# Characterizations of  Semihyperrings by Their (∈γ,∈γ∨qδ)-Fuzzy Hyperideals

Authors: Xiaokun Huang; Yunqiang Yin; Jianming Zhan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The concepts of (∈γ,∈γ∨qδ)-fuzzy bi-hyperideals and (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideals of a semihyperring are introduced, and some related properties of such (∈γ,∈γ∨qδ)-fuzzy hyperideals are investigated. In particular, the notions of hyperregular semihyperrings
and left duo semihyperrings are given, and their characterizations in terms of hyperideals and (∈γ,∈γ∨qδ)-fuzzy hyperideals are studied.


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## Body

## 1. Introduction

The concept of the fuzzy set, initiated by Zadeh in his pioneer paper [1] of 1965, is an important tool for modeling uncertainties in many complicated problems in engineering, economics, environment, medical science, and social science, due to information incompleteness, randomness, limitations of measuring instruments, and so forth. Many classical mathematics is extended to fuzzy mathematics, and various properties of them in the context of fuzzy sets are established. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [2], played a prominent role to generalize some basic concepts of fuzzy algebraic systems. Bhakat and Das [3, 4] gave the concepts of (α,β)-fuzzy subgroups by using the “belongs to” relation (∈) and “quasi-coincident with” relation (q) between a fuzzy point and a fuzzy subgroup and introduced the concept of (∈,∈∨q)-fuzzy subgroup. They also defined and investigated the (∈,∈∨q)-fuzzy subrings and ideals of a ring in [5]. Later on, Dudek et al. [6] introduced the notion of (∈,∈∨q)-fuzzy ideal and (∈,∈∨q)-fuzzy h-ideal in hemirings. Davvaz et al. [7] considered the concept of interval-valued (∈,∈∨q)-fuzzy Hv-submodules of Hv-modules. Recently, Yin and Zhan [8] further generalized the above relations (∈) and (q) by using a pair of thresholds γ and δ (γ<δ). They gave some new relations ∈γ and qδ between a fuzzy point and a fuzzy set and studied (α,β)-fuzzy filters in BL-algebras where α,β are two of {∈γ,qδ,∈γ∨qδ,∈γ∧qδ} with α≠∈γ∧qδ. Afterwards, this direction was continued by Ma and Zhan, and others (for instance, [9, 10]).Algebraic hyperstructure was introduced in 1934 by a French mathematician, Marty [11], at the 8th Congress of Scandinavian Mathematicians. Later on, people have observed that hyperstructures have many applications in both pure and applied sciences. A comprehensive review of the theory of hyperstructures can be found in [12–14]. In a recent book of Corsini and Leoreanu [15], the authors have collected numerous applications of algebraic hyperstructures, especially those from the last fifteen years to the following subjects: geometry, hypergraphs, binaryrelations, lattices, fuzzy sets and roughsets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities. The study of fuzzy hyperstructures is also an interesting research topic of hyperstructure. Many fuzzy theorems in hyperstructures have been discussed by several authors, for example, Corsini, Cristea, Davvaz, Kazanci, Leoreanu, Yin, and Zhan (see, e.g., [7, 8, 13, 16–24]). Hyperstructure, in particular semihyperring, is a generalization of classical algebra in which the ordinary operations are replaced by hyperoperations which map a pair of elements into a subset. In [25], Ameri and Hedayati gave the notions of semihyperrings and studied the k-hyperideals of them. Davvaz [26] gave the concepts of ternary semihyperrings and investigated their fuzzy hyperideals. Dehkordi and Davvaz introduced the notions of Γ-semihyperrings and discussed roughness and a kind of strong regular (equivalence) relations on Γ-semihyperrings (see, [27, 28]). However, one can see that these semihyperrings are based on a hyperoperation and an ordinary operation (or Γ-operation). In this paper, we consider another kind of semihyperring in which addition and multiplication are both hyperoperations. We define (∈γ,∈γ∨qδ)-fuzzy hyperideals, (∈γ,∈γ∨qδ)-fuzzy bi-hyperideals, and (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideals in a semihyperring and investigate some related properties of them. In the rest, we give the concepts of hyperregular semihyperrings and left duo semihyperrings and address their characterizations in terms of (∈γ,∈γ∨qδ)-fuzzy hyperideals, (∈γ,∈γ∨qδ)-fuzzy bi-hyperideals, and (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideals.


## 2. Preliminaries

LetH be a set and P(H) the family of all nonempty subsets of H and ∘ a hyperoperation or join operation; that is, ∘ is a map from H×H to P(H). If (x,y)∈H×H, its image under ∘ is denoted by x∘y. If A,B⊆H, then A∘B is given by A∘B=⋃‍{x∘y∣x∈A,y∈B}. x∘A is used for {x}∘A and A∘x for A∘{x}. Generally, the singleton x is identified by its element x.H together with a hyperoperation ∘ is called a semihypergroup if (x∘y)∘z=x∘(y∘z) for all x,y,z∈H. A semihypergroup H is said to be commutative if a∘b=b∘a for all a,b∈H. The concept of semihyperrings was introduced by Ameri and Hedayati in 2007 [25]. We formulate it as follows.Definition 1 (see [25]).
A semihyperring is an algebraic hyperstructure(H,+,·) satisfying the following axioms:(1)
(H,+) is a commutative semihypergroup with a zero element 0 satisfying x+0=0+x={x};(2)
(H,·) is semigroup; that is, (x·y)·z=x·(y·z) for all x,y,z∈H;(3)
the multiplication· is distributive over the hyperoperation +; that is, for any x,y,z∈H, we have x·(y+z)=x·y+x·z and (y+z)·x=y·x+z·x;(4)
the element0 is an absorbing element; that is, x·0=0·x=0 for all x∈H.In Definition1, if the multiplication is replaced by a hyperoperation, then we have the following definition.Definition 2.
A semihyperring is an algebraic hyperstructure(H,+,∘) consisting of a nonempty set H together with two binary hyperoperations on H satisfying the following axioms:(1)
(H,+) is a commutative semihypergroup;(2)
(H,∘) is semihypergroup;(3)
the hyperoperation∘ is distributive over the hyperoperation +; that is, for any x,y,z∈H, we have x∘(y+z)=x∘y+x∘z and (y+z)∘x=y∘x+z∘x;(4)
there exists an element0∈H such that x+0=0+x={x} and x∘0=0∘x={0}, which is called the zero of H.Let(H,+,∘) be a semihyperring; for any x1,x2,…,xn∈H, we write ∑i=1n‍xi=x1+x2+⋯+xn.Example 3.
