# On(∈,∈∨qk)-Fuzzy Hyperideals in Ordered LA-Semihypergroups

Authors: Muhammad Azhar; Naveed Yaqoob; Muhammad Gulistan; Mohammed M. Khalaf
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The concept of(∈,∈∨qk)-fuzzy hyperideal of an ordered LA-semihypergroup is introduced by the ordered fuzzy points, and related properties are investigated. We study the relations between different (∈,∈∨qk)-fuzzy hyperideal of an ordered LA-semihypergroup. Furthermore, we study some results on homomorphisms of (∈,∈∨qk)-fuzzy hyperideals.


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## Body

## 1. Introduction

Hyperstructures represent a natural extension of classical algebraic structures and they were introduced by the French mathematician Marty [1]. Since then, several authors continued their researches in this direction. Nowadays hyperstructures are widely studied from theoretical point of view and for their applications in many subjects of pure and applied mathematics. In a classical algebraic structure, the composition of two elements is an element, while, in an algebraic hyperstructure, the composition of two elements is a set. Some basic definitions and theorems about hyperstructures can be found in [2, 3]. The concept of a semihypergroup is a generalization of the concept of a semigroup. Many authors studied different aspects of semihypergroups, for instance, Bonansinga and Corsini [4], Corsini [5] Davvaz [6], Davvaz et al. [7], Davvaz and Poursalavati [8], Gutan [9], Hasankhani [10], Hila et al. [11], and Onipchuk [12]. Recently, Hila and Dine [13] introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semihypergroups. Yaqoob, Corsini, and Yousafzai [14] extended the work of Hila and Dine and characterized intraregular left almost semihypergroups by their hyperideals using pure left identities. Other results on LA-semihypergroups can be found in [15, 16].The concept of ordered semihypergroup was studied by Heidari and Davvaz in [17], where they used a binary relation “≤” on semihypergroup (H,∘) such that the binary relation is a partial order and the structure H,∘,≤ is known as ordered semihypergroup. There are several authors who study the ordering of hyperstructures, for instance, Bakhshi and Borzooei [18], Chvalina [19], Hoskova [20], Kondo and Lekkoksung [21], and Novak [22]. The ordering in LA-semihypergroups was introduced by Yaqoob and Gulistan [23].In 1965, the concept of fuzzy sets was first proposed by Zadeh [24], which has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics, and many more. Many papers on fuzzy sets have been published, showing the importance and their applications to set theory, group theory, real analysis, measure theory, and topology etc. There are several authors who applied the concept of fuzzy sets to algebraic hyperstructures. Ameri and others studied fuzzy sets in hypervector spaces [25], Γ-hyperrings [26], and polygroups [27]. Corsini et al. [28] studied semisipmle semihypergroups in terms of hyperideals and fuzzy hyperideals. Cristea [29] introduced hyperstructures and fuzzy sets endowed with two membership functions. In [30, 31], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup and Hv-semigroup. Recently in [32], Davvaz and Leoreanu-Fotea studied the structure of fuzzy Γ-hyperideals in Γ-semihypergroups. Also see [33, 34]. Fuzzy ordered hyperstructures have been considered by some researchers, for instance, Pibaljommee et al. [35, 36], Tang et al. [37–40], Tipachot and Pibaljommee [41].Murali [42] defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. In [43], the idea of quasi-coincidence of a fuzzy point with a fuzzy set is defined. The concept of a (α,β)-fuzzy subgroup was first considered by Bhakat and Das in [44, 45] by using the “belongs to” relation (∈) and “quasi coincident with” relation (q) between a fuzzy point and a fuzzy subgroup, where α,β∈{∈,q,∈∨q,∈∧q} and α≠∈∧q. Jun et al. [46] gave the concept of (∈,∈∨q)-fuzzy ordered semigroups. Moreover, (∈,∈∨qk)-fuzzy ideals, (∈,∈∨qk) -fuzzy quasi-ideals, and (∈,∈∨qk)-fuzzy bi-ideals of a semigroup are defined in [47]. Azizpour and Talebi characterized semihypergroups by (∈,∈∨qk)-fuzzy interior hyperideals [48, 49]. Huang [50] provided characterizations of semihyperrings by their (∈γ,∈γ,∨qδ) -fuzzy hyperideals. Shabir and Mahmood characterized semihypergroups by (∈,∈∨qk)-fuzzy hyperideals [51] and also by (∈γ,∈γ,∨qδ)-fuzzy hyperideals [52].Motivated by the work of Shabir et al. [47, 51], we applied the concept of (∈,∈∨qk)-fuzzy sets to LA-semihypergroups. In this paper, we introduce and study several types of (∈,∈∨qk)-fuzzy hyperideals in an ordered LA-semihypergroup. Further, we study some results on homomorphisms of (∈,∈∨qk)-fuzzy hyperideals.


## 2. Preliminaries and Basic Definitions

In this section, we provide some basic definitions needed for our further work.LetH be a nonempty set. Then the map ∘:H×H→P∗(H) is called hyperoperation or join operation on the set H, where P∗H=PH∖{∅} denotes the set of all nonempty subsets of H. A hypergroupoid is a set H together with a (binary) hyperoperation. For any nonempty subsets A,B of H, we denote (1)A∘B=⋃a∈A,b∈Ba∘bInstead of a∘A and B∘a, we write a∘A and B∘a, respectively.Recently, in [13, 14] authors introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semihypergroups. A hypergroupoid (H,∘) is called an LA-semihypergroup if for every x,y,z∈H, we have (x∘y)∘z=(z∘y)∘x. The law (x∘y)∘z=(z∘y)∘x is called a left invertive law. An element e∈H is called a left identity (resp., pure left identity) if for all x∈H, x∈e∘x (resp., x=e∘x). In an LA-semihypergroup, the medial law (x∘y)∘(z∘w)=(x∘z)∘(y∘w) holds for all x,y,z,w∈H. An LA-semihypergroup may or may not contain a left identity and pure left identity. In an LA-semihypergroup H with pure left identity, the paramedial law (x∘y)∘(z∘w)=(w∘z)∘(y∘x) holds for all x,y,z,w∈H. If an LA-semihypergroup contains a pure left identity, then by using medial law, we get x∘(y∘z)=y∘(x∘z) for all x,y,z∈H.Definition 1 (see [23]).
