Homomorphism and quotient of fuzzy k-hyperideals

R. Ameria

aDepartment of Mathematics, University of Mazandaran,

Babolsar, Iran

E-mail: ameri@umz.ac.ir

H. Hedayatib

bDepartment of Mathematics, Faculty of Basic Science,

Babol University of Technology, Babol, Iran

E-mail: h.hedayati@nit.ac.ir, hedayati143@yahoo.com

Abstract

In [15], we introduced the notion of weak (resp. strong) fuzzy k-

hyperideal. In this note we investigate the behavior of them under

homomorphisms of semihyperrings. Also we define the quotient of fuzzy

weak (resp. strong) k-hyperideals by a regular relation of semihyperring

and obtain some results.

Mathematics Subject Classification: 20N20

Keywords: (semi-) hyperring, homomorphism, fuzzy weak (strong) k-

hyperideals, regular relation, (fuzzy) quotient of k-hyperideals

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1 Introduction

Following the introduction of fuzzy set by L. A. Zadeh in 1965 ([26]), the fuzzy

set theory developed by Zadeh himself and can be found in mathematics and

many applied areas. The concept of a fuzzy group was introduced by A. Rosen-

feld in [24]. The notion of fuzzy ideals in a ring was introduced and studied

by W. J. Liu [20]. T.K. Dutta and B. K. Biswas studied fuzzy ideals, fuzzy

prime ideals of semirings in [14, 16] and they defined fuzzy ideals of semirings

and fuzzy prime ideals of semirings and characterized fuzzy prime ideals of

non-negative prime integers and determined all it’s prime ideals. Recently, Y.

B. Jun, J. Neggeres and H. S. Kim ([16]) extended the concept of a L-fuzzy

(characteristic) ideal left(resp. right) ideal of a ring to a semiring. S. I. Baik

and H. S. Kim introduced the notion of fuzzy k-ideals in semirings [6].

Also a hypergroup was introduced by F. Marty ([23]), today the literature

on hypergroups and related structures counts 400 odd items [8, 9, 25]. Among

the several contexts which they aries is hyperrings. First M. Krasner studied

hyperrings, which is a triple (R, +, .), where (R, +) is a canonical hypergroup

and (R, .) is a semigroup, such that for all a, b, c ∈ R, a(b + c) = ab + ac, (b +
c)a = ba + ca ([18]). Zahedi and others introduced and studied the notion of

fuzzy hyperalgebraic structures [3, 4, 5, 11, 12, 19, 27]. In [15] we introduced

the notion of fuzzy weak (strong) k-hyperideal and then we obtained some

related basic results. In this note we investigate the behavior of them under

homomorphisms of semihyperrings. Also we define the quotient of fuzzy weak

(strong) k-hyperideals by a regular relation of semihyperring and obtain some

results.

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2 Preliminaries

In this section we gather all definitions and simple properties we require of

semihyperrings and fuzzy subsets and set the notions.

A map ◦ : H × H −→ P∗(H) is called hyperoperation or join operation.
A hypergroupoid is a set H with together a (binary) hyperoperation ◦.

A hypergroupoid (H, ◦), which is associative, that is x ◦ (y ◦ z) = (x ◦ y) ◦
z, ∀x, y, z ∈ H is called a semi-hypergroup . A hypergroup is a semihypergroup
such that ∀x ∈ H we have x ◦ H = H = H ◦ x, which is called reproduction
axiom.

Let H be a hypergroup and K a nonempty subset of H. Then K is a

subhypergroup of H if itself is a hypergroup under hyperoperation restricted

to K. Hence it is clear that a subset K of H is a subhypergroup if and only if

aK = Ka = K, under the hyperoperation on H.

