(ii)
(R,∘) is a Hν-semigroup,(iii)
the hyperoperation “∘” is weak distributive over the hyperoperation “+”; that is, for each x,y,z∈R the following holds true:(10)x∘y+z∩x∘y+x∘z≠ϕ,x+y∘z∩x∘z+y∘z≠ϕ.Definition 16 (see [3]).
A nonempty subsetR′ of R is called asubhyperring of R,+,∘ if R′,+ is a subhypergroup of R,+ and for all x,y,z∈R′, x∘y∈P∗R′, where P∗R′ is the set of all nonempty subsets of R′.Definition 17 (see [3]).
LetR be a Hv-ring. A nonempty subset I of R is called aleft(resp.,right) Hv-ideal if the following axioms hold:(i)
(I,+) is a Hv-subgroup of R,+,(ii)
R∘I⊆I (resp.I∘R⊆I).If I is both left and right Hv-ideals of R, then I is said to be a Hv-ideal of R.


## 3. Vague Soft Hyperrings

In this section, the concept of vague soft hyperrings in Rosenfeld’s sense is introduced as a continuation to the theory of vague soft hypergroups introduced by [10]. We derive the definition of vague soft hyperrings by applying the theory of hyperrings to vague soft sets. The properties of this concept are studied and discussed.From now on, letR,+,∘ be a hyperring (or Hν-ring), let E be a set of parameters, and let A⊆E. For the sake of simplicity, we will denote R,+,∘ by R.Definition 18 (see [14]).
Let(F,A) be a nonnull soft set over R. Then (F,A) is called asoft hyperring over R if F(x) is a subhyperring of R for all x∈SuppF^,A.Definition 19 (see [10]).
Let(F^,A) be a nonnull vague soft set over a hypergroup H,∘. Then (F^,A) is called avague soft semihypergroup over H,∘ if the following conditions are satisfied for all a∈SuppF^,A:
For allx,y∈H,∘, min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x∘y and min⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x∘y⁡; that is, min⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y.Definition 20 (see [10]).
Let(F^,A) be a nonnull vague soft set over H. Then (F^,A) is called avague soft hypergroup over H if the following conditions are satisfied for all a∈SuppF^,A:(i)
For allx,y∈H,min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x∘y and min⁡{1-fF^ax,1-fF^a(y)}≤inf⁡{1-fF^az:z∈x∘y}; that is, min⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y.(ii)
For allw,x∈H, there exists y∈H such that x∈w∘y and min⁡tF^aw,tF^ax≤tF^a(y) and min⁡1-fF^aw,1-fF^ax≤1-fF^a(y); that is, min⁡F^aw,F^ax≤F^ay.(iii)
For allw,x∈H, there exists z∈H such that x∈z∘w and min⁡tF^aw,tF^ax≤tF^a(z) and min⁡1-fF^aw,1-fF^ax≤1-fF^a(z); that is, min⁡F^aw,F^ax≤F^az.Conditions (ii) and (iii) represent the left and right reproduction axioms, respectively. Then F^a is a nonnullvague subhypergroup of H,∘ (in Rosenfeld’s sense) for all a∈SuppF^,A.Definition 21.
Let(F^,A) be a nonnull vague soft set over R. Then (F^,A) is called avague soft hyperring over R if, for all a∈SuppF^,A, the following conditions are satisfied: (i)
For allx,y∈R, min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x+y and min⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x+y, which means min⁡F^ax,F^ay≤inf⁡F^az:z∈x+y.(ii)
For allw,x∈R, there exists y∈R such that x∈w+y and min⁡tF^aw,tF^ax≤tF^a(y) and min⁡1-fF^aw,1-fF^ax≤1-fF^ay; that is, min⁡F^aw,F^ax≤F^ay.(iii)
For allw,x∈R, there exists z∈R such that x∈z+w and min⁡tF^aw,tF^ax≤tF^a(z) and min⁡1-fF^aw,1-fF^ax≤1-fF^az; that is, min⁡F^aw,F^ax≤F^az.(iv)
For allx,y∈R, min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x∘y and min⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x∘y; that is, min⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y.That is, F^a is a nonnullvague subhyperring (in Rosenfeld’s sense) of R for each a∈SuppF^,A.
Conditions (i), (ii), and (iii) indicate thatF^a is a vague subhypergroup of (R,+) while condition (iv) indicates that F^a is a vague subsemihypergroup of R,∘. Therefore it can be concluded that (F^,A) is avague soft hyperring over R if (F^,A) is a vague soft hypergroup over (R,+) and a vague soft semihypergroup over R,∘.Theorem 22.
Let(F^,A) be a nonnull vague soft set over R. Then the necessary and sufficient condition for (F^,A) to be a vague soft hyperring over R is for (F^,A) to be a vague soft semihypergroup over R,∘ and also a vague soft hypergroup over R,+.Proof.
The proof is straightforward by Definition21.Example 23.
A vague soft hyperring(F^,A) for which A is a singleton is a vague subhyperring. Hence vague subhyperrings and classical hyperrings are a particular type of vague soft hyperrings.Example 24.