LetH be a nonempty set and ≤ be an ordered relation on H. The triplet H,∘,≤ is called an ordered LA-semihypergroup if the following conditions are satisfied.
1  H,∘ is an LA-semihypergroup.
2  H,≤ is a partially ordered set.
3 For every a,b,c∈H, a≤b implies a∘c≤b∘c and c∘a≤c∘b, where a∘c≤b∘c means that for x∈a∘c there exist y∈b∘c such that x≤y.Definition 2 (see [23]).
IfH,∘,≤ is an ordered LA-semihypergroup and A⊆H, then (A] is the subset of H defined as follows: (2)A=t∈H:t≤a,for  somea∈A.If x∈H and A is a nonempty subset of H, then Ax=y,z∈H×H∣x≤y∘z.Definition 3 (see [23]).
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called an LA-subsemihypergroup of H if (A∘A]⊆(A].Definition 4 (see [23]).
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called a right (resp., left) hyperideal of H if
1  A∘H⊆A (resp., H∘A⊆A),
2 for every a∈H, b∈A and a≤b implies a∈A.IfA is both right hyperideal and left hyperideal of H, then A is called a hyperideal (or two sided hyperideal) of H.Definition 5 (see [23]).
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called an interior hyperideal of H if
1  (H∘A)∘H⊆A,
2 for every a∈H, b∈A and a≤b implies a∈A.Definition 6 (see [23]).
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called a generalized bi-hyperideal of H if
1  (A∘H)∘A⊆A,
2 for every a∈H, b∈A and a≤b implies a∈A.IfA is both an LA-subsemihypergroup and a generalized bi-hyperideal of H, then A is called a bi-hyperideal of H.Definition 7.
LetH,∘,≤ be an ordered LA-semihypergroup, and a∈H. Then a is said to be regular element of H if there exists an element x∈H such that a≤(a∘x)∘a, or equivalently a≤(a∘H)∘a. If every element of H is regular then H is said to be a regular ordered LA-semihypergroup.Now, we give some fuzzy logic concepts.A fuzzy subsetf of a universe X is a function from X into the unit closed interval [0,1], i.e., f:X→[0,1] (see [24]). A fuzzy subset f in a universe X of the form (3)fy=t∈0,1ify=x0ify≠xis said to be a fuzzy point with support x and value t and is denoted by xt. For a fuzzy point xt and a fuzzy set f in a set X, Pu and Liu [43] introduced the symbol xtαf, where α∈{∈,q,∈∨q,∈∧q}, which means that(i)xt∈f if f(x)≥t,(ii)xtqf if f(x)+t>1,(iii)xt∈∨qf if xt∈f or xtqf,(iv)xt∈∧qf if xt∈f and xtqf,(v)xtα¯f if xtαf does not hold.For any two fuzzy subsetsf and g of H, f≤g means that, for all x∈H, f(x)≤g(x). The symbols f∧g and f∨g will mean the following fuzzy subsets: (4)f∧g:H⟶0,1∣x⟼f∧gx=fx∧gx=min⁡fx,gxf∨g:H⟶0,1∣x⟼f∨gx=fx∨gx=max⁡fx,gxfor all x∈H.The product of any fuzzy subsetsf and g of H is defined by (5)f∗g:H⟶0,1∣x⟼f∗gx=⋁y,z∈Axfy∧gz,ifAx≠∅0ifAx=∅.For ∅≠A⊆H, the characteristic function fA of A is a fuzzy subset of H, defined by (6)fA=1ifx∈A,0ifx∉A.For any fuzzy subset f of H and for any t∈(0,1], the set U(f;t)={x∈H:f(x)≥t} is called a level subset of f.Definition 8 (see [53]).
LetH,∘,≤ be an ordered LA-semihypergroup. A fuzzy subsetf:H→0,1 is called fuzzy LA-subsemihypergroup of H if the following assertion are satisfied.
(i)⋀z≤a∘bfz≥min⁡fa,fb,
(ii) ifa≤b implies fa≥fb, for every a,b∈H.Definition 9 (see [53]).
Let(H,∘,≤) be an ordered LA-semihypergroup. A fuzzy subset f:H→0,1 is called a fuzzy right (resp., left) hyperideal of H if
(1)⋀z≤a∘bfz≥fa (resp., ⋀z≤a∘bfz≥fb),
(2)a≤b implies fa≥fb, for every a,b∈H.Iff is both fuzzy right hyperideal and fuzzy left hyperideal of H, then f is called a fuzzy hyperideal of H.Definition 10 (see [54]).
Let(H,∘,≤) be an ordered LA-semihypergroup. A fuzzy subset f:H→0,1 is called a fuzzy bi-hyperideal of H if
(1)⋀z≤(a∘b)∘cfz≥min⁡fa,fc,
(2)a≤b implies fa≥fb, for every a,b∈H.Definition 11 (see [54]).
Let(H,∘,≤) be an ordered LA-semihypergroup. A fuzzy subset f:H→0,1 is called a fuzzy interior hyperideal of H if
(1)⋀z≤(a∘b)∘cfz≥fb,
(2)a≤b implies fa≥fb, for every a,b∈H.


## 3.(∈,∈∨qk)-Fuzzy Hyperideals

In what follows, letH denote an ordered LA-semihypergroup, k∈[0,1) and t,r∈(0,1] unless otherwise is specified. Note that for a fuzzy point xt and a fuzzy subset f, xt∈f means that f(x)≥t, while xtqkf means that f(x)+t+k>1. Notice that (∈,∈∨qk)-fuzzy hyperideals are particular types of (α,β)-fuzzy hyperideals.Definition 12.
A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H if for all x,y,z∈H and t∈(0,1], the following conditions hold:
(i)x≤y, yt∈f⇒xt∈∨qkf,
(ii)xt,yu∈f⇒zmin{t,u}∈∨qkf, for each z∈x∘y.Definition 13.
A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy left hyperideal of H if for all x,y,z∈H and t∈(0,1], the following conditions hold.
(i)x≤y, yt∈f⇒xt∈∨qkf,
(ii)yt∈f⇒zt∈∨qkf, for each z∈x∘y.Definition 14.
A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy right hyperideal of H if for all x,y,z∈H and t∈(0,1], the following conditions hold.