A set H together a hyperoperation ◦ is called a polygroup if the following
conditions are satisfied:

(1) (x ◦ y) ◦ z = x ◦ (y ◦ z) ∀x, y, z ∈ H;
(2) ∃e ∈ H as unique element such that e ◦ x = x = x ◦ e ∀x ∈ H;
(3) ∀x ∈ H there exists an unique element, say x′ ∈ H such that
e ∈ x ◦ x′ ∩ x′ ◦ x ( we denote x′ by x−1).
(4) ∀x, y, z ∈ H, z ∈ x ◦ y =⇒ x ∈ z ◦ y−1 =⇒ y ∈ x−1 ◦ z.
A non-empty subset K of a polygroup (H, ◦) is called a subpolygroup if

(K, ◦) is itself a polygroup. In this case we write K <P H.
A commutative polygroup is called canonical hypergroup.

Definition 2.1. A hyperalgebra (R, +, .) is called a semihyperring if and only

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if

(i) (R, +) is a semihypergroup and (R, .) is a semigroup;

(ii) a.(a + b) = a.b + a.c and (b + c).a = b.c + c.a ∀a, b, c ∈ R.
A semihyperring is called with zero, if there exists an element, say 0 ∈ R

such that 0.x = 0 = x.0 and 0 + x = x = x + 0 ∀x ∈ R.
Also a semihyperring (R, +, .) is called a hyperring provided (R, +) is a

canonical hypergroup.

A hyperring (R, +, .) is called

(i) commutative if and only if a.b = b.a ∀a, b ∈ R;
(ii)with identity, if there exists an element, say 1 ∈ R such that 1.x =

x.1 = x ∀x ∈ R.
Let (R, +, .) be a hyperring, a nonempty subset S of R is called a subhy-

perring of R if (S, +, .) is itself a hyperring.

Definition 2.2. A subhyperring I of a hyperring R is a (resp. left) right

hyperideal of R provided that ( resp. x.r ∈ I) r.x ∈ I ∀r ∈ R, ∀x ∈ I. I is
called a hyperideal if I is both left and right hyperideal.

We use I = [0, 1], the real unit interval as a chain with the usual ordering, in

which
∧

stands for infimum (inf ) (or intersection ) and
∨

stands for supremum

( sup ) (or union), for the degree of membership.

A fuzzy subset of a given set X is a mapping µ : X −→ I. We denote the
set of all fuzzy subsets of X by F S(X). For µ ∈ F S(X), the level subset of µ
is defined by

µt = {x ∈ X| µ(x) ≥ t} ∀t ∈ I.

For a fuzzy subset µ of X we denote by Im(µ) the image of µ. Let {µi |

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i ∈ I} be a family of fuzzy subsets, intersection of µi’s is defined by

(
⋂
i∈I

µi)(x) =
∧
i∈I

µi(x).

Definition 2.3. Let (G, .) be a group and µ ∈ F S(G). Then µ is said to be
a fuzzy subgroup of G if ∀x, y ∈ G we have :

(i) µ(xy) ≥ µ(x) ∧ µ(y);
(ii) µ(x−1) ≥ µ(x).

Definition 2.4. If f : X −→ Y be a function and µ ∈ F S(X) , then we say
µ is f−invariant if and only if

f (a) = f (b) =⇒ µ(a) = µ(b).

In the sequel by R we mean a semihyperring.

Definition 2.5.[1] A nonempty subset I of R is called

(i) a left (resp. right) hyperideal of R if and only if

(1) (I, +) is a semihypergroup of (R, +);

(2) rx ∈ I (resp. xr ∈ I), for all r ∈ R and for all x ∈ I.
(ii) a hyperideal of R if it is both a left and a right hyperideal of R. By

I <h R, we mean hyperideal of R.

(iii) a left hyperideal I of R is called weak left k-hyperideal of R if for a ∈ I
and x ∈ R we have

a + x ⊆ I or x + a ⊆ I =⇒ x ∈ I.

A left hyperideal I of R is called strong left k-hyperideal of R if for a ∈ I
and x ∈ R we have

a + x ≈ I or x + a ≈ I =⇒ x ∈ I,

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where by A ≈ B, we mean A ∩ B 6= ∅, for all nonempty subsets A and B
of R.