Let(F^,A) be a nonnull vague soft set over a hyperring R which is as defined below:(11)
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A
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LetF^a be a vague subhyperring of a hyperring R. This means that F^a satisfies the axioms stated in Definition 21. Now consider the family of α-level sets for F^a given by(12)F^,Aα=x∈H:F^ax≥α,for all a∈A and α∈0,1. Then, for all α∈0,1, F^a is a subhyperring of R. Hence F^,0,1 is a soft hyperring over R.Example 26.
Any soft hyperring is a vague subhyperring since any characteristic function of a subhyperring is a vague subhyperring.Proposition 27.
Let(F^,A) and (G^,B) be vague soft hyperrings over R. Then F^,A∧~(G^,B) is a vague soft hyperring over R if it is nonnull.Proof.
Suppose that(F^,A) and (G^,B) are vague soft hyperrings over R. Now let F^,A∧~G^,B=H^,A×B. Then for all (α,β)∈SuppH^,A×B and for all x∈R we have(13)tH^α,βx=min⁡tF^αx,tG^βx,1-fH^α,βx=min⁡1-fF^αx,1-fG^βx;that is,(14)H^α,βx=F^αx∩G^βx.Since (F^,A) and (G^,B) are both nonnull and (H^,A×B) represents the basic intersection between (F^,A) and (G^,B) it follows that (H^,A×B) must be nonnull. For the sake of similarity, we only show the proof for the truth membership function of H^(α,β) given by tH^(α,β). The proof for 1-fH^(α,β) can be derived in the same manner.
Hence for allx,y∈R and (α,β)∈SuppH^,A×B we obtain(15)min⁡tH^α,βx,tH^α,βy=min⁡tF^αx∩tG^βx,tF^αy∩tG^βy≤min⁡tF^αx∩tF^αy,tG^βx∩tG^βy≤min⁡tF^αx,tF^αy∩min⁡tG^βx,tG^βy≤inf⁡tF^αz:z∈x+y∩inf⁡tG^βz:z∈x+y≤inf⁡tF^αz∩tG^βz:z∈x+y=inf⁡tH^α,β:z∈x+y.Furthermore for all w,x∈R we have(16)min⁡tH^α,βw,tH^α,βx=min⁡tF^αw∩tG^βw,tF^αx∩tG^βx≤min⁡tF^αw∩tF^αx,tG^βw∩tG^βx≤min⁡tF^αw,tF^αx∩min⁡tG^βw,tG^βx≤tF^αy∩tG^βy=tH^α,βy,where y∈R such that x∈w+y. Similarly, it can also be proved that min⁡tH^α,βw,tH^α,βx≤tH^α,βz, where z∈R such that x∈z+w. Therefore it has been proved that H^(α,β) is a vague subhypergroup of (R,+), which means that (H^,A×B) is a vague soft hypergroup over R,+.
Lastly for allx,y∈R we have(17)min⁡tH^α,βx,tH^α,βy=min⁡tF^αx∩tG^βx,tF^αy∩tG^βy≤min⁡tF^αx∩tF^αy,tG^βx∩tG^βy≤min⁡tF^αx,tF^αy∩min⁡tG^βx,tG^βy≤inf⁡tF^αz:z∈x∘y∩inf⁡tG^βz:z∈x∘y≤inf⁡tF^αz∩tG^βz:z∈x∘y=inf⁡tH^α,β:z∈x∘y.This proves that H^(α,β) is a vague subhyperring of R for all α,β∈SuppH^,A×B. Hence H^,A×B=F^,A∧~(G^,B) is a vague soft hyperring over R.Theorem 28.
Let(F^,A) be a vague soft set over R. Then (F^,A) is a vague soft hyperring over R if and only if, for every t∈0,1, (F^,A)(t,t) is a soft hyperring over R.Proof.
(
⇒
) Since R is a hyperring, (R,+) is a commutative hypergroup and (R,∘) is a semihypergroup. Let (F^,A) be a vague soft hyperring over R,+,∘ and t∈0,1. Then, for all a∈SuppF^,A, the corresponding F^a is a vague subhyperring of R. Now let x,y∈(F^a)(t,t). Then tF^ax,tF^a(y)≥t and 1-fF^ax, 1-fF^ay≥t, which means that F^ax≥t and F^ay≥t. Furthermore since F^a is a vague subhypergroup of R,+, we have inf⁡F^az:z∈x+y≥min⁡F^ax,F^ay≥t. This implies that z∈(F^a)(t,t) and therefore, for every z∈x+y, we obtain x+y⊆(F^a)(t,t). As such, for every z∈(F^a)(t,t), we obtain z+(F^a)(t,t)⊆(F^a)(t,t). Now let x,z∈(F^a)(t,t). Then tF^ax,tF^a(z)≥t and 1-fF^ax,1-fF^az≥t, which means that F^ax≥t and F^az≥t. Due to the fact that F^a is a vague subhypergroup of R,+, there exists y∈R such that x∈z+y and min⁡F^az,F^ax≤F^ay. Since x,z∈(F^a)(t,t), we obtain min⁡F^az,F^ax≥min⁡t,t=t and this implies that F^a(y)≥t and as a result, y∈(F^a)(t,t). Therefore, we obtain (F^a)(t,t)⊆z+(F^a)(t,t). As such, we obtain z+(F^a)(t,t)=(F^a)(t,t). As a result, (F^a)(t,t) is a subhypergroup of R,+. Hence (F^,A)(t,t) is a soft hypergroup over R,+. Furthermore, since (R,∘) is a semihypergroup, F^a is a nonnull vague subsemihypergroup of (R,∘) for all a∈SuppF^,A and min⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y. Now let x,y∈(F^a)(t,t). Then tF^ax,tF^a(y)≥t and 1-fF^ax,1-fF^ay≥t, which means that F^ax≥t and F^ay≥t. Therefore it follows that min⁡F^ax,F^ay≥min⁡t,t=t and this implies that inf⁡F^az:z∈x∘y≥t. This in turn implies that z∈(F^a)(t,t) and consequently x∘y⊆(F^a)(t,t). Therefore, for every x,y∈(F^a)(t,t), we obtain x∘y∈P∗R. As such, (F^a)(t,t) is a subhyperring of R. Hence (F^,A)(t,t) is a soft hyperring over R.