(i)x≤y, yt∈f⇒xt∈∨qkf,
(ii)xt∈f⇒zt∈∨qkf, for each z∈x∘y.A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy hyperideal of H if it is a left hyperideal and a right hyperideal of H.Definition 15.
Consider a setH=a,b,c,d with the following hyperoperation “∘” and the order “≤”: (7)
∘
a
b
c
d
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d
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d
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a
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b
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. We give the covering relation “≺” and the figure of H as follows: (8)≺=a,b,a,c,d,a. Then (H,∘,≤) is an ordered LA-semihypergroup. Now let f be a fuzzy subset of H such that (9)f:H⟶0,1∣x⟼fx=0.6ifx=a0.4ifx∈b,c0.9ifx=d. Let t=0.4 and k∈[0,1). Then by routine calculations it is clear that f is an (∈,∈∨qk)-fuzzy hyperideal of H.Definition 16.
A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy interior hyperideal of H if for all x,y,z,w∈H and t∈(0,1], the following conditions hold:
(i)x≤y, yt∈f⇒xt∈∨qkf,
(ii)yt∈f⇒wt∈∨qkf, for each w∈(x∘y)∘z.Example 17.
Consider a setH=a,b,c,d with the following hyperoperation “∘” and the order “≤”: (10)
∘
a
b
c
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e
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c
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c
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d
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d
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e
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e
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e
. We give the covering relation “≺” and the figure of H as follows: (11)≺=a,b,a,d,d,c,d,e. Then (H,∘,≤) is an ordered LA-semihypergroup. Now let f be a fuzzy subset of H such that (12)f:H⟶0,1∣x⟼fx=0.8ifx=a0.4ifx=b0.5ifx∈c,e0.7ifx=d. Let t=0.3 and k∈[0,1). Then by routine calculations it is clear that f is an (∈,∈∨qk)-fuzzy interior hyperideal of H.Definition 18.
A fuzzy subsetf of H is called an (∈,∈∨qk) -fuzzybi-hyperideal of H if for all x,y,z,w∈H and t∈(0,1], the following conditions hold:
(i)x≤y, yt∈f⇒xt∈∨qkf,
(ii)xt,yu∈f⇒zmin{t,u}∈∨qkf, for each z∈x∘y.
(iii)xt,zu∈f⇒wmin{t,u}∈∨qkf, for each w∈(x∘y)∘z.A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy generalizedbi-hyperideal of H if it satisfies (i) and (iii) conditions of Definition 18.Theorem 19.
LetA be an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H and f a fuzzy subset of H defined by (13)fx=≥1-k2ifx∈A0otherwise. Then
(i)f is a (q,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H,
(ii)f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H.Proof.
(i) Consider thatA is an LA-subsemihypergroup of H. Let x,y∈H, x≤y and t∈(0,1] be such that ytqf. Then y∈A, f(y)+t>1. Since A is an LA-subsemihypergroup of H and x≤y∈A, we have x∈A. Thus f(x)≥1-k/2. If t≤1-k/2, then f(x)≥t and so xt∈f. If t>1-k/2, then f(x)+t+k>1-k/2+1-k/2+k=1 and so xtqkf. Therefore xt∈∨qkf. Let x,y,z∈H and t,r∈(0,1] be such that xtqf and yrqf. Then x,y∈A, f(x)+t>1 and f(x)+r>1. Since A is an LA-subsemihypergroup of H, we have x∘y⊆A. Thus for every z∈x∘y, f(z)≥1-k/2. If min⁡{t,r}≤1-k/2, then f(z)≥min⁡{t,r} and so zmin{t,r}∈f. If min⁡{t,r}>1-k/2, then f(z)+min⁡{t,r}+k>1-k/2+1-k/2+k=1 and so zmin{t,r}qkf. Therefore zmin{t,r}∈∨qkf, for all z∈x∘y.
(ii) Letx,y∈H, x≤y and t∈(0,1] be such that yt∈f. Then f(y)≥t and y∈A. Since x≤y∈A, we have x∈A. Thus f(x)≥1-k/2. If t≤1-k/2, then f(x)≥t and so xt∈f. If t>1-k/2, then f(x)+t+k>1-k/2+1-k/2+k=1 and so xtqkf. Therefore xt∈∨qkf. Let x,y,z∈H and t,r∈(0,1] be such that xt∈f and yr∈f. Then f(x)≥t>0 and f(y)≥r>0. Thus f(x)≥1-k/2 and f(y)≥1-k/2, this implies that x,y∈A. Since A is an LA-subsemihypergroup of H, we have x∘y⊆A. Thus for every z∈x∘y, f(z)≥1-k/2. If min⁡{t,r}≤1-k/2, then f(z)≥min⁡{t,r} and so zmin{t,r}∈f. If min⁡{t,r}>1-k/2, then f(z)+min⁡{t,r}+k>1-k/2+1-k/2+k=1 and so zmin{t,r}qkf. Therefore zmin{t,r}∈∨qkf, for all z∈x∘y.
The cases for left hyperideal, right hyperideal, hyperideal, interior hyperideal and bi-hyperideal can be seen in a similar way.Corollary 20.
If a nonempty subsetA of H is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H, then the characteristic function of A is a (q,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H.Corollary 21.
A nonempty subsetA of H is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H if and only if fA is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi- and hyperideal) of H.If we takek=0 in Theorem 19, then we have the following corollary.Corollary 22.
LetA be an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H and f a fuzzy subset of H defined by (14)fx=≥0.5ifx∈A0otherwise. Then
(i)f is a (q,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H,
(ii)f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H.Theorem 23.
Letf be a fuzzy subset of H. Then, f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H if and only if
(i)x≤y⇒f(x)≥min⁡{f(y),1-k/2}, for all x,y∈H;
(ii)⋀z∈x∘yf(z)≥min⁡{f(x),f(y),1-k/2}, for all x,y,z∈H.Proof.
Assume thatf is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H. Let x,y∈H be such that x≤y. If f(y)=0, then f(x)≥min⁡{f(y),1-k/2}. Let f(y)≠0 and assume on the contrary that f(x)<min⁡{f(y),1-k/2}. Choose t∈(0,1] such that f(x)<t≤min⁡{f(y),1-k/2}. If f(y)<1-k/2, then f(x)<t<f(y), so yt∈f, but f(x)+t+k<1-k/2+1-k/2+k=1. This implies that xtqk¯f, therefore xt∈∨qk¯f, which is a contradiction. Hence f(x)≥min⁡{f(y),1-k/2}, for all x,y∈H.