A right(resp. strong) weak k-hyperideal is defined dually. A two sided

(resp. strong) weak k-hyperileal or simply a (resp. strong) weak k-hyperideal

is both left and right (resp. strong) weak k-hyperideal. We denote I <w.k.h R

(resp. I <s.k.h R) for weak (resp. strong) k-hyperideal of R.

Clearly, every (strong) weak k− hyperideal is a hyperideal, but the converse
is not true.

Example. Consider Z, the set of integer numbers. Define new hyperopera-

tions ⊕ and ◦ on Z as follow

m ⊕ n = {m, n} and m ◦ n = mn ∀m, n ∈ Z.

Clearly (Z, ⊕, ◦) is a semihyperring. Now it is easy to verify that I =<
2 >= {2k | k ∈ Z}, is a hyperideal of Z, but it isn’t strong k−hyperideal, since
3 ⊕ 2 = {3, 2} ≈ I and 2 ∈ I but 3 6∈ I.

Definition 2.6 .[7] Let R and S be semihyperrings. A mapping f : R −→ S
is said to be

(i) homomorphism if and only if

f (x + y) ⊆ f (x) + f (y) and
f (x.y) = f (x).f (y) ∀x, y ∈ R.

(ii) good homomorphism if and only if

f (x + y) = f (x) + f (y) and

f (x.y) = f (x).f (y) ∀x, y ∈ R.

Definition 2.7 .[15] A fuzzy subset µ of a semihyperring R is called a fuzzy

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left hyperideal of R if and only if

(i)
∧

z∈x+y
µ(z) ≥ µ(x)

∧
µ(y) ∀x, y ∈ R;

(ii) µ(xy) ≥ µ(y) ∀x, y ∈ R.
A fuzzy right hyperideal is defined dually. A fuzzy left and right hyperideal

is called a fuzzy hyperideal. We denote µ <f.h R for fuzzy hyperideal of R.

Definition 2.8.[15] A fuzzy hyperideal µ of R is called

(i) a weak fuzzy k-hyperideal of R if and only if

µ(x) ≥ [(
∧

u∈x+y
µ(u))

∨
(

∧
v∈y+x

µ(v))]
∧

µ(y) ∀x, y ∈ R.

(ii) a strong fuzzy k-hyperideal of R if and only if

µ(x) ≥ (µ(z) ∨ µ(z′)) ∧ µ(y) ∀z ∈ x + y, ∀z′ ∈ y + x.

Note that if (R, +) is a commutative semihyperring, then the above condi-

tions reduce to the following conditions:

µ(x) ≥ (
∧

u∈x+y
µ(u))

∧
µ(y) ∀x, y ∈ R.

and

µ(x) ≥ µ(z) ∧ µ(y) ∀z ∈ x + y.

We denote by µ <w.f.k.h R (resp. µ <s.f.k.h R), for a weak fuzzy k−hyperideal
(resp. strong fuzzy k−hyperideal) of R.

Proposition 2.9.[15] Let R be a semihyperring and µ ∈ F S(R). Then
(i) µ is a fuzzy hyperideal of R if and only if every nonempty level subset,

µt is a hyperideal of R.

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(ii) µ is a weak fuzzy k-hyperideal of R if and only if every nonempty level

subset, µt is a weak k-hyperideal of R.

(iii) µ is a strong fuzzy k-hyperideal of R if and only if every nonempty

level subset, µt is a strong k-hyperideal of R.

Lemma 2.10. Let R be a semihyperring with zero and µ be a fuzzy hyperideal

of R. Then µ(x) ≤ µ(0) for all x ∈ R.