(
⇐
) Assume that (F^,A)(t,t) is a soft hyperring over R. Thus for every t∈0,1, (F^a)(t,t) is a nonnull subhyperring of R. Now for every x,y∈R, we have min⁡F^ax,F^ay≤F^a(x) or min⁡F^ax,F^ay≤F^ay. So if we let min⁡F^ax,F^ay=t0, then x,y∈(F^a)t0,t0 and therefore x+y⊆(F^a)t0,t0. Therefore, for every z∈x+y, we have F^a(z)≥t0 which implies that min⁡F^ax,F^ay≤inf⁡F^az:z∈x+y. As such, condition (i) of Definition 21 is verified. Next, let w,x∈R and min⁡F^aw,F^ax=t1. This implies that w,x∈(F^a)t1,t1 which means that there exists y∈(F^a)t1,t1 such that x∈w∘y. Since y∈(F^a)t1,t1, we have tF^a(y)≥t1 and 1-tF^a(y)≥t1 which means that F^a(y)≥t1 and thus min⁡F^aw,F^ax=t1≤F^a(y) which means that min⁡F^aw,F^ax≤F^ay. Therefore condition (ii) of Definition 21 has been verified. Condition (iii) of Definition 21 can be verified in a similar manner. Thus it has been proven that F^a is a vague subhypergroup of R,+. Now (F^a)(t,t) is a subsemihypergroup of the semihypergroup R,∘. Let min⁡F^ax,F^ay=t2 for every x,y∈R. Thus x,y∈(F^a)t2,t2 and therefore we have x∘y⊆(F^a)t2,t2. Thus, for every z∈x∘y, we obtain F^az≥t2. As a result, we obtain min⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y which proves that condition (iv) of Definition 21 has been verified. As such, F^a is a vague subhyperring of R and therefore (F^,A) is a vague soft hyperring over R.Theorem28 proves that there exists a one-to-one correspondence between vague soft hyperrings and soft hyperrings over a hyperring.Theorem 29.
Let(F^,A) be a vague soft hyperring over R. Then, for each α,β∈0,1, (F^,A)(α,β) is a subhyperring of R.Proof.
Let(F^,A) be a vague soft hyperring over R,+,∘. Then for each a∈SuppF^,A, the corresponding F^a is a vague subhyperring of R. Now let x,y∈(F^,A)(α,β). Then tF^ax≥α, 1-fF^a(x)≥β and tF^ay≥α, 1-fF^ay≥β. Thus the following is obtained:(18)min⁡tF^ax,tF^ay≥min⁡α,α=α,min⁡1-fF^ax,1-fF^ay≥min⁡β,β=β.However, since F^a is a vague subhypergroup of R,+, we have inf⁡tF^az:z∈x+y≥α and inf⁡1-fF^az:z∈x+y≥β. Therefore, for every z∈x+y, we obtain z∈(F^,A)(α,β) and thus x+y⊆(F^,A)(α,β). As such, it can be concluded that z+(F^,A)(α,β)⊆(F^,A)(α,β). Furthermore, since F^a is a vague subhypergroup of R,+, there exists y∈R such that x∈z+y and min⁡F^ax,F^az≤F^ay. Now let x,z∈(F^,A)(α,β). Then the following is obtained:(19)min⁡tF^ax,tF^az≥min⁡α,α=α,min⁡1-fF^ax,1-fF^ay≥min⁡β,β=β.This means that tF^a(y)≥α and 1-fF^a(y)≥β which implies that y∈(F^,A)(α,β). Therefore for all x,z∈(F^,A)(α,β), we have x∈z+y and y∈(F^,A)(α,β). As such, it can be concluded that (F^,A)(α,β)⊆z+(F^,A)(α,β) and this implies that, for any z∈(F^,A)(α,β), z+(F^,A)(α,β)=(F^,A)(α,β). Hence it has been proven that (F^,A)(α,β) is a subhypergroup of R,+. Furthermore (R,∘) is a semihypergroup and F^a is also a vague subsemihypergroup of (R,∘) for all a∈SuppF^,A. Now let x,y∈(F^,A)(α,β). Then min⁡tF^ax,tF^ay≥min⁡α,α=α and min⁡1-fF^ax,1-fF^ay≥min⁡β,β=β and this results in(20)inf⁡tF^az:z∈x∘y≥min⁡tF^ax,tF^ay≥α,inf⁡1-fF^az:z∈x∘y≥min⁡1-fF^ax,1-fF^ay≥β.This implies that, for all z∈x∘y, we obtain z∈(F^,A)(α,β) and therefore x∘y⊆(F^,A)(α,β). Thus, for every x,y∈(F^,A)(α,β), we obtain x∘y∈P∗F^,Aα,β. As such, (F^,A)(α,β) is a subsemihypergroup of the semihypergroup R,∘. Hence (F^,A)(α,β) is a subhyperring of R.As a result of Theorem29, we obtain Corollary 30.Corollary 30.
