# Fuzzy Ideals in Pseudo-Hoop Algebras

Authors: Teferi Getachew Alemayehu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebraH with conditions pDN and the set of all fuzzy congruences relation on H. Additionally, we investigate the relationship between fuzzy ideals and fuzzy filters in pseudo-hoop algebras by considering the set of complement elements of pseudo-hoop algebras. Lastly, we study fuzzy prime ideals and fuzzy ⊙-prime ideals and their relation.


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## Body

## 1. Introduction

Bosbach [1, 2] introduced Hoop-algebras which are ordered commutative residualed integral monoids. Georgescu et al. [3] introduce the concepts of pseudo-hoop algebras, which is non-commutative generalizations of hoop algebras, following after Georgescu and Iorgulescu [4] introduce the notions of pseudo-MV algebras, also Nola et al. [5] present pseudo-BL algebras. Pseudo-hoop algebras are weaker structures than Pseudo-MV algebras and pseudo-BL algebras. They are particular cases of pseudo-hoop algebras. In recent years, the investigation of hoop algebras and pseudo-hoop algebras has made great progress. Many researchers study in filter and ideals of pseudo-hoop algebras in [6–9] and ideals of pseudo-hoop algebras in [10].Fuzzy set theory has been studied by many scholars in many directions. Rosenfeld [11] introduced fuzzy subgroups and his work led to a new field in fuzzy mathematics. Since then many mathematicians have been used this idea and applied in different algebras such as in group [12–14], ring [15–17], Universal algebra [18–22], lattice [23–26], Ockham algebra [27, 28], MS-algebra [29–32], BL-algebra [33–37], BE-algebra [38, 39] etc.In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relatonship between the set of all normal fuzzy ideals of a pseudo-hoop algebraH with conditions pDN and the set of all fuzzy congruences relation on H. For a pseudo-hoop algebra, the pre image of the (normal) fuzzy ideals is the (normal) fuzzy ideals under homomorphism, and the image is the (normal) fuzzy ideals under isomorphism.Finally, by taking into account the set of complement elements of pseudo-hoop algebras, we investigate the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. Additionally, we investigate the relationship between fuzzy prime ideals and⊙-prime ideals.