LetH={0,a,b} be a set with two hyperoperations (+) and (∘) as follows:
(1)+0ab0{0}{a}{b}a{a}{0,a}{0,a,b}b{b}{0,a,b}{0,b}∘0ab0{0}{0}{0}a{0}{0}{0}b{0}{0}{0,a}
Then H is a semihyperring with a zero.A nonempty subsetL (resp., R) of a semihyperring H is called a left (resp., right) hyperideal of H if it satisfies L+L⊆L and H∘L⊆L (resp., R∘H⊆R). A nonempty subset B of a semihyperring H is called a bi-hyperideal of H if it satisfies B+B⊆B, B∘B⊆B and B∘H∘B⊆B. A nonempty subset Q of a semihyperring H is called a quasi-hyperideal of H if it satisfies Q+Q⊆Q and Q∘H∩H∘Q⊆Q.A fuzzy subset of a semihyperringH, by definition, is an arbitrary mapping μ:H→[0,1], where [0,1] is the usual interval of real numbers. We denote by F(H) the set of all fuzzy subsets of H.In what follows letγ,δ∈[0,1] be such that γ<δ and H a semihyperring with a zero 0. For any A⊆H, we define κAγ,δ to be the fuzzy subset of H by κAγ,δ(x)≥δ for all x∈A and κAγ,δ(x)≤γ otherwise. Clearly, κAγ,δ is the characteristic function of A if γ=0 and δ=1.A fuzzy subsetμ in H defined by
(2)μ(y)={r(≠0)ify=x,0otherwise
is said to be a fuzzy point with support x and value r and is denoted by xr.For a fuzzy pointxr and a fuzzy subset μ of H, we say that(1)
xr∈γμ if μ(x)≥r>γ,(2)
xrqδμ if μ(x)+r>2δ,(3)
xr∈γ∨qδμ if xr∈γμ or xrqδμ.In the sequel, unless otherwise stated,α¯ means that α does not hold, where α∈{∈γ,qδ,∈γ∨qδ}. For any μ,ν∈F(H), by μ⊆(γ,δ)ν, we mean that xr∈γμ implies xr∈γ∨qδν for all x∈H and r∈(γ,1].Lemma 4.
Letμ and ν be two fuzzy subsets of H, and then the following conditions are equivalent.(1)
μ⊆(γ,δ)ν.(2)
max{ν(x),γ}≥min{μ(x),δ} for all x∈H.(3)
μr⊆νr for all r∈[γ,δ], where μr={x∈H∣μ(x)≥r}.Proof.
(1)⇒(2) Assume that (1) holds. Let μ and ν be any fuzzy subsets of H. If max{ν(x),γ}<min{μ(x),δ} for some x∈H, then there exists r such that max{ν(x),γ}<r<min{μ(x),δ}; it follows that xr∈γμ but xr∈γ∨qδ¯ν, a contradiction. Hence max{ν(x),γ}≥min{μ(x),δ} for all x∈H.
(2)⇒(1) Assume that (2) holds. If μ⊆(γ,δ)ν does not hold, then there exists xr∈γμ but xr∈γ∨qδ¯ν, which contradicts max{ν(x),γ}≥min{μ(x),δ}. Therefore, (1) holds.
It is easy to check that (1)⇔(3). This completes the proof.According to Lemma4, it can be easily seen that μ⊆(γ,δ)κHγ,δ for all μ∈F(H). Also, the following result can be easily deduced.Lemma 5.
Letμ,ν, and ω be any fuzzy subsets of H such that μ⊆(γ,δ)ν and ν⊆(γ,δ)ω, and then μ⊆(γ,δ)ω.We will writeμ≈(γ,δ)ν if μ⊆(γ,δ)ν and ν⊆(γ,δ)μ. Let μ≈(γ,δ)ν be the relation which is defined in the above two fuzzy sets of a hypersemigorup H. Then it satisfies reflexivity, symmetry, and transitivity. That is, it is an equivalence relation on the F(H).For two fuzzy subsetsμ and ν of a semihyperring H; the sum μ⊕ν is defined by
(3)μ⊕ν(x)=supx∈a+bmin{μ(a),ν(b)}
and μ⊕ν(x)=0 if x cannot be expressible as an element in a+b for all x∈H.Definition 6.
Letμ and ν be fuzzy subsets of a semihyperring H, the intrinsic product μ⊙ν is defined by:
(4)μ⊙ν(x)=supx∈∑i=1n‍ai∘bimin{μ(ai),ν(bi)}
and μ⊙ν(x)=0 if x cannot be expressible as an element in ∑i=1n‍ai∘bi for all x∈H.Lemma 7.
Letμ1,μ2,ν1, and ν2 be any fuzzy subsets of H such that μ1⊆(γ,δ)μ2 and ν1⊆(γ,δ)ν2, and then(1)
μ1⊙ν1⊆(γ,δ)μ2⊙ν2,(2)
μ1∩ν1⊆(γ,δ)μ2∩ν2.Proof.
(1) For anyx∈H, by Lemma 4, we have
(5)max{(μ2⊙ν2)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{μ2(ai),ν2(bi)},γ}=supx∈∑i=1n‍ai∘bimin{max{μ2(ai),γ},max{ν2(bi),γ}}≥supx∈∑i=1n‍ai∘bimin{min{μ1(ai),δ},min{ν1(bi),δ}}=supx∈∑i=1n‍ai∘bimin{min{μ1(ai),ν1(bi)},δ}=min{supx∈∑i=1n‍ai∘bimin{μ1(ai),ν1(bi)},δ}=min{(μ1⊙ν1)(x),δ}.
This implies that μ1⊙ν1⊆(γ,δ)μ2⊙ν2.
(2) It is straightforward by Lemma4.Lemma 8.
Letμ,ν, and ω be any fuzzy subsets of H. Then(1)
(μ⊙ν)⊙ω=μ⊙(ν⊙ω),(2)
μ⊙(ν∪ω)=μ⊙ω∪ν⊙ω, (μ∪ν)⊙ω=μ⊙ω∪ν⊙ω,(3)
μ⊙(ν∩ω)⊆(γ,δ)μ⊙ω∩ν⊙ω, (μ∩ν)⊙ω⊆(γ,δ)μ⊙ω∩ν⊙ω.Proof.
It is straightforward.Lemma 9.
LetA,B be any subsets in H. Then one has(1)
A⊆B if and only if κAγ,δ⊆(γ,δ)κBγ,δ,(2)
κAγ,δ∩κBγ,δ≈(γ,δ)κA∩Bγ,δ,(3)
κAγ,δ⊙κBγ,δ≈(γ,δ)κA∘Bγ,δ.Proof.
(1) Assume thatκAγ,δ⊆(γ,δ)κBγ,δ. If A⊈B, then there exists x∈A but x∉B. This implies that x1∈γκAγ,δ and x1∈γ∨qδ¯κBγ,δ, which contradicts κAγ,δ⊆(γ,δ)κBγ,δ. Conversely, assume that A⊆B. Let x∈H and r∈(γ,1] be such that xr∈γκAγ,δ. Then κAγ,δ(x)≥r>γ, and so x∈A. Thus x∈B and κBγ,δ(x)≥δ. This gives that xr∈γ∨qδκBγ,δ. Hence κAγ,δ⊆(γ,δ)κBγ,δ.
(2) It is straightforward.
(3) Letx∈H. If x∈A∘B, then κA∘Bγ,δ(x)≥δ and x∈y∘z for some y∈A and z∈B. Thus we have
(6)(κAγ,δ⊙κBγ,δ)(x)=supx∈∑i=1n‍ai∘bimin{κAγ,δ(ai),κBγ,δ(bi)}≥min{κAγ,δ(y),κBγ,δ(z)}≥δ.
This implies that κAγ,δ⊙κBγ,δ≈(γ,δ)κA∘Bγ,δ.