Let(F^,A) be a vague soft hyperring over R. Then, for each α∈0,1, (F^,A)(α,α) is a subhyperring of R.Theorem 31.
LetT be a nonempty subset of R, let (F^,A) be a nonnull vague soft set over R, and let (F^,A)T be a vague soft characteristic set over T. If (F^,A) is a vague soft hyperring over R, then T is a subhyperring of R.Proof.
Let(F^,A) be a vague soft hyperring over R. Then, by Definition 21, (F^,A) is a vague soft hypergroup over (R,+) and a vague soft semihypergroup over (R,∘) and for each a∈SuppF^,A, the corresponding vague subset F^a is a vague subhyperring of R. Furthermore, since (R,+,∘) is a hyperring, (R,+) is a commutative hypergroup and (R,∘) is a semihypergroup. Now let x,y∈T. Then tF^aTx=tF^aTy=s and fF^aTx=fF^aTy=w, which means that F^aTx=F^aTy=s. Therefore, min⁡F^aTx,F^aTy=min⁡s,s=s and so inf⁡F^aTz:z∈x+y≥s (∵F^a is a vague subhyperring of R and a vague subhypergroup of (R,+)). Thus, for every z∈x+y, we have z∈T and so it follows that x+y⊆T. As such, for every z∈T, we have z+T⊆T.
Now letx, z∈T. Then tF^aTx=tF^aTz=s and fF^aTx=fF^aTz=w and so F^aTx=F^aTz=s. Since F^a is a vague subhypergroup of (R,+) for all a∈SuppF^,A, there exists y∈R such that x∈z+y and min⁡F^aTx,F^aTz≤F^aTy. Thus we obtain min⁡F^aTx,F^aTz=min⁡s,s=s and therefore F^aT(y)≥s which implies that y∈T. Since x,y∈T, this implies that T⊆z+T and so we have z+T=T for all z∈T. Therefore it has been proven that (T,+) is a subhypergroup of R,+. Let x,y∈T. Then tF^aTx=tF^aTy=s and fF^aTx=fF^aTy=w and thus F^aTx=F^aTy=s. Now since R,∘ is a semihypergroup, F^a is a vague subhypergroup of (R,∘) for all a∈SuppF^,A. Thus it follows that min⁡F^aTx,F^aTy=min⁡s,s=s and therefore inf⁡F^aTz:z∈x∘y≥s too. As a result, for every z∈x∘y, we have z∈T and therefore x∘y⊆T. As such, for every x,y∈T, we have x∘y∈P∗T. Hence T is a subhyperring of R.Theorem 32.
LetT be a nonempty subset of R, let (F^,A) be a nonnull vague soft set over R, and let (F^,A)T be a vague soft characteristic set over T. Then (F^,A) is a vague soft hyperring over R if and only if T is a subhyperring of R.Proof.
The proof is similar to that of Theorem31.Theorem32 implies that there exists a one-to-one correspondence between vague soft hyperrings and classical subhyperrings of a hyperring.


## 4. Vague Soft Hyperideals

The ideal of a ring is an important concept in the study of rings. The concept of a fuzzy ideal of a ring was introduced by Liu (see [9]). Davvaz (see [8]) introduced the notion of a fuzzy hyperideal (resp., fuzzy Hν-ideal) of a hyperring (resp., Hν-ring). In this section the notion of vague soft hyperideals of a hyperring is defined using Rosenfeld’s approach.Definition 33.
Let(F,A) be a nonnull soft set over R. Then (F,A) is called asoft left (resp.,right)hyperidealof R if F(x) is a left (resp., right) hyperideal of R for all x∈SuppF^,A.Definition 34.
Let(F,A) be a nonnull soft set over R. Then (F,A) is called asoft hyperideal of R if F(x) is a hyperideal of R; that is, F(x) is both right and left hyperideals of R for all x∈SuppF^,A.Definition 35.