Suppose on the contrary that there existx,y,z∈H such that ⋀z∈x∘yf(z)<min⁡{f(x),f(y),1-k/2}. Then there exists z∈x∘y such that f(z)<min⁡{f(x),f(y),1-k/2}. Choose t∈(0,1] such that f(z)<t<min⁡{f(x),f(y),1-k/2}. Then f(x)>t and f(y)>t implies that xt∈f and yt∈f, but f(z)<t and f(z)+t+k<1-k/2+1-k/2+k=1. So zt∈∨qk¯f, which is a contradiction. Hence ⋀z∈x∘yf(z)≥min⁡{f(x),f(y),1-k/2}.
Conversely, letyt∈f for t∈(0,1]. Then f(y)≥t. Now f(x)≥min⁡fy,1-k/2≥min⁡{t,1-k/2}. If t>1-k/2, then f(x)≥1-k/2 and f(x)+t+k>1-k/2+1-k/2+k=1, it follows that xtqkf. If t≤1-k/2, then f(x)≥t and so xt∈f. Thus xt∈∨qkf. Now, assume that ⋀z∈x∘yf(z)≥min⁡{f(x),f(y),1-k/2} for all x,y,z∈H. Let xt∈f and yr∈f for all t,r∈(0,1]. Then f(x)≥t and f(y)≥r. Thus ⋀z∈x∘yf(z)≥min⁡fx,fy,1-k/2≥min⁡{t,r,1-k/2}. Now if min⁡{t,r}>1-k/2, then ⋀z∈x∘yf(z)≥1-k/2. So for every z∈x∘y, f(z)+min⁡{t,r}+k>1-k/2+1-k/2+k=1, which implies that zmin{t,r}qkf. If min⁡{t,r}<1-k/2, then ⋀z∈x∘yf(z)≥min⁡{t,r}. So for every z∈x∘y, zmin{t,r}∈f. Thus zmin{t,r}qkf. Therefore f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H.Theorem 24.
Letf be a fuzzy subset of H. Then, f is an (∈,∈∨qk)-fuzzy left (resp., right) hyperideal of H if and only if
(i)x≤y⇒f(x)≥min⁡{f(y),1-k/2}, for all x,y∈H.
(ii)⋀z∈x∘yf(z)≥min{f(y),1-k/2} (resp., ⋀z∈x∘yf(z)≥min⁡{f(x),1-k/2}), for all x,y,z∈H.Proof.
The proof is similar to the proof of Theorem 23.Theorem 25.
Letf be a fuzzy subset of H. Then, f is an (∈,∈∨qk) -fuzzy interior hyperideal of H if and only if
(i)x≤y⇒f(x)≥min⁡{f(y),1-k/2}, for all x,y∈H.
(ii)⋀w∈x∘y∘zf(t)≥min⁡{f(y),1-k/2}, for all x,y,z,w∈H.Proof.
The proof is similar to the proof of the Theorem23.Theorem 26.
Letf be a fuzzy subset of H. Then, f is an (∈,∈∨qk)-fuzzy bi-hyperideal of H if and only if
(i)x≤y⇒f(x)≥min⁡{f(y),1-k/2}, for all x,y∈H.
(ii)⋀z∈x∘yf(z)≥min⁡{f(x),f(y),1-k/2}, for all x,y,z∈H.
(ii)⋀w∈x∘y∘zf(t)≥min⁡{f(x),f(z),1-k/2}, for all x,y,z,t∈H.Proof.
Theproof is similar to the proof of Theorem 23.A fuzzy subsetf of H is called an (∈,∈∨qk)-fuzzy generalizedbi-hyperideal of H if it satisfies (i) and (iii) conditions of Theorem 26.Theorem 27.
A fuzzy subsetf of an ordered LA-semihypergroup H is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H if and only if U(f;t)(≠0) is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of H for all t∈(0,1-k/2].Proof.
Assume thatf is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H. Let x,y,z∈H be such that x≤y∈U(f;t) where t∈(0,1-k/2]. Then f(y)≥t and by Theorem 23, f(x)≥min⁡fy,1-k/2≥min⁡{t,1-k/2}=t. It follows that x∈U(f;t). Let x,y∈U(f;t) where t∈(0,1-k/2]. Then f(x)≥t and f(y)≥t. It follows from Theorem 23 that ⋀z∈x∘yf(z)≥min⁡fx,fy,1-k/2≥min⁡{t,t,1-k/2}=t. Thus for every z∈x∘y, f(z)≥t and so z∈U(f;t), that is x∘y⊆U(f;t). Hence U(f;t) is an LA-subsemihypergroup of H.
Conversely, assume thatU(f;t)(≠0) is an LA-subsemihypergroup of H for all t∈(0,1-k/2]. If there exist x,y∈H with x≤y such that f(x)<min⁡{f(y),1-k/2}. Then f(x)<t<min⁡{f(y),1-k/2} for some t∈(0,1-k/2]. Then, y∈U(f;t) but x∉U(f;t), a contradiction. Thus f(x)≥min⁡{f(y),1-k/2} for all x,y∈H with x≤y. Now, suppose that there exist x,y∈H such that ⋀z∈x∘yf(z)<min⁡{f(x),f(y),1-k/2}. Thus there exist z∈x∘y such that f(z)<min⁡{f(x),f(y),1-k/2}. Choose t∈(0,1-k/2] such that f(z)<t<min⁡{f(x),f(y),1-k/2}. Then x,y∈U(f;t) but z∉U(f;t) i.e., x∘y⊈U(f;t), which is a contradiction. Hence ⋀z∈x∘yf(z)≥min⁡{f(x),f(y),1-k/2} and so f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H.Corollary 28.
A fuzzy subsetf of an ordered LA-semihypergroup H is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of H if and only if U(f;t)(≠0) is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of H for all t∈(0,0.5].Example 29.