Ifx∉A∘B, then (κA∘Bγ,δ)(x)≤γ and x cannot be expressible as a element in y∘z for any y∈A and z∈B. Thus (κAγ,δ⊙κBγ,δ)(x)=0≤γ. This implies that κAγ,δ⊙κBγ,δ≈(γ,δ)κA∘Bγ,δ.Definition 10.
A fuzzy subsetμ in a semihyperring H is called an (∈γ,∈γ∨qδ)-fuzzy left (resp., right) hyperideal if it satisfies(F1a)
μ⊕μ⊆(γ,δ)μ,(F2a)
κHγ,δ⊙μ⊆(γ,δ)μ (resp., μ⊙κHγ,δ⊆(γ,δ)μ).A fuzzy subset of a semihyperringH is called an (∈γ,∈γ∨qδ)-fuzzy hyperideal if it is both an (∈γ,∈γ∨qδ)-fuzzy left hyperideal and an (∈γ,∈γ∨qδ)-fuzzy right hyperideal.Definition 11.
A fuzzy subsetμ in a semihyperring H is called an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal if it satisfies conditions (F1a) and(F3a)
μ⊙μ⊆(γ,δ)μ,(F4a)
μ⊙κHγ,δ⊙μ⊆(γ,δ)μ.Definition 12.
A fuzzy subsetμ in a semihyperring H is called an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal if it satisfies conditions (F1a) and(F5a)
μ⊙κHγ,δ∩κHγ,δ⊙μ⊆(γ,δ)μ.Theorem 13.
Letμ be a fuzzy subset of a semihyperring H.Then μ is an (∈γ,∈γ∨qδ)-fuzzy left (resp., right) hyperideal if and only if for any x,y∈H, one has(F1b)
max{infz∈x+yμ(z),γ}≥min{μ(x),μ(y),δ} for all x,y∈H.(F2b)
max{infz∈x∘yμ(z),γ}≥min{μ(y),δ} (resp., max{infz∈x∘yμ(z),γ}≥min{μ(x),δ}).Proof.
Assume that (F1a) holds. For anyx,y∈H, if possible, let max{infz∈x+yμ(z),γ}<min{μ(x),μ(y),δ}. Choose r such that max{infz∈x+yμ(z),γ}<r<min{μ(x),μ(y),δ}. Then there exists z∈x+y such that μ(z)<r<min{μ(x),μ(y),δ}; that is, zr∈γ∨qδ¯μ. Then (μ⊕μ)(z)=supz∈a+bmin{μ(a),μ(b)}≥min{μ(x),μ(y),δ}≥r; hence zr∈γμ⊕μ, which contradicts (F1a). Therefore max{infz∈x+yμ(z),γ}≥min{μ(x),μ(y),δ}. That is, (F1b) holds.
Conversely, assume that (F1b) holds. Letx∈H and r∈(γ,1] be such that zr∈γμ⊕μ, if possible; let zr∈γ∨qδ¯μ. Then μ(x)<r and μ(x)+r≤2δ. Hence μ(x)<δ. Now, r≤(μ⊕μ)(x)=supx∈y+zmin{μ(y),μ(z)}≤supx∈y+zμ(z)≤μ(x) (since x∈y+z and the assumption imply that δ>μ(x)≥min{μ(z),δ}=μ(y)), a contradiction. Hence, xr∈∨qμ. This implies that μ⊕μ⊆(γ,δ)μ. Therefore (F1a) holds.
In a similar way, we can prove that (F2a)⇔(F2b).Theorem 14.
Letμ be a fuzzy subset of a semihyperring H. Then μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal if and only if it satisfies (F1b) and for any x,y,z∈H, one has(F3b)
max{infw∈x∘yμ(w),γ}≥min{μ(x),μ(y),δ},(F4b)
max{infw∈x∘y∘zμ(w),γ}≥min{μ(x),μ(z),δ}.Proof.
The proof is analogous to that of Theorem13.Remark 15.
Letμ be any fuzzy subsets of H. If μ is an (∈γ,∈γ∨qδ)-fuzzy left (resp., right) hyperideal of H, then a∈x∘y implies that max{μ(a),γ}≥min{μ(y),δ} (resp., max{μ(a),γ}≥min{μ(x),δ}). If μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H, then a∈x∘y∘z implies that max{μ(a),γ}≥min{μ(x),μ(z),δ}.Theorem 16.
(1)μ∩ν is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H for every (∈γ,∈γ∨qδ)-fuzzy left hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy right hyperideal ν of H.
(2) Any(∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal.Proof.
(1) Letμ and ν be any (∈γ,∈γ∨qδ)-fuzzy left hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then κHγ,δ⊙μ⊆(γ,δ)μ and ν⊙κHγ,δ⊆(γ,δ)ν. By Lemma 7, we have (μ∩ν)⊙κHγ,δ∩κHγ,δ⊙(μ∩ν)⊆(γ,δ)ν⊙κHγ,δ∩κHγ,δ⊙μ⊆(γ,δ)μ∩ν, and so μ∩ν is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.
(2) Letμ be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. Then, μ⊙μ⊆(γ,δ)κHγ,δ⊙μ, μ⊙μ⊆(γ,δ)μ⊙κHγ,δ, μ⊙κHγ,δ⊙μ⊆(γ,δ)κHγ,δ⊙μ and μ⊙κHγ,δ⊙μ⊆(γ,δ)μ⊙κHγ,δ, and so μ⊙μ⊆(γ,δ)κHγ,δ⊙μ∩μ⊙κHγ,δ⊆(γ,δ)μ and μ⊙κHγ,δ⊙μ⊆(γ,δ)κHγ,δ⊙μ∩μ⊙κHγ,δ⊆(γ,δ)μ. Hence μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H.One may easily see that any(∈γ,∈γ∨qδ)-fuzzy left hyperideal and any (∈γ,∈γ∨qδ)-fuzzy right hyperideal of a semihyperring H are an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. However, the converse of the property and Theorem 16 do not hold in general as shown in the following examples.Example 17.
LetH={0,a,b,c} be a set with two hyperoperations (+) and (∘) as follows:
(7)+0abc0{0}{a}{b}{c}a{a}{a}{a}{a}b{b}{a}{0,b}{0,b,c}c{c}{a}{0,b,c}{0,c}∘0abc0{0}{0}{0}{0}a{0}{a}{0,b}{0}b{0}{0}{0}{0}c{0}{0,c}{0}{0}
Then H is a semihyperring and Q={0,a} is a quasi-hyperideal of H, and it is not a left (right) hyperideal of H. Define the fuzzy set μ of H as follows:
(8)μ(0)=μ(a)=0.6,μ(b)=μ(c)=0.3.
Then μ is an (∈0.4,∈0.4∨q0.6)-fuzzy quasi-hyperideal of H, and it is not an (∈0.4,∈0.4∨q0.6)-fuzzy left (right) hyperideal of H.Example 18.