Let(F^,A) be a nonnull vague soft set over R. Then (F^,A) is called avague soft left (resp.,right)hyperideal of R if the following axioms hold for all a∈SuppF^,A:(i)
for allx,y∈R, min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x+y and min⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x+y, which means that min⁡F^ax,F^ay≤inf⁡F^az:z∈x+y,(ii)
for allw,x∈R, there exists y∈R such that x∈w+y and min⁡tF^aw,tF^ax≤tF^a(y) and min⁡1-fF^aw,1-fF^ax≤1-fF^ay; that is, min⁡F^aw,F^ax≤F^ay,(iii)
for allw,x∈R, there exists z∈R such that x∈z+w and min⁡tF^aw,tF^ax≤tF^a(z) and min⁡1-fF^aw,1-fF^ax≤1-fF^az; that is, min⁡F^aw,F^ax≤F^az,(iv)
for allx,y∈R, tF^ay≤inf⁡tF^az:z∈x∘y (resp., tF^ax≤inf⁡{tF^az:z∈x∘y}) and 1-fF^ay≤inf⁡1-fF^az:z∈x∘y (resp., 1-fF^ax≤inf⁡1-fF^az:z∈x∘y); that is, F^ay≤inf⁡F^az:z∈x∘y (resp., F^ax≤inf⁡F^az:z∈x∘y).That is, F^a is a vague left (resp., right) hyperideal of R (in Rosenfeld’s sense) for all a∈SuppF^,A.
Conditions (i), (ii), and (iii) prove thatF^a is a vague subhypergroup of R,+. This means that (F^,A) is avague soft left (resp.,right)hyperideal of R if, for all a∈SuppF^,A, F^a is a vague subhypergroup of (R,+) and F^a also satisfies the condition F^ay≤inf⁡F^az:z∈x∘y (resp., F^ax≤inf⁡F^az:z∈x∘y).Definition 36.
Let(F^,A) be a nonnull vague soft set over R. Then (F^,A) is called avague soft hyperideal of R if the following axioms hold for all a∈SuppF^,A:(i)
for allx,y∈R, min⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x+y and min⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x+y, which means that min⁡F^ax,F^ay≤inf⁡F^az:z∈x+y,(ii)
for allw,x∈R, there exists y∈R such that x∈w+y and min⁡tF^aw,tF^ax≤tF^a(y) and min⁡1-fF^aw,1-fF^ax≤1-fF^ay; that is, min⁡F^aw,F^ax≤F^ay,(iii)
for allw,x∈R, there exists z∈R such that x∈z+w and min⁡tF^aw,tF^ax≤tF^a(z) and min⁡1-fF^aw,1-fF^ax≤1-fF^az; that is, min⁡F^aw,F^ax≤F^az,(iv)
for allx,y∈R, max⁡tF^ax,tF^ay≤inf⁡tF^az:z∈x∘y and max⁡1-fF^ax,1-fF^ay≤inf⁡1-fF^az:z∈x∘y; that is, max⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y.That is, F^a is avague hyperideal of R (in Rosenfeld’s sense) for all a∈SuppF^,A.
Similar to Definition35, conditions (i), (ii), and (iii) of Definition 36 indicate that F^a is a vague subhypergroup of R,+ while condition (iv) is a combination of both the conditions given in Definition 35 (condition (iv)).Theorem 37.
Let(F^,A) be a nonnull vague soft set over R. Then the necessary and sufficient condition for (F^,A) to be a vague soft hyperideal is for (F^,A) to be a vague soft hypergroup of (R,+) and (F^,A) is both a vague soft left hyperideal and a vague soft right hyperideal of R.Proof.
The proof is straightforward by Definitions35 and 36.Proposition 38.
Let(F^,A) and (G^,B) be two vague soft hyperideals of R. Then F^,A∧~(G^,B) is a vague soft hyperideal of R if it is nonnull.Proof.
Suppose that(F^,A) and (G^,B) are vague soft hyperideals of R. Now to prove that H^,A×B is a vague soft hyperideal of R, we have to prove that all the conditions of Definition 36 have been satisfied. However, in Proposition 27, it has been proven that conditions (i), (ii), and (iii) have been satisfied. Therefore it is only necessary to prove that (H^,A×B) satisfies condition (iv) of Definition 36.
Thus for allx,y∈R and α,β∈SuppH^,A×B we have(21)max⁡tH^α,βx,tH^α,βy=max⁡tF^αx∩tG^βx,tF^αy∩tG^βy≤max⁡tF^αx,tF^αy∩max⁡tG^βx,tG^βy≤inf⁡tF^αz:z∈x∘y∩inf⁡tG^βz:z∈x∘y≤inf⁡tF^αz∩tG^βz:z∈x∘y=inf⁡tH^α,β:z∈x∘y.Similarly, it can be proven that the condition max⁡1-fH^α,βx,1-fH^α,βy≤inf⁡1-fH^α,β:z∈x∘y is satisfied. This means that max⁡H^(α,β)x,H^(α,β)y≤inf⁡H^α,βz:z∈x∘y. Thus it has been proven that H^α,β is a vague hyperideal of R for all α,β∈SuppH^,A×B and as such H^,A×B=F^,A∧~(G^,B) is a vague soft hyperideal of R.Theorem 39.