3 Homomorphisms of Fuzzy k-Hyperideals

In this section we investigate the behavior of fuzzy weak (strong) k-hyperideals

under homomorphisms of semihyperrings.

proposition 3.1. Let f : R −→ R′ be a homomorphism of semihyperrings. If
ν <s.f.k.h R

′, then f−1(ν) <s.f.k.h R.

proof. We know that f−1(ν)(x) = ν(f (x)). Let x, y ∈ R and z ∈ x + y, then
we have f (z) ∈ f (x + y) ⊆ f (x) + f (y), and since ν <f.h R′, it concluded that
ν(f (z)) ≥ ν(f (x)) ∧ ν(f (y)).

Also

ν(f (xy)) = ν(f (x)f (y)) ≥ ν(f (x)) ∨ ν(f (y)).

Therefore f−1(ν) <f.h R.

Now let z ∈ x + y and z′ ∈ y + x, thus f (z) ∈ f (x) + f (y) and f (z′) ∈
f (y) + f (x), then ν <s.f.k.h R

′ implies that

ν(f (x)) ≥ [ν(f (z)) ∨ ν(f (z′))] ∧ ν(f (y))

as required.

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proposition 3.2. Let f : R −→ R′ be a good homomorphism of semihyper-
rings. If ν <w.f.k.h R

′
, then f−1(ν) <w.f.k.h R.

proof. We know that f−1(ν)(x) = ν(f (x)). First we prove that f−1(ν) is a

fuzzy hyperideal of R. Let x, y ∈ R and z ∈ x + y, we should prove that

ν(f (z)) ≥ ν(f (x)) ∧ ν(f (y)) (1)

(1) is valid because ν is a fuzzy hyperideal and f is a good homomorphism,

then for z ∈ x + y we have f (z) ∈ f (x + y) = f (x) + f (y).
Also similar previous proposition

ν(f (xy)) ≥ ν(f (x)) ∨ ν(f (y)).

Therefore f−1(ν) <f.h R.

Now we prove that f−1(ν) <w.f.k.h R, that is

f−1(ν)(x) ≥ {(
∧

t∈x+y
f−1(ν)(t))

∨
(

∧

t
′∈y+x

f−1(ν)(t
′
))}

∧
f−1(ν)(y) (2).

Note that since f is a good homomorphism, then t ∈ x + y if and only if
f (t) ∈ f (x) + f (y), and also ν <w.f.k.h R′, we have

ν(f (x)) ≥ {(
∧

f (t)∈f (x)+f (y)
ν(f (t)))

∨
(

∧

f (t
′
)∈f (y)+f (x)

ν(f (t
′
)))}

∧
ν(f (y)).

The last relation implies (2), and this complete the proof.

proposition 3.3. Let f : R −→ R′ be a good epimorphism of semihyperrings.
If µ <w.f.k.h R (resp. µ <s.f.k.h R) and µ be f−invariant, then f (µ) <w.f.k.h R′

(resp. f (µ) <s.f.k.h R).

proof. First we show that f (µ) <f.h R
′
.

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Let a, b ∈ R′ and c ∈ a + b, we should prove that

f (µ(c)) ≥ f (µ(a)) ∧ f (µ(b)).

We have

f (µ(c)) =
∨

z∈f−1(c)
µ(z),

f (µ(a)) =
∨

x∈f−1(a)
µ(x),

f (µ(b)) =
∨

y∈f−1(b)
µ(y).

Since µ is f−invariant, then

∃z0 ∈ f−1(c), f (µ(c)) = µ(z0),

∃x0 ∈ f−1(a), f (µ(a)) = µ(x0),

∃y0 ∈ f−1(b), f (µ(b)) = µ(y0),

therefore

f (z0) = c, f (x0) = a, f (y0) = b =⇒ f (z0) ∈ f (x0) + f (y0)
=⇒ z0 ∈ x0 + y0 ( f is a good homomorphism )
=⇒ µ(z0) ≥ µ(x0) ∧ µ(y0) (µ <f.h R)
=⇒ f (µ(c)) ≥ f (µ(a)) ∧ f (µ(b)).