LetH={0,a,b,c} be a set with two hyperoperations (+) and (∘) as follows:
(9)+0abc0{0}{a}{b}{c}a{a}{a}{0,a,b}{0,a,c}b{b}{0,a,b}{0,b}{0,b,c}c{c}{0,a,c}{0,b,c}{0,c}∘0abc0{0}{0}{0}{0}a{0}{0}{0}{0}b{0}{0}{0}{0,a}c{0}{0}{0,a}{0,b}
Then B={0,b} is a bi-hyperideal of H, and it is not a quasi-hyperideal of H. Define the fuzzy set μ of H as follows:
(10)μ(0)=μ(b)=0.6,μ(a)=μ(c)=0.4.
Then μ is an (∈0.4,∈0.4∨q0.6)-fuzzy bi-hyperideal of H, and it is not an (∈0.4,∈0.4∨q0.6)-fuzzy quasi-hyperideal of H.Lemma 19.
LetA be any subset in H. Then the following conditions hold.(1)
A is a left (resp., right) hyperideal of H if and only if κAγ,δ is an (∈γ,∈γ∨qδ)-fuzzy left (resp., right) hyperideal of H.(2)
A is a bi-hyperideal of H if and only if κAγ,δ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H.(3)
A is a quasi-hyperideal of H if and only if κAγ,δ is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.Proof.
LetA be any subset in H. According to Lemma 9, H∘A⊆A if and only if κHγ,δ⊙κAγ,δ≈(γ,δ)κH∘Aγ,δ⊆(γ,δ)κAγ,δ. Hence A is a left hyperideal of H if and only if κAγ,δ is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H.
The case for bi-hyperdeals and quasi-hyperideals can be similarly proven.


## 3. Characterizations of Hyperregular Semihyperrings

Definition 20.
A semihyperringH is said to be hyperregular if for each x∈H, there exists a∈H such that x∈x∘a∘x. Equivalent definitions: (1) x∈x∘H∘x for all x∈H; (2) A⊆A∘H∘A for all A⊆H.Example 21.
LetH={0,a,b} be a set with two hyperoperations (+) and (∘) as follows:
(11)+0ab0{0}{a}{b}a{a}{0,a}{0,a,b}b{b}{0,a,b}{0,b}∘0ab0{0}{0}{0}a{0}{0,a}{0,b}b{0}{0,b}{0,a}
Then H is a hyperregular semihyperring. Since 0∈0∘0∘0, a∈a∘a∘a and b∈b∘b∘b.Remark 22.
IfH is a hyperregular semihyperring, then H∘H=H and κHγ,δ⊙κHγ,δ≈(γ,δ)κHγ,δ.Theorem 23.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(3)
R∩L=R∘L for every right hyperideal R and every left hyperideal L of H.Proof.
(1)⇒ (2) Let H be a hyperregular semihyperring, with μ being any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and ν any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then μ⊙ν⊆(γ,δ)μ⊙κHγ,δ⊆(γ,δ)μ and μ⊙ν⊆(γ,δ)κHγ,δ⊙ν⊆(γ,δ)ν. Hence μ⊙ν⊆(γ,δ)μ∩ν. Let x be any element of H. Then, since H is hyperregular, there exists a∈H such that x∈x∘a∘x, and then x∈x∘b for some b∈x∘a. Now, since μ is (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H, by Remark 15, we have max{μ(b),γ}≥min{μ(x),δ}. Hence,
(12)max{μ⊙ν(x),γ}=max{supx∈∑i=1n‍ai∘bimin{μ(ai),ν(bi)},γ}≥max{min{μ(b),ν(x)},γ}=min{max{μ(b),γ},max{ν(x),γ}}≥min{min{μ(x),δ},max{ν(x),γ}}=min{μ∩ν(x),δ}.
This implies that μ∩ν⊆(γ,δ)μ⊙ν. Hence (2) holds.
(2)⇒(3) It is straightforward by Lemmas 9 and 19.
(3)⇒(1) Let x be any element of H. Then x∘H+Mx and H∘x+Nx, where M={1,2,…} and N={1,2,…} are the principal right hyperideal and principal left hyperideal generated by x, respectively. By the assumption, we have
(13)x∈(x∘0+x)∩(0∘x+x)⊆(x∘H+Mx)∩(H∘x+Nx)=(x∘H+Mx)∘(H∘x+Nx)=x∘H∘x+(x∘H)∘(Nx)+(Mx)∘(H∘x)+(Mx)∘(Nx).
This implies that x∈x∘a∘x for some a∈H. Hence H is hyperregular.Corollary 24.
LetH be semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ∩ν⊆(γ,δ)μ⊙ν for every fuzzy subset μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(3)
μ∩ν⊆(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ and every fuzzy subset ν of H.Corollary 25.
LetH be a hyperregular semihyperring. Then every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.Proof.
Letμ be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H. Evidently, μ⊙κHγ,δ and κHγ,δ⊙μ are an (∈γ,∈γ∨qδ)-fuzzy right hyperideal and an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Now, by Theorem 23 and Remark 22, we have
(14)μ⊙κHγ,δ∩κHγ,δ⊙μ≈(γ,δ)μ⊙κHγ,δ⊙κHγ,δ⊙μ≈(γ,δ)μ⊙κHγ,δ⊙μ⊆(γ,δ)μ.
This implies that μ is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.Lemma 26.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
B=B∘H∘B for every bi-hyperideal B of H.(3)
Q=Q∘H∘Q for every quasi-hyperideal Q of H.Proof.
(1)⇒(2) Assume that (1) holds. Let B be any bi-hyperideal of H and x any element of B. Then there exists a∈H such that x∈x∘a∘x. It is easy to see that x∘a∘x⊆B∘H∘B and so x∈B∘H∘B. Hence, B⊆B∘H∘B. On the other hand, since B is an bi-hyperideal of H, we have B∘H∘B⊆B. Therefore, B=B∘H∘B.
(2)⇒(3) Evidently, every quasi-hyperideal of H is a bi-hyperideal of H. Then by the assumption, we have Q=Q∘H∘Q for every quasi-hyperideal Q of H.
(3)⇒(1) Assume that (3) holds. Let R and L be any right hyperideal and any left hyperideal of H, respectively. Then we have (R∩L)∘S∩S∘(R∩L)⊆R∘S∩S∘L⊆R∩L, and so it is easy to see that R∩L is a quasi-hyperideal of H. By the assumption and Theorem 23, we have R∩L=(R∩L)∘S∘(R∩L)⊆R∘S∘L⊆R∘L⊆R∩L. Hence, R∘L=R∩L and so S is hyperregular.Theorem 27.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ≈(γ,δ)μ⊙κHγ,δ⊙μ for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ of H.(3)
μ≈(γ,δ)μ⊙κHγ,δ⊙μ for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H and x any element of H. Then μ⊙κHγ,δ⊙μ⊆(γ,δ)μ. On the other hand, since H is hyperregular, there exists a∈H such that x∈x∘a∘x, and then x∈b∘x for some b∈x∘a. Thus, by Remark 15, we have
(15)max{(μ⊙κHγ,δ⊙μ)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙κHγ,δ)(ai),μ(bi)},γ}≥max{min{(μ⊙κHγ,δ)(b),μ(x)},γ}=max{min{supb∈∑j=1mcj∘djmin{μ(cj),κHγ,δ(dj)},μ(x)},γ}≥max{min{μ(x),κHγ,δ(a),μ(x)},γ}≥max{min{μ(x),δ,μ(x)},γ}=min{μ(x),δ}.