Let(F^,A) be a nonnull vague soft set over R. Then (F^,A) is a vague soft hyperideal of R if and only if, for each t∈0,1, (F^,A)t,t is a soft hyperideal of R.Proof.
(
⇒
) Let (F^,A) be a vague soft hyperideal of R and t∈0,1. Then F^a is a vague subhypergroup of R,+. Now let x,y∈(F^a)(t,t). Then tF^ax,tF^a(y)≥t and 1-fF^ax,1-fF^ay≥t, which means that F^a(x)≥t and F^ay≥t. In Theorem 28, it was proved that if F^a is a vague subhypergroup of R,+, then (F^a)(t,t) is a subhypergroup of R,+. This means that it is only necessary to prove that (F^a)(t,t) satisfies the following conditions:(22)R∘F^at,t⊆F^at,t,F^at,t∘R⊆F^at,t.Now let x∈R and y∈(F^a)(t,t). Then tF^a(y)≥t and 1-fF^a(y)≥t which means that F^ay≥t. Since F^a is a vague hyperideal of R for all a∈SuppF^,A, F^a must satisfy the following condition:(23)tF^ay≤inf⁡tF^az:z∈x∘y,1-fF^ay≤inf⁡1-fF^az:z∈x∘y,which means F^ay≤inf⁡F^az:z∈x∘y. Therefore we obtain inf⁡F^az:z∈x∘y≥F^ay≥t. This proves that z∈(F^a)(t,t) for all z∈x∘y and as a result we obtain x∘y⊆(F^a)(t,t). Thus for every x∈R and y∈(F^a)(t,t), we obtain R∘(F^a)(t,t)⊆(F^a)(t,t) and this proves that (F^a)(t,t) is a left hyperideal of R. This implies that (F^,A)t,t is a soft left hyperideal of R. Similarly, it can been proven that (F^,A)t,t is a soft right hyperideal of R. Thus (F^,A)t,t is a soft right hyperideal of R. Hence (F^,A)t,t is a soft left hyperideal of R and a soft right hyperideal of R which proves that (F^,A)t,t is a soft hyperideal of R.
(
⇐
) Let (F^,A)t,t be a soft hyperideal of R. Then, for each a∈SuppF^,A, (F^a)(t,t) is a nonnull hyperideal of R. Therefore by Definition 16, (F^a)(t,t) is a subhypergroup of R,+. This implies that (F^,A)t,t is a soft hypergroup over R,+. In [10], it was proven that if (F^,A)t,t is a soft hypergroup, then (F^,A) is a vague soft hypergroup. Thus it can be concluded that (F^,A) is a vague soft hypergroup over R,+. Furthermore, since (F^a)(t,t) is both a left hyperideal and a right hyperideal of R, the following conditions are satisfied:(24)R∘F^at,t⊆F^at,t,F^at,t∘R⊆F^at,t.Now let x∈R and y∈(F^a)(t,t) or x∈(F^a)(t,t) and y∈R. Thus we obtain x∘y⊆(F^a)(t,t) and y∘x⊆(F^a)(t,t) which implies that z∈x∘y, where z∈(F^a)(t,t). As such, we obtain inf⁡F^az:z∈x∘y≥t. Now let x,y∈(F^a)(t,t). Then F^a(x)≥t and F^ay≥t. Therefore, we obtain inf⁡F^az:z∈x∘y≥F^a(x)≥t as well as inf⁡F^az:z∈x∘y≥F^a(y)≥t which means that F^a(x)≤inf⁡F^az:z∈x∘y and F^ay≤inf⁡F^az:z∈x∘y. Combining these conditions, we obtain max⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y. Thus condition (iv) of Definition 36 has also been verified. This means that F^a is a vague hyperideal of R and hence it proves that (F^,A) is a vague soft hyperideal of R.Theorem39 proves that there exists a one-to-one correspondence between vague soft hyperideal and the soft hyperideal of a hyperring.Theorem 40.
Let(F^,A) be a nonnull vague soft set over R left (resp., right) hyperideal of R. Then, for all α,β∈0,1, (F^,A)α,β is a left (resp., right) hyperideal of R.Proof.