For proving the second condition of a fuzzy hyperideal, we should prove

that

f (µ)(r
′
x
′
) ≥ f (µ)(x′) ∨ f (µ)(r′) ∀ r′, x′ ∈ R′

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Since f is onto, then r′ = f (r) and x′ = f (x) for some r and x in R, Thus

f (µ)(r
′
x
′
) =

∨

rx∈f−1(r′x′)
µ(rx)

= µ(r0x0) ∃r0 ∈ f−1(r′), x0 ∈ f−1(x′) (µ is f -invariant)
≥ µ(x0) ∨ µ(r0) ( µ <f.h R)
= f (µ)(x

′
) ∨ f (µ)(r′) (µ is f -invariant).

Therefore

f (µ)(r′x′) ≥ f (µ)(r′) ∨ f (µ)(x′).

Now we prove that f (µ) <w.f.k.h R
′
. Let a, b ∈ R, we show that

f (µ)(a) ≥ [(
∧

t∈a+b
f (µ)(t))

∨
(

∧

t
′∈b+a

f (µ)(t
′
))]

∧
f (µ)(b) (1)

Since f is onto and µ is f−invariant, then

f (µ)(a) = µ(x0), f (µ)(t) = µ(z0), f (µ)(t
′
) = µ(z

′
0), f (µ)(b) = µ(y0),

where

x0 ∈ f−1(a), y0 ∈ f−1(b), z0 ∈ f−1(t), z
′ ∈ f−1(t′).

Hence (1) reduced to the form

µ(x0) ≥ [(
∧

t∈a+b
µ(z0))

∨
(

∧

t
′∈b+a

µ(z
′
0))]

∧
µ(y0) (2)

On the other hand from above discussion and since f is a good homomor-

phism t ∈ a + b if and only if f (z0) ∈ f (x0) + f (y0) if and only if z0 ∈ x0 + y0.
Similarly, t′ ∈ b + a if and only if z′0 ∈ y0 + x0.

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Therefore by (2), it is enough that we prove that

µ(x0) ≥ [(
∧

z0∈x0+y0
µ(z0))

∨
(

∧

z
′
0∈y0+x0

µ(z
′
0))]

∧
µ(y0),

but clearly the last statement is true, since µ <w.f.k.h R. This complete the

proof.

In this part we define the quotient of fuzzy weak (strong) k-hyperideals by

a regular relation of semihyperring

Let R be a semihyperring and θ be an equivalence relation on R. Naturally

we can extend θ to θ to the subsets of R as follow:

Let A, B be nonempty subsets of R. Define

AθB ⇐⇒ ∀a ∈ A ∃b ∈ B : aθb, ∀b ∈ B ∃ a ∈ A : bθa.

An equivalence relation θ on R is said to be regular if for all a, b, x ∈ R
we have

(i) aθb =⇒ (a + x)θ(b + x) and (x + a)θ(x + b),
(ii) aθb =⇒ (ax)θ(bx) and (xa)θ(bx).
By R : θ we mean the set of all equivalence classes with respect to θ, that

is

R : θ = {rθ|r ∈ R}.

Remark 3.4. We know that if R is a semihyperring and θ is a regular equiv-

alence relation on R , then R : θ by hyperoperations ⊕ and ¯ is defined as
follow

xθ ⊕ yθ = {xθ|z ∈ x + y},

xθ ¯ yθ = (xy)θ.

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is a semihyperring. For µ ∈ F S(R), define (µ : θ)(xθ) =
∨

y∈xθ
µ(y). Also

we know that the mapping ϕ : R −→ R : θ defined by ϕ(a) = aθ is a good
epimorphism. Now if µ <w.f.k.h R and µ be ϕ−invariant then by proposition
3.3 it concludes that ϕ(µ) = µ : θ <w.f.k.h R : θ .