Consider a setH=a,b,c,d with the following hyperoperation “∘” and the order “≤”: (15)
∘
a
b
c
d
a
a
a
a
a
b
a
b
c
d
c
a
d
c
,
d
c
,
d
d
a
c
c
,
d
c
,
d
≤
≔
a
,
a
,
a
,
b
,
a
,
c
,
a
,
d
,
b
,
b
,
c
,
c
,
d
,
d
. We give the covering relation “≺” and the figure of H as follows: (16)≺=a,b,a,c,a,d. Then (H,∘,≤) is an ordered LA-semihypergroup. Here, {a}, {b}, {a,b}, {a,c,d}, {b,c,d} and {c,d} are LA-subsemihypergroups of H. Now let f be a fuzzy subset of H such that (17)f:H⟶0,1∣x⟼fx=0.8ifx=a0.2ifx=b0.6ifx=c0.7ifx=d and (18)Uf;t=Hif0<t≤0.2a,c,dif0.2<t≤0.6aif0.6<t≤0.8∅if0.8<t≤1. Then by Theorem 27, f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H for k∈[0,1).Lemma 30.
The intersection of any family of(∈,∈∨qk)-fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, interior hyperideals, bi-hyperideals, generalized bi-hyperideals) of H is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of H.Proof.
Let{fi}i∈I be a family of (∈,∈∨qk)-fuzzy left hyperideals of H and x,y∈H. Then (19)⋀z∈x∘y∧i∈Ifiz=∧i∈I⋀z∈x∘yfiz. Since fi is a (∈,∈∨qk)-fuzzy left hyperideals of H. Thus (20)⋀z∈x∘y∧i∈Ifiz≥∧i∈Ifiy∧1-k2=∧i∈Ifiy∧1-k2=∧i∈Ifiy∧1-k2. Hence ∧i∈Ifi is an (∈,∈∨qk)-fuzzy left hyperideal of H. The other cases can be seen in a similar way.Lemma 31.
The union of any family of(∈,∈∨qk)-fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, interior hyperideals, bi-hyperideals) of an ordered LA-semihypergroup H is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, interior hyperideal, bi-hyperideal) of H.Proof.
The proof is similar to the proof of the Lemma30.Proposition 32.
LetH be an ordered LA-semihypergroup with pure left identity e. Then every (∈,∈∨qk)-fuzzy right hyperideal of H is an (∈,∈∨qk)-fuzzy left hyperideal of H.Proof.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of H. For x,y,z∈H, we have (21)⋀z∈x∘yfz=⋀z∈e∘x∘yfz=⋀z∈y∘x∘efz≥min⁡⋀w∈y∘xfw,1-k2≥min⁡fy,1-k2,1-k2=min⁡fy,1-k2. Thus f is an (∈,∈∨qk)-fuzzy left hyperideal of H.Proposition 33.
LetH be an ordered LA-semihypergroup with pure left identity e. Then f is an (∈,∈∨qk)-fuzzy right hyperideal of H if and only if it is an (∈,∈∨qk)-fuzzy interior hyperideal of H.Proof.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of H. For x,y,z∈H, we have (22)⋀a∈x∘y∘zfa≥min⁡⋀w∈x∘yfw,1-k2=min⁡⋀w∈e∘x∘yfw,1-k2=min⁡⋀w∈y∘x∘efw,1-k2≥min⁡⋀s∈y∘xfs,1-k2,1-k2≥min⁡fy,1-k2,1-k2,1-k2=min⁡fy,1-k2, which implies that f is an (∈,∈∨qk)-fuzzy interior hyperideal. Conversely, for any x and y in H we have (23)⋀a∈x∘yfa=⋀a∈e∘x∘yfa≥min⁡fx,1-k2. This shows that f is an (∈,∈∨qk)-fuzzy right hyperideal of H. This completes the proof.Proposition 34.
Letf be an (∈,∈∨qk)-fuzzy left hyperideal of an ordered LA-semihypergroup H with pure left identity e. If f is an (∈,∈∨qk)-fuzzy interior hyperideal of H, then it is an (∈,∈∨qk)-fuzzy bi-hyperideal of H.Proof.
Letf be an (∈,∈∨qk)-fuzzy left hyperideal of H. Then for every z∈H, there exist x,y∈H such that (24)⋀z∈x∘yfz≥min⁡fy,1-k2. Let e be a pure left identity in H. So, (25)⋀z∈x∘yfz=⋀z∈e∘x∘yfz≥min⁡fx,1-k2. This implies that (26)⋀z∈x∘yfz≥min⁡fx,fy,1-k2. Thus f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup of H. Now for any x,y,z∈H, (27)⋀a∈x∘y∘zfa=⋀a∈x∘e∘y∘zfa=⋀a∈e∘x∘y∘zfa≥min⁡⋀t∈x∘yft,1-k2=min⁡⋀t∈e∘x∘yft,1-k2≥min⁡fx,1-k2,1-k2=min⁡fx,1-k2. Also (28)⋀a∈x∘y∘zfa=⋀a∈z∘y∘xfa=⋀a∈e∘z∘y∘xfa≥min⁡⋀t∈z∘yft,1-k2=min⁡⋀t∈e∘z∘yft,1-k2≥min⁡fz,1-k2,1-k2=min⁡fz,1-k2. Hence we get (29)⋀a∈x∘y∘zfa≥min⁡fx,fz,1-k2. This shows that f is an (∈,∈∨qk)-fuzzy bi-hyperideal of H.Proposition 35.
LetH be an ordered LA-semihypergroup with pure left identity. If f is an (∈,∈∨qk)-fuzzy subset of H and g is an (∈,∈∨qk)-fuzzy left hyperideal of H, then f∗g is an (∈,∈∨qk)-fuzzy left hyperideal of H.Proof.
Letx,y∈H such that x≤y. Let (a,b)∈Ay then y≤a∘b. Since x≤y, so x≤a∘b implies (a,b)∈Ax. Hence Ay⊆Ax. Now (30)f∗gy=⋁a,b∈Aymin⁡fa,gb=⋁a,b∈Ay⊆Axfa,gb≤⋁a,b∈Axfa,gb=f∗gx. Thus (f∗g)(x)≥(f∗g)(y). Let x,y∈H. Then (31)min⁡f∗gy,1-k2=min⁡⋁p,q∈Aymin⁡fp,gq,1-k2=⋁p,q∈Aymin⁡fp,gq,1-k2. If y∈p∘q, then x∘y⊆x∘(p∘q)=p∘(x∘q). Now for each z∈x∘y, there exist a∈x∘q such that z∈p∘a. Since g is an (∈,∈∨qk)-fuzzy left hyperideal of H, ⋀a∈x∘qg(a)≥min⁡g(q),1-k/2, that is g(a)≥min⁡g(q),1-k/2. Thus (32)min⁡f∗gy,1-k2≤⋁p,a∈Azmin⁡fp,ga=f∗gzfor  everyz∈x∘y⊆p∘a. So ⋀z∈x∘yf∗g(z)≥min⁡f∗g(y),1-k/2. Thus f∗g is an (∈,∈∨qk)-fuzzy left hyperideal of H.Definition 36.