This implies that μ⊆(γ,δ)μ⊙κHγ,δ⊙μ. Hence we have μ≈(γ,δ)μ⊙κHγ,δ⊙μ.
(2)⇒(3) It is straightforward by Theorem 16.
(3)⇒(1) Assume that (3) holds. Let Q be any quasi-hyperideal of H. Then by Lemma 19, κQγ,δ is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. Then, by the assumption and Lemma 9, we have
(16)κQγ,δ⊆(γ,δ)κQγ,δ⊙κHγ,δ⊙κQγ,δ≈(γ,δ)κQ∘H∘Qγ,δ.
It follows from Lemma 9 that Q⊆Q∘H∘Q. Now, since Q is a quasi-hyperideal of H, we have Q∘H∘Q=Q∘H∩H∘Q⊆Q, and so Q∘H∘Q=Q. Therefore H is hyperregular by Lemma 26.Corollary 28.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ⊆(γ,δ)μ⊙κHγ,δ⊙μ for every fuzzy subset μ of H.Theorem 29.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ∩ν≈(γ,δ)μ⊙ν⊙μ for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy hyperideal ν of H.(3)
μ∩ν≈(γ,δ)μ⊙ν⊙μ for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy hyperideal ν of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, respectively. Then it is clear that μ⊙ν⊙μ⊆(γ,δ)μ∩ν. Now let x be any element of H. Then since H is hyperregular, there exists a∈H such that x∈x∘a∘x⊆(x∘a∘x)∘a∘x=x∘(a∘x∘a)∘x. Then there exist b∈a∘x∘a, c∈x∘(a∘x∘a) such that c∈x∘b and x∈c∘x. Now, since ν is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, by Remark 15, we have max{ν(b),γ}≥min{ν(x),δ}. Thus,
(17)max{(μ⊙ν⊙μ)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙ν)(ai),μ(bi)},γ}≥max{min{(μ⊙ν)(c),μ(x)},γ}=max{min{supc∈∑j=1m‍cj∘djmin{μ(cj),ν(dj)},μ(x)},γ}≥max{min{μ(x),ν(b),μ(x)},γ}=min{max{μ(x),γ},max{ν(b),γ},max{μ(x),γ}}≥min{max{μ(x),γ},min{ν(x),δ},max{μ(x),γ}}=min{(μ∩ν)(x),δ}.
This implies that μ∩ν⊆(γ,δ)μ⊙ν⊙μ. Hence, (2) holds.
(2)⇒(3) This is straightforward by Theorem 16.
(3)⇒(1) Assume that (3) holds. Let μ be any fuzzy quasi-hyperideal of H. Then since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, we have
(18)μ≈(γ,δ)μ∩κHγ,δ≈(γ,δ)μ⊙κHγ,δ⊙μ.
Then it follows from Theorem 27 that H is hyperregular.Corollary 30.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
B∩A=B∘A∘B for every bi-hyperideal B and every hyperideal A of H.(3)
Q∩A=Q∘A∘Q for every quasi-hyperideal Q and every hyperideal A of H.Proof.
(1)⇒(2) Assume that (1) holds. Let B and A be any bi-hyperideal and any hyperideal of H, respectively. Then by Lemma 19, κBγ,δ and κAγ,δ are an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, respectively. Thus, by Theorem 29, we have
(19)κB∩Aγ,δ≈(γ,δ)κBγ,δ∩κAγ,δ≈(γ,δ)κB⊙κAγ,δ⊙κBγ,δ≈(γ,δ)κB∘A∘Bγ,δ.
Then it follows from Lemma 9 that B∩A=B∘A∘B.
(2)⇒(3) It is straightforward.
(3)⇒(1) Assume that (3) holds. Let Q be any quasi-hyperideal of H. Then since H itself is a hyperideal of H, we have
(20)Q=Q∩H=Q∘H∘Q.
Then it follows from Lemma 26 that H is hyperregular.Theorem 31.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
μ∩ν⊆(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(3)
μ∩ν⊆(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(4)
μ∩ν⊆(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal ν of H.(5)
μ∩ν⊆(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal ν of H.(6)
μ∩ν∩ω⊆(γ,δ)μ⊙ν⊙ω for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ, every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal ν, and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ω of H.(7)
μ∩ν∩ω⊆(γ,δ)μ⊙ν⊙ω for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ, every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal ν, and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ω of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Now let x be any element of H. Since H is hyperregular, there exists a∈H such that x∈x∘a∘x, and then x∈x∘b for some b∈a∘x. Since ν is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, we have max{ν(b),γ}≥min{ν(x),δ}. Thus,
(21)max{(μ⊙ν)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{μ(ai),ν(bi)},γ}≥max{min{μ(x),ν(b)},γ}=min{max{μ(x),γ},max{ν(b),γ}}≥min{max{μ(x),γ},min{ν(x),δ}}=min{(μ∩ν)(x),δ}.
This implies that μ∩ν⊆(γ,δ)μ⊙ν.
(2)⇒(1) Assume that (2) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then it is easy to check that μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H. By the assumption, we have μ∩ν⊆(γ,δ)μ⊙ν⊆(γ,δ)μ⊙κHγ,δ∩κHγ,δ⊙ν⊆(γ,δ)μ∩ν. Hence, μ∩ν≈(γ,δ)μ⊙ν. So H is hyperregular by Theorem 23.
Similarly, we can show that (1)⇔(3), (1)⇔(4), and (1)⇔(5).
(1)⇒(6) Assume that (1) holds. Let μ, ν, and ω be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal, any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal, and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Now let x be any element of H. Since H is hyperregular, there exists a∈H such that x∈x∘a∘x⊆(x∘a∘x)∘a∘x. Then there exist b∈x∘a, c∈a∘x and d∈b∘x such that x∈d∘c. Now, since μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal and ω is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, by Remark 15, we have max{μ(b),γ}≥min{μ(x),δ} and max{ω(c),γ}≥min{ω(x),δ}. Thus, we have
(22)max{(μ⊙ν⊙ω)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙ν)(ai),ω(bi)},γ}≥max{min{(μ⊙ν)(d),ω(c)},γ}=max{min{supd∈∑j=1m‍cj∘djmin{μ(cj),ν(dj)},ω(c)},γ}≥max{min{μ(b),ν(x),ω(c)},γ}=min{max{μ(b),γ},max{ν(x),γ},max{ω(c),γ}}≥min{min{μ(x),δ},max{ν(x),γ},min{ω(x),δ}}=min{(μ∩ν∩ω)(x),δ}.
This implies that μ∩ν∩ω⊆(γ,δ)μ⊙ν⊙ω.
(6)⇒(1) Assume that (6) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H, by the assumption and Lemma 9, we have
(23)μ∩ν≈(γ,δ)μ∩κHγ,δ∩ν⊆(γ,δ)μ⊙ν⊆(γ,δ)μ⊙κHγ,δ∩κHγ,δ⊙ν⊆(γ,δ)μ∩ν.
Hence, μ∩ν≈(γ,δ)μ⊙ν. Therefore, H is hyperregular by Theorem 23.