Proposition 3.5. If µ <w.f.k.h R and R has zero, then µ∗ = {x ∈ R | µ(x) =
µ(0)} is a weak k−hyperideal of R.
Proof. First we prove that µ∗ <h R. For x, y ∈ µ∗ and z ∈ x + y, then
µ(z) ≥ µ(x) ∧ µ(y) = µ(0), hence by Lemma 2.10 µ(z) = µ(0), therefore
z ∈ µ∗.

Let r ∈ R and x ∈ µ∗, then we have

µ(rx) ≥ µ(r) ∨ µ(x)
= µ(r) ∨ µ(0) ( x ∈ µ∗ )
= µ(0) ( by Lemma 2.10 )

=⇒ µ(rx) = µ(0) ( by Lemma 2.10 )
=⇒ rx ∈ µ∗.

Now suppose r + x ⊆ µ∗ or x + r ⊆ µ∗ and x ∈ µ∗, we show that r ∈ µ∗.
From µ <w.f.k.h R then we have :

µ(r) ≥ [(
∧

z∈r+x
µ(z))

∨
(

∧

z′∈x+r
µ(z′))]

∧
µ(x).

Since µ(x) = µ(0) and
∧

z∈r+x
µ(z) = µ(0) and

∧

z′∈x+r
µ(z′) = µ(0), then

µ(r) ≥ µ(0), and then by Lemma 2.10, µ(r) = µ(0). Therefore µ∗ <w.k.h R.

Proposition 3.6. If µ <s.f.k.h R, then µ
∗ = {x ∈ R | µ(x) > 0} is a strong

k-hyperideal of R.

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Proof. Let x, y ∈ µ∗ and z ∈ x + y, then by hypothesis yields

µ(z) ≥ µ(x) ∧ µ(y) > 0,

thus z ∈ µ∗.
If r ∈ R and x ∈ µ∗, then we have

µ(rx) ≥ µ(r) ∨ µ(x) ≥ µ(x) > 0,

therefore rx ∈ µ∗. Similarly xr ∈ µ∗. Thus µ∗ <h R.
Now if r + x ≈ µ∗ or x + r ≈ µ∗ and x ∈ µ∗.
By hypothesis we have

µ(r) ≥ (µ(z) ∨ µ(z′)) ∧ µ(x) > 0 ∀z ∈ r + x ≈ µ∗, ∀z′ ∈ x + r ≈ µ∗,

that is r ∈ µ∗, and hence µ∗ <s.k.h R.

Proposition 3.7. Let R be a semihyperring with zero and x, y ∈ R:
(i) If µ <w.f.k.h R and µ(t) = µ(0) = µ(t

′) for all t ∈ x+y and t′ ∈ y +x,
then µ(x) = µ(y).

(ii) If µ <s.f.k.h R and µ(u) = µ(0) = µ(v) for some u ∈ x + y and
v ∈ y + x, then µ(x) = µ(y).
Proof. (i) Since µ <w.f.k.h R and µ(t) = µ(0) = µ(t

′) for all t ∈ x + y and
t′ ∈ y + x, then

∧
t∈x+y

µ(t) = µ(0) =
∧

t′∈y+x
µ(t′), thus

µ(x) ≥ [(
∧

t∈x+y
µ(t))

∨
(

∧

t′∈y+x
µ(t′))]

∧
µ(y)

= µ(0) ∧ µ(y)
= µ(y) (by Lemma 2.10)

=⇒ µ(x) ≥ µ(y).

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Similarly we conclude that µ(y) ≥ µ(x). Therefore µ(x) = µ(y).
(ii) Suppose u ∈ x + y and v ∈ y + x such that µ(u) = µ(0) = µ(v), since

µ <s.f.k.h R, then

µ(y) ≥ (µ(u) ∨ µ(v)) ∧ µ(x) = µ(0) ∧ µ(x) ( by hypothesis )
= µ(x) (by Lemma 2.10)

=⇒ µ(y) ≥ µ(x).

Similarly we obtain µ(x) ≥ µ(y). Therefore µ(x) = µ(y).

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