## 2. Preliminaries

We retain significant results and fundamental terminology from this area that we will require in the follow-up. To learn more, go to [1–3, 10, 40, 41].Definition 1 (see [1]).
A structureH,⊙,1,≤ is a partially ordered monoid if(1)
H,⊙,1 is a monoid, i.e., ⊙ is associative and 1⊙s=s=s⊙1 for all s∈H,(2)
≤ is a partial order on H,(3)
for alls,t,u∈H, s≤t⇒s⊙u≤t⊙u and u⊙s≤u⊙t.
The largest element (under≤) of the set u∈H:u⊙s≤t, if it exists, is called the left-residual of s relative to t, and is denoted by s→t. Thus s→t can be defined by the condition ∀xx⊙s≤t⇔x≤s→t.Definition 2.
Reference [1] A,⊙,1,≤ is left-residuated if s→t exists for all s,t∈H.
In this event the enriched structureH,⊙,→,1,≤ is called a left-residuated partially ordered monoid.
A left-residuated partially ordered monoid can be thought of as an algebraH,⊙,→,1, since the partial order can be retrieved via s≤t if and only if s→t=1. The inverse right divisibility relation ≤r on a monoid H,⊙,1 is defined by s≤rt⇔∃us=u⊙t.Definition 3 (see [1]).
An algebraH,⊙,→,1 is a left-complemented monoid if H,⊙,→,1,≤r is a partially ordered monoid with left-residuation →.
The notions of right-residual, right-residuated a partially ordered monoid, inverse left divisibility relation≤l, and right-complemented monoid are defined similarly.Lemma 1 (see [1, 2]).
(1)
IfH,⊙,→,1 is a left-complemented monoid, then ≤r is a meet-semilattice order, where s∧t=s→t⊙a for all s,t∈H.(2)
IfH,⊙,→,1 is a right-complemented monoid, then ≤l is a meet-semilattice order, where s∧t=s⊙s→t for all s,t∈H.
A variety of left- (and right-) complemented monoids exist, and they have a straightforward equational characterization.Theorem 1 (see [1, 2]).
An algebraH,⊙,→,1 is a left-complemented monoid iff the following identities hold:(1)
s⊙1=1⊙s=s,(2)
s→s=1,(3)
s→t⊙s=t→s⊙t,(4)
s⊙t→u=s→t→u.Theorem 2 (see [3]).
An algebraH,⊙,→,1 is a right-complemented monoid iff the following identities hold:(1)
s⊙1=1⊙s=s,(2)
s→s=1,(3)
s⊙s→t=t⊙t→s,(4)
s⊙t→u=t→s→u.Definition 4.
(see [3]). If an algebra H,⊙,→,⇝,1 of type 2,2,2,0 satisfies the following requirements or any s,t,u∈H, it is known to be a pseudo-hoop:(1)
H,⊙,→,1 is a left-complemented monoid,(2)
H,⊙,⇝,1 is a right-complemented monoid,(3)
≤r=≤l.
We will agree in the next section that⊙ has priority towards the operations →,⇝. The inverse left and right divisibility relations are coincident in a pseudo-hoop by (3) and are indicated by the symbol ≤.
It follows that for anys,t∈H, s≤t⇔∃us=t⊙u⇔∃us=u⊙t.
We specify that for any natural integern on H, s0=1 and sn=sn−1⊙s.
A partial order on H is represented by the relation denoted bys t s t s t = 1 s t = 1. If is commutative or, alternatively, if = H, then H is referred to as a hoop algebra. Additionally, H is said to be bounded if 0 s for any s H.
The relation≤ described by s≤t⇔s→t=1⇔s⇝t=1 is a partial order on H. If ⊙ is commutative or alternatively →=⇝H is said to be a hoop algebra. Additionally, H is said to be bounded if 0≤s for any s∈H. In this instance, s−=s→0 and u∼=u⇝0 on H are defined. If s−∼=s∼− for all s∈H, then the bounded pseudo-hoop algebra is called good (see [42]). In a bounded pseudo-hoop algebra H, if s−∼=s∼−=s for all s∈H, then H is called satisfying the pDN condition (see [42]). A good pseudo-hoop algebra H is known as normal if it satisfies s⊙t−∼=s−∼⊙t−∼ for each s,t∈H.Theorem 3 (see [42]).
Let’s consider the pseudo-hoop algebrH,⊙,→,⇝,1. For any s,t,u∈H, the subsequent characteristics are true:(1)
s⊙t≤u⇔s≤t→u⇔t≤s⇝u,(2)
ifs≤t, then s⊙w≤t⊙w and w⊙s≤w⊙t,(3)
H,⊙,1 is a monoid,(4)
s∧t=s→t⊙t=t→s⊙t=t⊙s⇝t=t⊙t⇝s,(5)
s≤t, then t→u≤s→u and t⇝u≤s⇝u,(6)
s≤t, then u→s≤u→t and u⇝t≤u⇝s,(7)
t→u⊙s→t≤s→u and t⇝u⊙s⇝t≤s⇝u.Theorem 4 (see [42]).
LetH be a pseudo-hoop algebra with bounds. The following statements are valid for every s,t,u∈H:(1)
s⊙0=0⊙s=0,(2)
s−⊙s=0,s⊙s∼=0,(3)
s⊙t=0⇔s≤t−⇔t≤s∼,(4)
s≤s−∼,s≤s∼−,(5)
s−∼−=s−,s∼−∼=s∼,(6)
ifH is good, then s→t−∼=s−∼→t−∼ and s⇝t−∼=s−∼⇝t−∼,(7)
ifH is good, then s→t−=s−∼→t− and s⇝t∼=s−∼⇝t∼.Definition 5 (see [3]).
A filterF of a pseudo-hoop algebra H is a nonempty subset of H which satisfies(1)
s,t∈F⇒s⊙t∈F,(2)
for anys,t∈H if s≤t and t∈F, then s∈F.Definition 6 (see [3]).
A filterF of a pseudo-hoop algebra H is known as normal if s→t∈F⇔s⇝t∈F.Theorem 5 (see [10]).
LetH,⊙,→,⇝,1 be pseudo-hoop algebra with bounds. For every s,t∈H , we describe left addition ⊲ and right addition ⊳ as follows: s⊲t=t−⇝s and s⊳t=s∼→t.Theorem 6 (see [10]).
LetH be a pseudo-hoop algebra. For all t,u,m,n∈H, if t≤u and m≤n, then t⊲m≤u⊲n and t⊳m≤u⊳n.Theorem 7 (see [10]).
LetH be a pseudo-hoop algebra. If H is normal, then left addition ⊲ and right addition ⊳ are associative.Theorem 8 (see [10]).
LetI be a nonempty subset of a bounded pseudo-hoop algebra H. Then I is called a left ideal of H if:(1)
t,u∈I⇒t⊲u∈I,(2)
for anyt,u∈H,t≤u and y∈I⇒t∈I. Similarly, I is known as a right ideal of H if:(3)
t,u∈I⇒t⊳u∈I,(4)
for anyt,u∈H,t≤u and u∈I⇒t∈I.
IfI is a left ideal and a right ideal of H, we say I to be an ideal of H.
For every idealI of H, we get 0∈I. For each t∈H, we get u∈I⇔t−∼∈I⇔t∼−∈I.
An idealI of H is known as proper if I≠H. An ideal I of H is known as normal if t−⊙u∈I⇔u⊙t∼∈I for every t,u∈H.
LetH1 and H2 be pseudo-hoop algebras. In [10], a map f:H1→H2 is known as a pseudo-hoop homomorphism if f preserves the operations ⊙, → and ⇝ The pseudo-hoop homomorphism f:H1→H2 is called a bounded pseudo-hoop homomorphism if H1, H1 are bounded and f0=0.