Similarly, we can show that (1)⇔(7). This completes the proof.Corollary 32.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is hyperregular.(2)
B∩L⊆B∘L for every bi-hyperideal B and every left hyperideal L of H.(3)
Q∩L⊆Q∘L for every quasi-hyperideal Q and every left hyperideal L of H.(4)
R∩B⊆R∘B for every right hyperideal R and every bi-hyperideal B of H.(5)
R∩Q⊆R∘Q for every right hyperideal R and every quasi-hyperideal Q of H.(6)
R∩B∩L⊆R∘B∘L for every right hyperideal R, every bi-hyperideal B, and every left hyperideal L of H.(7)
R∩Q∩L⊆R∘Q∘L for every right hyperideal R, every quasi-hyperideal Q, and every left hyperideal L of H.Definition 33.
A subsetA in a semihyperring H is called idempotent if A=A∘A. A fuzzy subset μ in a semihyperring H is called (∈γ,∈γ∨qδ)-idempotent if μ≈(γ,δ)μ⊙μ.Example 34.
Consider Example21. Let A={0,a}. Then A∘A=A and so A is idempotent. Define a fuzzy subset of H by μ(0)=0.6,μ(a)=0.5, and μ(b)=μ(c)=0. Then μ⊙μ≈(0.4,0.6)μ and so μ is (∈0.4,∈0.4∨q0.6)-fuzzy idempotent.Theorem 35.
A semihyperringH is hyperregular if and only if all (∈γ,∈γ∨qδ)-fuzzy right and (∈γ,∈γ∨qδ)-fuzzy left hyperideals of H are (∈γ,∈γ∨qδ)-fuzzy idempotent and for every (∈γ,∈γ∨qδ)-fuzzy right hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H, the fuzzy subset μ⊙ν is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.Proof.
Assume thatH is hyperregular. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then we have μ⊙μ⊆(γ,δ)μ⊙κHγ,δ⊆(γ,δ)μ. Since H is hyperregular, by Theorem 31, we have μ⊆(γ,δ)μ⊙μ and so μ≈(γ,δ)μ⊙μ. Hence μ is (∈γ,∈γ∨qδ)-idempotent. In a similar way, we may prove that every (∈γ,∈γ∨qδ)-fuzzy left hyperideal is (∈γ,∈γ∨qδ)-idempotent. Now let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. By Theorem 23, we have μ∩ν≈(γ,δ)μ⊙ν and so
(24)κHγ,δ⊙(μ⊙ν)∩(μ⊙ν)⊙κHγ,δ≈(γ,δ)κHγ,δ⊙(μ∩ν)∩(μ∩ν)⊙κHγ,δ⊆(γ,δ)κHγ,δ⊙μ∩ν⊙κHγ,δ⊆(γ,δ)μ∩ν≈(γ,δ)μ⊙ν.
Therefore μ⊙ν is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.
Conversely, assume that the given conditions hold. Letμ be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. It is easy to check that μ∪κHγ,δ⊙μ is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H; then by the assumption, it is (∈γ,∈γ∨qδ)-fuzzy idempotent. Thus, we have
(25)μ⊆(γ,δ)μ∪κHγ,δ⊙μ≈(γ,δ)(μ∪κHγ,δ⊙μ)⊙(μ∪κHγ,δ⊙μ)≈(γ,δ)μ⊙μ∪μ⊙κHγ,δ⊙μ∪κHγ,δ⊙μ⊙μ∪κHγ,δ⊙μ⊙κHγ,δ⊙μ⊆(γ,δ)κHγ,δ⊙μ.
That is, μ⊆(γ,δ)κHγ,δ⊙μ. Similarly, we can show that μ⊆(γ,δ)μ⊙κHγ,δ. Thus,
(26)μ⊆(γ,δ)κHγ,δ⊙μ∩μ⊙κHγ,δ⊆(γ,δ)μand  soμ≈(γ,δ)κHγ,δ⊙μ∩μ⊙κHγ,δ.
On the other hand, it is clear that μ⊙κHγ,δ and κHγ,δ⊙μ are an (∈γ,∈γ∨qδ)-fuzzy right hyperideal and an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. By the assumption, we have (μ⊙κHγ,δ)⊙(μ⊙κHγ,δ)≈(γ,δ)μ⊙κHγ,δ, (κHγ,δ⊙μ)⊙(κHγ,δ⊙μ)≈(γ,δ)κHγ,δ⊙μ, and (μ⊙κHγ,δ)⊙(κHγ,δ⊙μ) is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. Thus, we have
(27)μ≈(γ,δ)μ⊙κHγ,δ∩κHγ,δ⊙μ≈(γ,δ)(μ⊙κHγ,δ)⊙(μ⊙κHγ,δ)∩(κHγ,δ⊙μ)⊙(κHγ,δ⊙μ)(sinceκHγ,δis  an(∈γ,∈γ∨qδ)-fuzzy  hyperidealofHand  soκHγ,δ⊙κHγ,δ≈(γ,δ)κHγ,δ)≈(γ,δ)μ⊙(κHγ,δ⊙κHγ,δ)⊙μ⊙κHγ,δ∩κHγ,δ⊙μ⊙(κHγ,δ⊙κHγ,δ)⊙μ≈(γ,δ)((μ⊙κHγ,δ)⊙(κHγ,δ⊙μ))⊙κHγ,δ∩κHγ,δ⊙((μ⊙κHγ,δ)⊙(κHγ,δ⊙μ))≈(γ,δ)(μ⊙κHγ,δ)⊙(κHγ,δ⊙μ)≈(γ,δ)μ⊙(κHγ,δ⊙κHγ,δ)⊙μ⊆(γ,δ)μ⊙κHγ,δ⊙μ⊆(γ,δ)μ,
which implies that μ≈(γ,δ)μ⊙κHγ,δ⊙μ. By Theorem 27, H is hyperregular.Corollary 36.
A semihyperringH is hyperregular if and only if all right hyperideals and all left hyperideals of H are idempotent and for every right hyperideal R and every left hyperideal L of H, the set R∘L is a quasi-hyperideal of H.Proof.
Assume thatH is hyperregular. Let R and L be any right hyperideal and any left hyperideal of S, respectively. Then by Lemma 19, κRγ,δ and κLγ,δ are an (∈γ,∈γ∨qδ)-fuzzy right hyperideal and an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. By the assumption and Theorem 35, we know that κRγ,δ⊙κLγ,δ≈(γ,δ)κR∘Lγ,δ is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H and κRγ,δ≈(γ,δ)κRγ,δ⊙κRγ,δ,κLγ,δ≈(γ,δ)κLγ,δ⊙κLγ,δ. Consequently, R∘L is a quasi-hyperideal of H and R=R∘R, L=L∘L.
Conversely, assume that the given conditions hold. LetQ be any quasi-hyperideal of H. Analogous to the proof of Theorem 35, we may show Q=Q∘H∘Q. Therefore, H is hyperregular by Lemma 26.Corollary 37.
LetH be a hyperregular semihyperring and μ any fuzzy subset of H. Then the following conditions are equivalent.(1)
μ is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H.(2)
μ≈(γ,δ)ν⊙ω for some (∈γ,∈γ∨qδ)-fuzzy right hyperideal ν and (∈γ,∈γ∨qδ)-fuzzy left hyperideal ω of H.Proof.