Let(F^,A) be a vague soft left (resp., right) hyperideal of R. Then, by Definition 35, (F^,A) is a vague soft hypergroup over R,+. Now in Theorem 29 it was proven that if (F^,A) is a vague soft hypergroup over R,+, then (F^,A)α,β is a subhypergroup of R,+. Therefore, in order to prove that (F^,A)α,β is a left (resp., right) hyperideal of R, we only have to prove that R∘(F^,A)α,β⊆(F^,A)α,β (resp., F^,Aα,β∘R⊆F^,Aα,β). Now let x∈R and y∈(F^,A)α,β. Then tF^a(y)≥α and 1-fF^a(y)≥β. Since F^a is a vague left ideal of R for all a∈SuppF^,A, F^a satisfies the conditions tF^a(y)≤inf⁡tF^az:z∈x∘y and 1-fF^ay≤inf⁡1-fF^az:z∈x∘y. Therefore we obtain inf⁡tF^az:z∈x∘y≥tF^a(y)≥α and inf⁡1-fF^az:z∈x∘y≥1-fF^ay≥β which implies that, for all z∈x∘y, z∈F^,Aα,β. As such, we obtain x∘y⊆F^,Aα,β. Thus for every x∈R and y∈F^,Aα,β, we obtain R∘F^,Aα,β⊆F^,Aα,β. Hence F^,Aα,β is a left hyperideal of R. Next let x∈F^,Aα,β and y∈R. Then tF^a(x)≥α and 1-fF^ax≥β. Similarly, we obtain inf⁡tF^az:z∈x∘y≥tF^a(x)≥α and inf⁡1-fF^az:z∈x∘y≥1-fF^ax≥β. This implies that z∈F^,Aα,β for all z∈x∘y and therefore we obtain x∘y⊆F^,Aα,β. As such, for every x∈F^,Aα,β and y∈R, we obtain F^,Aα,β∘R⊆F^,Aα,β which means F^,Aα,β is a right hyperideal of R.Theorem 41.
Let(F^,A) be a nonnull vague soft set over R. Then (F^,A) is a vague soft left (resp., right) hyperideal of R if and only if, for all α,β∈0,1, (F^,A)α,β is a left (resp., right) hyperideal of R.Proof.
The proof is straightforward.Theorem 42.
LetM be a nonempty subset of R and (F^,A)M be a vague soft characteristic set over M. If (F^,A) is a vague soft hyperideal of R, then M is a hyperideal of R.Proof.
Let(F^,A) be a vague soft hyperideal of R. Therefore, by Definition 36, (F^,A) is a vague soft hypergroup over (R,+) and satisfies the condition max⁡F^ax,F^ay≤inf⁡F^az:z∈x∘y. In order to prove that M is a hyperideal of R, we need to prove that (M,+) is a subhypergroup of (R,+) and R∘M⊆M and M∘R⊆M. In Theorem 31, it was proven that (M,+) is a subhypergroup of (R,+) if F^,A is a vague soft hypergroup over R,+. Now let x∈R and y∈M. Then tF^ay=1-fF^ay=s which means that F^ay=s. Since F^a is a vague hyperideal of R for all a∈SuppF^,A, it must satisfy the following conditions:(25)F^ay≤inf⁡F^az:z∈x∘y,F^ax≤inf⁡F^az:z∈x∘y.Therefore since F^ay=s for all a∈SuppF^,A and y∈M, we obtain inf⁡F^az:z∈x∘y≥F^ay=s which means that inf⁡F^az:z∈x∘y≥s. This implies that z∈M for all z∈x∘y and this proves that x∘y⊆M for all x∈R and y∈M. Therefore we obtain R∘M⊆M and this implies that M is a left hyperideal of R. Now let x∈M and y∈R. Then tF^ax=1-fF^a=s which means that F^ax=s. Similarly, we obtain inf⁡F^az:z∈x∘y≥F^ax=s which means that inf⁡F^az:z∈x∘y≥s. This implies that z∈M for all z∈x∘y and consequently we obtain x∘y⊆M for all x∈M and y∈R. This proves that M∘R⊆M. As such, M is a right hyperideal of R. Hence it has been proven that M is a hyperideal of R.Theorem 43.
LetM be a nonempty subset of R, let (F^,A) be a nonnull vague soft set over R, and let (F^,A)M be a vague soft characteristic set over M. Then (F^,A) is a vague soft hyperideal of R if and only if M is a hyperideal of R.Proof.
The proof is straightforward.


## 5. Vague Soft Hyperring Homomorphism

In this section, we introduce the notion of vague soft hyperring homomorphism by applying the concept of vague soft functions as well as the notion of the image and preimage of a vague soft set under a vague soft function introduced by [10] in the context of vague soft hyperrings. Lastly, it is proven that the vague soft hyperring homomorphism preserves vague soft hyperrings.Definition 44 (see [2]).
Let(R1,+1,∘1) and (R2,+2,∘2) be two hyperrings. A map f:R1→R2 is called ahyperring homomorphism if, for all x,y∈R1, one has fx+1y⊆fx+2f(y) and fx∘1y⊆fx∘2fy.Definition 45 (see [10]).
Letφ:X→Y and ψ:A→B be two functions, where A and B are parameter sets for the classical sets X and Y, respectively. Let (F^,A) and (G^,B) be vague soft sets over X and Y, respectively. Then the ordered pair (φ,ψ) is called avague soft function from (F^,A) to G^,B and is denoted by φ,ψ:F^,A→G^,B.Definition 46 (see [10]).
Let(F^,A) and (G^,B) be vague soft sets over X and Y, respectively. Let φ,ψ:F^,A→(G^,B) be a vague soft function. (i)
Theimage of (F^,A) under the vague soft function φ,ψ, which is denoted by φ,ψF^,A, is a vague soft set over Y, which is defined as (26)φ,ψF^,A=φF^,ψA,where(27)φF^aφx=φF^ψay,for alla∈A, x∈X, and y∈Y.(ii)
Thepreimage of (G^,B) under the vague soft function φ,ψ, which is denoted by φ,ψ-1G^,B, is a vague soft set over X, which is defined as (28)φ,ψ-1G^,B=φ-1G^,ψ-1B,where(29)φ-1G^bφ-1y=φ-1G^ψ-1bx,for allb∈B, x∈X, and y∈Y.If φ and ψ are injective (surjective), then the vague soft function (φ,ψ) is said to be injective (surjective).Definition 47.