Letf and g be any two fuzzy subsets of H. We define the product f∗kg by (33)f∗kgx=⋁y,z∈Axmin⁡fy,gz,1-k2ifAx≠∅0ifAx=∅.Definition 37.
The symbolsfk, f∩kg and f∪kg mean the following fuzzy subsets of H: (34)fkx=min⁡fx,1-k2,∀x∈H,f∩kgx=min⁡fx,gx,1-k2,∀x∈H,f∪kgx=min⁡fx,gx,1-k2,∀x∈H.Denote byF(H) the family of all fuzzy subsets in H.Theorem 38.
LetH be an ordered LA-semihypergroup. Then the set (F(H),∗k,⊆) is an ordered LA-semihypergroup.Proof.
ClearlyF(H) is closed. Let f, g and h be in F(H) and let x be any element of H such that it is not expressible as product of two elements in H. Then we have (35)f∗kg∗khx=0=h∗kg∗kfx. Let Ax≠∅. Then there exist y and z in H such that (y,z)∈Ax. Therefore by using left invertive law, we have (36)f∗kg∗khx=⋁y,z∈Axf∗kgy∧hz∧1-k2=⋁y,z∈Ax⋁p,q∈Ayfp∧gq∧1-k2∧hz∧1-k2=⋁x≤p∘q∘zfp∧gq∧hz∧1-k2∧1-k2=⋁x≤z∘q∘phz∧gq∧fp∧1-k2∧1-k2=⋁w,p∈Ax⋁z,q∈Awhz∧gq∧fp∧1-k2∧1-k2=⋁w,p∈Axh∗kgw∧fp∧1-k2=h∗kg∗kfx. Hence (F(H),∗k) is an ordered LA-semihypergroup. Assume that f⊆gand let Ax=∅ for any x∈H; then (37)f∗khx=0=g∗khx⇒f∗kh⊆g∗kh. Similarly we can show that f∗kh⊇g∗kh. Let Ax≠∅. Then there exist y and z in H such that (y,z)∈Ax, therefore (38)f∗khx=⋁y,z∈Axfy∧hz∧1-k2≤⋁y,z∈Axgy∧hz∧1-k2=g∗khx, Similarly we can show that f∗kh⊇g∗kh. It is easy to see that F(H) is a poset. Thus (F(H),∗,⊆) is an ordered LA-semihypergroup.Theorem 39.
LetH be an ordered LA-semihypergroup. Then the property (39)f∗kg∗kh∗kk=f∗kh∗kg∗kk holds in F(H), for all f,g,h and k in F(H).Proof.
T h e proof is straightforward.Theorem 40.
If an ordered LA-semihypergroupH has a pure left identity, then the following properties hold in F(H).
(i)f∗kg∗k(h∗kk)=(k∗kh)∗kg∗kf,
(ii)f∗kg∗kh=g∗kf∗kh,
for allf,g,h and kin F(H).Proof.
Theproof is straightforward.Proposition 41.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of H and g is an (∈,∈∨qk)-fuzzy left hyperideal of H. Then f∗kg⊆f∩kg.Proof.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of H and g an (∈,∈∨qk)-fuzzy left hyperideal of H. Let z∈H and suppose that there exist x,y∈H such that z∈x∘y. Then (40)f∗kgz=⋁x,y∈Azmin⁡fx,gy,1-k2=⋁x,y∈Azmin⁡fx,gy,1-k2,1-k2,1-k2≤⋁x,y∈Azmin⁡⋀z∈x∘yfx,⋀z∈x∘ygy,1-k2=min⁡fx,gy,1-k2=f∩kg. Let us suppose that there do not exist any x,y∈H such that z∈x∘y. Then, (f∗kg)(z)=0≤f∩kg. Hence f∗kg⊆f∩kg.Lemma 42.
Letf be an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of H. Then fk is a fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of H.Proof.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of H. Then for all x,y,z∈H, we have x≤y⇒f(x)≥min⁡{f(y),1-k/2} and ⋀z∈x∘yf(z)≥min⁡{f(x),1-k/2}. This implies that (41)min⁡fx,1-k2≥min⁡fy,1-k2. So x≤y⇒fk(x)≥fk(y). Also (42)min⁡⋀z∈x∘yfz,1-k2≥min⁡fx,1-k2. So ⋀z∈x∘yfk(z)≥fk(x). Thus fk is a fuzzy right hyperideal of H. Other cases can be seen in a similar way.For an ordered LA-semihypergroupsH, the fuzzy subset H is defined as follows: (43)H:H⟶0,1∣x⟼Hx≔1.Proposition 43.
For a fuzzy subsetf of H, the following conditions are true for all x,y∈H.
(i) Iff is a fuzzy LA-subsemihypergroup of H, then f∗kf⊆fk and x≤y⇒fk(x)≥fk(y),
(ii) Iff is a fuzzy left hyperideal of H, then H∗kf⊆fk and x≤y⇒fk(x)≥fk(y),
(iii) Iff is a fuzzy right hyperideal of H, then f∗kH⊆fk and x≤y⇒fk(x)≥fk(y),
(iv) Iff is a fuzzy interior hyperideal of H, then (H∗kf)∗kH⊆fk and x≤y⇒fk(x)≥fk(y),
(v) Iff is a fuzzy generalized bi-hyperideal of H, then (f∗kH)∗kf⊆fk and x≤y⇒fk(x)≥fk(y),
(vi) Iff is a fuzzy bi-hyperideal of H, then f∗kf≤fk, (f∗kH)∗kf⊆fk and x≤y⇒fk(x)≥fk(y).Proof.