(1)⇒(2) Let μ be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. Since H is hyperregular, by Remark 22 and Theorem 27, we have μ≈(γ,δ)μ⊙κHγ,δ⊙μ≈(γ,δ)μ⊙(κHγ,δ⊙κHγ,δ)⊙μ=(μ⊙κHγ,δ)⊙(κHγ,δ⊙μ). Evidently, μ⊙κHγ,δ and κHγ,δ⊙μ are an (∈γ,∈γ∨qδ)-fuzzy right hyperideal and an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Hence, (2) holds.
(2)⇒(1) It is straightforward by Theorem 35.


## 4. Characterizations of Hyperregular and Left Duo Semihyperrings

Definition 38.
A semihyperringH is called left duo if every left hyperideal of H is a hyperideal of H. A semihyperring H is called (∈γ,∈γ∨qδ)-fuzzy left duo if every (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H.Lemma 39.
A hyperregular semihyperringH is left duo if and only if for any x∈H one has x∘H⊆H∘x.Proof.
Assume that the hyperregular semihyperringH is left duo. Let x be any element of H. Clearly, H∘x is a left hyperideal of H. Since H is both hyperregular and left duo, then H∘x is also a right hyperideal of H. Then by the assumption, we have
(28)x∘H⊆(x∘H∘x)∘H⊆(H∘x)∘H⊆H∘x.
Conversely, assume that the given condition holds. LetL be any left hyperideal of H. Then L∘H⊆H∘L⊆L. That is, L is also a right hyperideal of H. Therefore, H is left duo.Theorem 40.
A hyperregular semihyperringH is left duo if and only if it is (∈γ,∈γ∨qδ)-fuzzy left duo.Proof.
Assume that the hyperregular semihyperringH is left duo. Let μ be any (∈γ,∈γ∨qδ)-fuzzy left hyperideal and x, y any elements of H. Then since H is both hyperregular and left duo, by Lemma 39, there exists z∈H such that x∘y⊆z∘x. Thus, we have
(29)max{infa∈x∘yμ(a),γ}≥max{infa∈z∘xμ(a),γ}≥min{μ(x),δ}.
This implies that μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Therefore, H is (∈γ,∈γ∨qδ)-fuzzy left duo.
Conversely, assume that the hyperregular semihyperringH is (∈γ,∈γ∨qδ)-fuzzy left duo. Let L be any left hyperideal of H. Then, by Lemma 19, κLγ,δ is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H. By the assumption, κLγ,δ is also an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Using Lemma 19 again, L is also a right hyperideal of H. Therefore, H is left duo.Lemma 41.
LetH be a semihyperring which is both hyperregular and left duo. Then every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H.Proof.
Letμ be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and x, y any elements of H. Since H is both hyperregular and left duo, there exists elements a,b∈H such that x∈x∘a∘x and x∘y⊆b∘x. Thus, we have
(30)x∘y⊆(x∘a∘x)∘y=(x∘a)∘(x∘y)⊆(x∘a)∘(b∘x)=x∘(a∘b)∘x.
Hence, there exists c∈H such that x∘y⊆x∘c∘x. Since μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H, we have
(31)max{infz∈x∘yμ(z),γ}≥max{infz∈x∘c∘xμ(z),γ}≥min{μ(x),δ}.
This implies that μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H.Theorem 42.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is both hyperregular and left duo.(2)
H is both hyperregular and left (∈γ,∈γ∨qδ)-fuzzy duo.(3)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H.(4)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H.(5)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy hyperideal of H.(6)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy hyperideal of H.(7)
μ∩ν≈(γ,δ)μ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy left or right hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy hyperideal of H.Proof.
Firstly, by Theorem40, (1) and (2) are equivalent. Assume that (1) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then since H is hyperregular, it follows from Theorem 31 that μ∩ν⊆(γ,δ)μ⊙ν. On the other hand, since μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H, by Lemma 41, μ is also an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Thus we have μ⊙ν⊆(γ,δ)μ⊙κHγ,δ⊆(γ,δ)μ and μ⊙ν⊆(γ,δ)κHγ,δ⊙ν⊆(γ,δ)ν. Hence, μ⊙ν⊆(γ,δ)μ∩ν and so μ⊙ν≈(γ,δ)μ⊙ν. Therefore, (3) holds. It is clear that (3)⇒(4)⇒(6)⇒(7) and (3)⇒(5)⇒(7). Assume that (7) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy right hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, by the assumption, we have ν≈(γ,δ)ν∩κHγ,δ≈(γ,δ)ν⊙κHγ,δ. Thus, ν is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H; that is, ν is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H. Hence, H is left duo. On the other hand, by the assumption, we have μ∩ν≈(γ,δ)μ⊙ν. Then it follows from Theorem 23 that H is hyperregular. This completes the proof.Theorem 43.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is both hyperregular and left duo.(2)
μ∩ν≈(γ,δ)μ⊙ν⊙μ for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(3)
μ∩ν≈(γ,δ)μ⊙ν⊙μ for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then
(32)μ⊙ν⊙μ⊆(γ,δ)μ⊙κHγ,δ⊙μ⊆(γ,δ)μ.
On the other hand, since H is left duo, ν is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Hence, we have
(33)μ⊙ν⊙μ⊆(γ,δ)κHγ,δ⊙ν⊙κHγ,δ⊆(γ,δ)ν⊙κHγ,δ⊆(γ,δ)ν.
Therefore, μ⊙ν⊙μ⊆(γ,δ)μ∩ν. Now let x be any element of H. Since H is both hyperregular and left duo, there exist elements a and b of S such that x∈x∘a∘x and x∘a⊆b∘x. Then we have
(34)x∈x∘a∘x⊆(x∘a∘x)∘a∘(x∘a∘x)=x∘a∘x∘a∘(x∘a)∘x⊆x∘a∘x∘a∘(b∘x)∘x=(x∘a∘x)∘(a∘b∘x)∘x.
Thus, there exist elements c,d,e, and f such that c∈x∘a∘x,  d∈a∘b,  e∈d∘x, f∈c∘e, and x∈f∘x. Since μ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and ν an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of S, by Remark 15, we have max{μ(c),γ}≥min{μ(x),δ} and max{ν(e),γ}≥min{ν(x),δ}. Thus, we have
(35)max{(μ⊙ν⊙μ)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙ν)(ai),ω(bi)},γ}≥max{min{(μ⊙ν)(f),μ(x)},γ}=max{supf∈∑j=1m‍cj∘djmin{min{μ(cj),ν(dj)},μ(x)},γ}≥max{min{μ(c),ν(e),μ(x)},γ}=min{max{μ(c),γ},max{ν(e),γ},max{μ(x),γ}}≥min{min{μ(x),δ},min{ν(x),δ},max{μ(x),γ}}=min{(μ∩ν)(x),δ}.
This implies that μ∩ν⊆(γ,δ)μ⊙ν⊙μ. Therefore, μ∩ν≈(γ,δ)μ⊙ν⊙μ.
(2)⇒(3) It is straightforward.