LetR1,+1,∘1 and R2,+2,∘2 be two hyperrings. Also let (F^,A) and (G^,B) be two vague soft hyperrings over R1 and R2, respectively, and let φ,ψ:(F^,A)→G^,B be a vague soft function. Then φ,ψ:(F^,A)→G^,B is called avague soft hyperring homomorphism if the following conditions are satisfied:(i)
φ is a hyperring homomorphism from R1 to R2,(ii)
ψ is a function from A to B,(iii)
φF^x=G^ψx for all x∈SuppF^,A.Theorem 48.
LetR1,+1,∘1 and R2,+2,∘2 be two hyperrings. Let (F^,A) and (G^,B) be two vague soft hyperrings over R1 and R2, respectively. Let φ,ψ:(F^,A)→G^,B be a vague soft hyperring homomorphism and let ψ:A→B be an injective mapping. Then φF^,ψA is a vague soft hyperring over R2,+2,∘2.Proof.
Since(φ,ψ) is a vague soft hyperring homomorphism, it can be seen that(30)SuppφF^,ψA⊆ψSuppF^,A.Now let w1,w2∈R2. Then there exists x,y∈R1 such that φx=w1 and φy=w2. Therefore, for all x,y∈R1, a∈SuppF^,A we obtain(31)min⁡φF^aφx,φF^aφy=min⁡φF^ψaw1,φF^ψaw2≤inf⁡φF^ψau:u∈w1+2w2=inf⁡φF^aφz,where z∈R1 such that z∈x+1y and φz=u.
Furthermore, for allw,x∈R1, there exists u1,u2∈R2 such that φw=u1 and φx=u2. Therefore, for all w,x∈R1 and for each a∈SuppF^,A, we obtain(32)min⁡φF^aφw,φF^aφx=min⁡φF^ψau1,φF^ψau2≤inf⁡φF^ψab:u2∈u1+2b=inf⁡φF^aφy,where y∈R1 such that x∈w+1y and φy=b and also(33)min⁡φF^aφw,φF^aφx=min⁡φF^ψau1,φF^ψau2≤inf⁡φF^ψac:u2∈c+2u1=inf⁡φF^aφz,where z∈R1 such that x∈z+1w and φz=c.
Thus it has been proven thatφF^a is a nonnull subhypergroup of R,+2 and consequently φF^,ψA is a vague soft hypergroup over R,+2.
Lastly, letx,y∈R1. Then there exists w1,w2∈R2 such that φx=w1 and φy=w2. Therefore for all x,y∈R1 and a∈SuppF^,A we obtain(34)min⁡φF^aφx,φF^aφy=min⁡φF^ψaw1,φF^ψaw2≤inf⁡φF^ψap:p∈w1∘2w2=inf⁡φF^aφz,where z∈R1 such that z∈x∘1y and φz=p.
Therefore it has been proven thatφF^a is a nonnull subsemihypergroup of R,∘2 and consequently φF^,ψA is a vague soft semihypergroup over R,∘2. As such, φF^a is a vague subhyperring of R2,+2,∘2. Hence φF^,ψA is a vague soft hyperring over R2,+2,∘2.Theorem 49.
LetR1,+1,∘1 and R2,+2,∘2 be two hyperrings. Let (F^,A) and (G^,B) be two vague soft hypergroups over R1 and R2, respectively. Let φ,ψ:(F^,A)→G^,B be a vague soft hyperring homomorphism. Then φ-1G^,ψ-1B is a vague soft hyperring over R1,+1,∘1.Proof.
The proof is similar to that of Theorem48.Theorems48 and 49 prove that the homomorphic image and preimage of a vague soft hyperring are also a vague soft hyperring. This implies that the vague soft hyperring homomorphism preserves vague soft hyperrings.Theorem 50.
LetR1, R2, and R3 be nonnull hyperrings and let F^,A, (G^,B), and (J^,C) be vague soft hyperrings over R1, R2, and R3, respectively. Let φ,ψ:F,A→G,B and φ′,ψ′:G,B→(J,C) be vague soft hyperring homomorphisms. Then φ′∘φ,ψ′∘ψ:F,A→J,C is a vague soft hyperring homomorphism.Proof.
The proof is straightforward.


## 6. Conclusion

In this paper we introduced and developed the initial theory of vague soft hyperrings as a continuation to the theory of vague soft hypergroups. It was proved that there exists a one-to-one correspondence between the concepts introduced here and the corresponding concepts in soft hyperring theory and classical hyperring theory. Lastly, it was proved that the notion of vague soft hyperring homomorphism introduced here preserves vague soft hyperrings. This is an important step in the formulation and advancement of knowledge in the area of vague soft hyperalgebra.



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