(i) Letf be a fuzzy LA-subsemihypergroup of H. If Ax={(y,z)∈H×H:x∈y∘z}=∅, then (44)f∗kfx=0≤fkx. If Ax≠∅, then (45)f∗kfx=min⁡f∗fx,1-k2=min⁡⋁x∈y∘zmin⁡fy,fz,1-k2≤min⁡⋁x∈y∘z⋀x∈y∘zfx,1-k2fis  a  fuzzy  LA-subsemihypergroup=min⁡fx,1-k2=fkx. Thus f∗kf⊆fk. For x≤y⇒f(x)≥f(y), we have min⁡f(x),1-k/2≥min⁡f(y),1-k/2. This implies fk(x)≥fk(y). The proofs of (ii) to (vi) can be seen in a similar way.Proposition 44.
LetH be an ordered LA-semihypergroup with pure left identity. Then for any (∈,∈∨qk)-fuzzy left hyperideal f, which is idempotent in C(H), the following properties hold:
(i)fk is an (∈,∈∨qk)-fuzzy interior hyperideal;
(ii)fk is an (∈,∈∨qk)-fuzzy bi-hyperideal.Proof.
Letf be an (∈,∈∨qk)-fuzzy left hyperideal of H and f∗kf=fk.
(i) We have(46)H∗kf∗kH⊆fk∗kH=f∗kf∗kH=H∗kf∗kf⊆fk∗kf=f∗kf=fk. This implies that fk is an (∈,∈∨qk)-fuzzy interior hyperideal H.
(ii) We have(47)f∗kH∗kf=f∗kH∗kfk=f∗kH∗kf∗kf=f∗kf∗kH∗kf⊆fk∗kfk=fk. This implies that fk is an (∈,∈∨qk)-fuzzy bi-hyperideal of H.Theorem 45.
LetH be a regular ordered LA-semihypergroup. Then (f∗kH)∗kf=fk for every (∈,∈∨qk)-fuzzy generalized bi-hyperideal f of H.Proof.
Let us suppose thatH is a regular ordered LA-semihypergroup and let f be an (∈,∈∨qk)-fuzzy generalized bi-hyperideal of H. Let a∈H and so a∈(a∘x)∘a for some x∈H. Thus we have (48)f∗kH∗kfa=min⁡f*H∗fa,1-k2=min⁡⋁a∈y∘zmin⁡f*Hy,fz,1-k2≥min⁡min⁡⋁t∈a∘xf*Ht,fa,1-k2≥min⁡min⁡⋁a∘x=p∘qmin⁡fp,Hq,fa,1-k2≥min⁡min⁡fa,1,fa,1-k2=min⁡fa,1-k2=fka. Therefore, fk⊆(f∗kH)∗kf. Since (f∗kH)∗kf⊆fk, by Proposition 43(v), we have (f∗kH)∗kf=fk for every (∈,∈∨qk)-fuzzy generalized bi-hyperideal f of H.Proposition 46.
Iff is an (∈,∈∨qk)-fuzzy right hyperideal of a regular ordered LA-semihypergroup H with pure left identity e, then fk(x∘y)=fk(y∘x) holds for all x,y∈H.Proof.
Letf be an (∈,∈∨qk)-fuzzy right hyperideal of a regular ordered LA-semihypergroup H with pure left identity e. Let x,y∈H. Since H is regular, x∈(x∘a)∘x and y∈(y∘b)∘y for some a,b∈H. Now by using the medial and paramedial laws, we get (49)x∘y⊆x∘a∘x∘y∘b∘y=y∘x∘y∘b∘x∘a. Since f is an (∈,∈∨qk)-fuzzy right hyperideal, for every w∈x∘y⊆(y∘x)∘((y∘b)∘(x∘a)), we have (50)⋀w∈x∘y⊆y∘x∘y∘b∘x∘afw≥min⁡⋀s∈y∘xfs,1-k2=⋀s∈y∘xfks. Again by using the medial and paramedial laws, we get (51)y∘x⊆y∘b∘y∘x∘a∘x=x∘y∘x∘a∘y∘b. Since f is an (∈,∈∨qk)-fuzzy right hyperideal, for every t∈y∘x⊆(x∘y)∘((x∘a)∘(y∘b)), we have (52)⋀t∈y∘x⊆x∘y∘x∘a∘y∘bft≥min⁡⋀p∈x∘yfp,1-k2=⋀p∈x∘yfkp. This shows that fk(x∘y)=fk(y∘x) holds for all x,y∈H.Theorem 47.
LetH be a regular ordered LA-semihypergroup with pure left identity e. Then f is an (∈,∈∨qk)-fuzzy left hyperideal of H if and only if f is an (∈,∈∨qk)-fuzzy bi-hyperideal of H.Proof.
Letf be an (∈,∈∨qk)-fuzzy left hyperideal of H. Let x,y,z∈H. We have (53)⋀w∈x∘y∘zfw=⋀w∈z∘y∘xfw≥min⁡fx,1-k2≥min⁡fx,fz,1-k2. This shows that f is an (∈,∈∨qk)-fuzzy generalized bi-hyperideal of H, and clearly f is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup. Therefore f is an (∈,∈∨qk)-fuzzy bi-hyperideal of H.
Conversely, assume thatf is an (∈,∈∨qk)-fuzzy bi-hyperideal of H. Let x,y∈H. Since H is regular, by the left invertive law and medial law, we have (54)x∘y⊆x∘y∘z∘y=y∘z∘x∘y=x∘y∘z∘y=x∘y∘e∘z∘y=x∘e∘y∘z∘y=y∘x∘e∘z∘y. Thus for every w∈x∘y⊆(y∘((x∘e)∘z))∘y, we have (55)⋀w∈x∘yfw≥⋀s∈y∘x∘e∘z∘yfs≥min⁡fy,fy,1-k2=min⁡fy,1-k2. Hence f is an (∈,∈∨qk)-fuzzy left hyperideal of H.


## 4. Homomorphism

Let(H1,∘1,≤1) and (H2,∘2,≤2) be two ordered LA-semihypergroups and ψ a mapping from H1 into H2. ψ is called isotone if x,y∈H1, x≤1y implies ψ(x)≤2ψ(y). ψ is said to be inverse isotone if x,y∈H1, ψ(x)≤2ψ(y) implies x≤1y [each inverse isotone mapping is 1-1]. ψ is called a homomorphism if it is isotone and satisfies ψ(x∘1y)=ψ(x)∘2ψ(y), for all x,y∈H1. Moreover, ψ is said to be isomorphism if it is onto homomorphism and inverse isotone.Definition 48.