(3)⇒(1) Assume that (3) holds. Let μ be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal of H. Since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, by the assumption, we have μ≈(γ,δ)μ∩κHγ,δ≈(γ,δ)μ⊙κHγ,δ⊙μ. Then it follows from Theorem 27 that H is hyperregular. Next, let ν be any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H. Then, since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal of H, by the assumption, we have
(36)ν≈(γ,δ)κHγ,δ∩ν⊆(γ,δ)κHγ,δ⊙ν⊙κHγ,δ.
Hence,
(37)ν⊙κHγ,δ⊆(γ,δ)(κHγ,δ⊙ν⊙κHγ,δ)⊙κHγ,δ≈(γ,δ)κHγ,δ⊙ν⊙κHγ,δ⊆(γ,δ)ν.
This implies that μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Therefore, μ is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H. It follows from Theorem 40 that H is left duo.Theorem 44.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is both hyperregular and left duo.(2)
μ∩ν≈(γ,δ)μ⊙κHγ,δ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.(3)
μ∩ν≈(γ,δ)μ⊙κHγ,δ⊙ν for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ν of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then by Lemma 41, μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Thus, we have
(38)μ⊙κHγ,δ⊙ν⊆(γ,δ)μ⊙ν⊆(γ,δ)μ⊙κHγ,δ⊆(γ,δ)μ,μ⊙κHγ,δ⊙ν⊆(γ,δ)μ⊙ν⊆(γ,δ)κHγ,δ⊙ν⊆(γ,δ)ν.
Therefore, μ⊙κHγ,δ⊙ν⊆(γ,δ)μ∩ν. Now let x be any element of H. Since H is hyperregular, there exists an element a of S such that x∈x∘a∘x⊆(x∘a∘x)∘a∘x. Thus, there exist elements b and c of H such that b∈a∘x∘a, c∈x∘b, and x∈c∘x. Then we have
(39)max{(μ⊙κHγ,δ⊙ν)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙κHγ,δ)(ai),ν(bi)},γ}≥max{min{(μ⊙κHγ,δ)(c),ν(x)},γ}=max{supc∈∑j=1m‍cj∘djmin{μ(cj),κHγ,δ(dj),ν(x)},γ}≥max{min{μ(x),δ,ν(x)},γ}≥min{μ(x),ν(x),δ}=min{(μ∩ν)(x),δ}.
This implies that μ∩ν⊆(γ,δ)μ⊙κHγ,δ⊙ν. Therefore, μ⊙κHγ,δ⊙ν≈(γ,δ)μ∩ν.
(2)⇒(3) It is straightforward.
(3)⇒(1) Assume that (3) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H. Then by the assumption, we have
(40)μ∩ν≈(γ,δ)μ⊙κHγ,δ⊙ν=μ⊙(κHγ,δ⊙ν)⊆(γ,δ)μ⊙ν.
Hence, it follows from Theorem 31 that H is hyperregular. Next, since ν is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal, then it is an (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal. It is clear that κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H; by the assumption, we have ν≈(γ,δ)ν∩κHγ,δ≈(γ,δ)ν⊙κHγ,δ⊙κHγ,δ. Hence,
(41)ν⊙κHγ,δ≈(γ,δ)ν⊙κHγ,δ⊙κHγ,δ⊙κHγ,δ⊆(γ,δ)ν⊙κHγ,δ⊙κHγ,δ≈(γ,δ)ν.
This implies that μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Therefore, μ is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H. It follows from Theorem 40 that H is left duo.Theorem 45.
LetH be a semihyperring. Then the following conditions are equivalent.(1)
H is both hyperregular and left duo.(2)
μ∩ν∩ω≈(γ,δ)μ⊙ν⊙ω for every (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal μ, every (∈γ,∈γ∨qδ)-fuzzy hyperideal ν, and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ω of H.(3)
μ∩ν∩ω≈(γ,δ)μ⊙ν⊙ω for every (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal μ, every (∈γ,∈γ∨qδ)-fuzzy hyperideal ν, and every (∈γ,∈γ∨qδ)-fuzzy left hyperideal ω of H.Proof.
(1)⇒(2) Assume that (1) holds. Let μ, ν, and ω be any (∈γ,∈γ∨qδ)-fuzzy bi-hyperideal, any (∈γ,∈γ∨qδ)-fuzzy hyperideal, and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Then by Lemma 41, μ is an (∈γ,∈γ∨qδ)-fuzzy right hyperideal of H. Thus, we have
(42)μ⊙ν⊙ω⊆(γ,δ)κHγ,δ⊙κHγ,δ⊙ω⊆(γ,δ)κHγ,δ⊙ω⊆(γ,δ)ω,μ⊙ν⊙ω⊆(γ,δ)κHγ,δ⊙ν⊙κHγ,δ⊆(γ,δ)ν,μ⊙ν⊙ω⊆(γ,δ)(μ⊙κHγ,δ)⊙κHγ,δ⊆(γ,δ)μ⊙κHγ,δ⊆(γ,δ)μ.
Therefore, μ⊙ν⊙ω⊆(γ,δ)μ∩ν∩ω. Now let x be any element of H. Since H is both hyperregular and left duo, analogous to the proof of Theorem 43, there exist elements a,b,c,d,e, and f such that c∈x∘a∘x, d∈a∘b, e∈d∘x, f∈c∘e, and x∈f∘x. It follows from Remark 15 that max{μ(c),γ}≥min{μ(x),δ} and max{ν(e),γ}≥min{ν(x),δ}. Thus, we have
(43)max{(μ⊙ν⊙ω)(x),γ}=max{supx∈∑i=1n‍ai∘bimin{(μ⊙ν)(ai),ω(bi)},γ}≥max{min{(μ⊙ν)(f),ω(x)},γ}=max{supf∈∑j=1m‍cjdjmin{μ(cj),ν(dj),ω(x)},γ}≥max{min{μ(c),ν(e),ω(x)},γ}=min{max{μ(c),γ},max{ν(e),γ},max{ω(x),γ}}≥min{min{μ(x),δ},min{ν(x),δ},max{ω(x),γ}}=min{(μ∩ν∩ω)(x),δ}.
This implies that μ∩ν∩ω⊆(γ,δ)μ⊙ν⊙ω. Therefore, μ∩ν∩ω≈(γ,δ)μ⊙ν⊙ω.
(2)⇒(3) It is straightforward.
(3)⇒(1) Assume that (3) holds. Let μ and ν be any (∈γ,∈γ∨qδ)-fuzzy quasi-hyperideal and any (∈γ,∈γ∨qδ)-fuzzy left hyperideal of H, respectively. Since κHγ,δ is an (∈γ,∈γ∨qδ)-fuzzy hyperideal of H, then by the assumption, we have
(44)μ∩ν≈(γ,δ)μ∩κHγ,δ∩ν≈(γ,δ)μ⊙κHγ,δ⊙ν.
Then it follows from Theorem 44 that H is both hyperregular and left duo.


## 5. Conclusions

In this paper, we investigate some new characterizations of some kinds of semihyperrings. Semihyperrings owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be semihyperrings. For this reason, the basic concepts introduced here have exhibited some universality and are applicable in so many diverse contexts. These concepts are important and effective tools in (hyper)algebraic systems, automata, and artificial intelligence. Our future work on this topic will focus on studying intuitionistic (soft) or interval-valued fuzzy sets in semihyperrings and other algebraic constructions of semihyperrings.



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