Let(H1,∘1,≤1) and (H2,∘2,≤2) be two ordered LA-semihypergroups and ψ a mapping from H1 into H2. Let f1 and f2 be fuzzy subsets of H1 and H2, respectively. Then the image ψ(f1) of f1 is defined by (56)ψf1:H2⟶0,1∣y2⟶⋁y1∈ψ-1y2f1y1ifψ-1y2≠∅0otherwise. And the inverse image ψ-1(f2) of f2 is defined by (57)ψ-1f2:H1⟶0,1∣y1⟶f2ψy1.Theorem 49.
Let(H1,∘1,≤1) and (H2,∘2,≤2) be two ordered LA-semihypergroups and ψ a mapping from H1 onto H2. Let f and g be (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of H1 and H2, respectively. Then
(i)ψ(f) is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of H2, provided ψ is inverse isotone.
(ii)ψ-1(g) is an (∈,∈∨qk)-fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of H1.
(iii) The mappingf→ψ(f) defines a one-to-one correspondence between the set of all (∈,∈∨qk)-fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, hyperideals, interior hyperideals, bi-hyperideals) of H1 and the set of all (∈,∈∨qk)-fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, hyperideals, interior hyperideals, bi-hyperideals) of H2, provided ψ is inverse isotone.Proof.
The proof isstraightforward.Theorem 50.
Letψ:(H1,∘1,≤1)→(H2,∘2,≤2) be a surjective homomorphism from an ordered LA-semihypergroup H1 to an ordered LA-semihypergroup H2. If H1 contains a pure left identity e, then
(i) the image of(∈,∈∨qk)-fuzzy interior hyperideal of H1 is an (∈,∈∨qk)-fuzzy right hyperideal of H2, provided ψ is inverse isotone.
(ii) the preimage of an(∈,∈∨qk)-fuzzy right hyperideal of H2 is an (∈,∈∨qk)-fuzzy interior hyperideal of H1.Proof.
(i) Letf be an (∈,∈∨qk)-fuzzy interior hyperideal of H1 and let x2,y2,z2∈H2. Then there exist x1,y1,z1∈H1 such that ψ(x1)=x2, ψ(y1)=y2 and ψ(z1)=z2. Now, we have (58)⋀z2∈x2∘2y2ψfz2=⋀z2∈x2∘2y2⋁t∈ψ-1z2ft≥⋀z2∈x2∘2y2fz1=⋀ψz1∈ψx1∘2ψy1fz1=⋀ψz1∈ψx1∘1y1fz1=⋀z1∈x1∘1y1fz1=⋀z1∈e∘1x1∘1y1fz1≥min⁡fx1,1-k2≥min⁡⋁x1∈ψ-1x2fx1,1-k2≥min⁡ψfx2,1-k2. Let x2≤2y2. Since ψ is inverse isotone, there exist unique x1,y1∈H1 such that ψ(x1)=x2, ψ(y1)=y2 and x1≤1y1. Thus we have (59)ψfx2=⋁t∈ψ-1x2ft=fx1≥min⁡fy1,1-k2=min⁡⋁t∈ψ-1y2ft,1-k2=min⁡ψfy2,1-k2. This shows that the image of an (∈,∈∨qk)-fuzzy interior hyperideal of H1 is an (∈,∈∨qk)-fuzzy right hyperideal of H2. The proof of (ii) can be seen in a similar way.In the following example we show that in an ordered LA-semihypergroupH1 without pure left identity, the image of an (∈,∈∨qk)-fuzzy subset f under ψ can be or cannot be an (∈,∈∨qk) -fuzzy right hyperideal of H2.Example 51.
LetH1={x,y,z,w} and H2={a,b,c} be two ordered LA-semihypergroups defined by the following hyperoperations “∘1,∘2” and the orders “≤1,≤2”: (60)
∘
1
x
y
z
w
x
x
x
x
x
y
x
y
,
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x
,
z
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,
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,
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1
≔
a
,
a
,
a
,
b
,
a
,
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,
a
,
d
,
b
,
b
,
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,
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∘
2
a
b
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a
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a
,
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≤
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≔
a
,
a
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a
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a
,
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a
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b
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d
. We define a homomorphism ψ:(H1,∘1,≤1)→(H2,∘2,≤2) by(61)ψ:H1⟶H2∣r⟶ψr≔cifr∈x,zbifr=yaifr=w. We take a fuzzy subset f of H1 as f(x)=0.1, f(y)=0.6, f(z)=0.7 and f(w)=0.5. By routine calculation one can see that the image of f under ψ is an (∈,∈∨qk)-fuzzy right hyperideal of H2 for k∈[0,1), but f is not an (∈,∈∨qk)-fuzzy interior hyperideal of H1.Theorem 52.
Letψ:(H1,∘1,≤1)→(H2,∘2,≤2) be a homomorphism from an ordered LA-semihypergroup H1 to an ordered LA-semihypergroup H2. If H1 contains a pure left identity e, then the preimage of every (∈,∈∨qk)-fuzzy interior hyperideal of H2 is an (∈,∈∨qk)-fuzzy generalized bi-hyperideal of H1, provided ψ is inverse isotone.Proof.
The proof is similar to the proof of Theorem50(i).


## 5. Conclusions

Fuzzification of algebraic hyperstructures plays an important role in mathematics with wide range of applications in many disciplines such as computer sciences, chemistry, engineering, and medical diagnosis. In this paper, we have introduced the concept of(∈,∈∨qk)-fuzzy hyperideals in ordered LA-semihypergroup and investigated their related properties. We hope that this work would offer foundation for further study of the theory on ordered LA-semihypergroup and fuzzy ordered LA-semihypergroup. The obtained results probably can be applied in various fields such as computer sciences, control engineering, coding theory, theoretical physics, and chemistry. In our future research, we will consider the characterization of intraregular ordered LA-semihypergroup in terms of (∈,∈∨qk)-fuzzy hyperideals.



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*Source: 9494072-2018-07-08.xml*