## 3. Fuzzy Ideals in Pseudo-hoop Algebras

In this subsection, we discuse the principles of fuzzy ideals in pseudo-hoop algebras and two types of fuzzy ideals (left and right fuzzy ideals). We provide several similar characterizations of fuzzy ideals of good pseudo-hoop algebras.Definition 7.
Letξ be a fuzzy subset of a bounded pseudo-hoop algebra H. Then ξ is called the left fuzzy ideal of H, if(1)
ξt⊲u≥ξt∧ξu, for every t,u∈H,(2)
t≤u⇒ξt≥ξu, for every t,u∈H.Similarly,ξ is called the right fuzzy ideal of H, if.(3)
ξt⊳u≥ξt∧ξu, for every t,u∈H,(4)
t≤u⇒ξt≥ξu, for every t,u∈H.
Ifξ is a left fuzzy ideal and a right fuzzy ideal of H, we say ξ to be a fuzzy ideal of H.Lemma 2.
Letξ be every fuzzy ideal of a bounded pseudo-hoop H. Then(1)
ξ0≥ξt,(2)
ξt=ξt−∼=ξt∼− for every t∈H.Proof.
(1)
Since0≤t for every t∈H. Then by (2) ξ0≥ξt by Definition 7. (2) and (4).(2)
ξt=ξt∧ξ0≤ξt⊳0=ξt∼−. Conversely, since t≤t∼−. Then ξt≥ξt∼−. Thus ξt=ξt∼−. Similarly, ξt=ξt−∼.
Obviously,χA,χ0 are fuzzy ideals of H.
The set of all fuzzy ideals ofH is denoted by ℱℐℋ.ξ is known as a proper fuzzy ideal if λ is a nonconstant fuzzy ideal of H.Definition 8.
LetH be bounded pseudo-hoop algebra. A fuzzy ideal ξ of H is called fuzzy normal ideal if ξs−⊙t=ξt−⊙s for any s,t∈H.Example 1.
LetH=0,w,x,y,z,1. Define the operations →,⇝ and ⊙ on H as follows:
ThenH,⊙,→,⇝,1 is a bounded hoop algebra. It is easy to see that b⊲c=c−⇝b=a⇝b=b and. c⊳a=c∼→a=a→a=1.
Define a fuzzy subsetξ:H→0,1 as ξ0=1=ξy, ξw=ξy=ξz=0.7 and ξ1=0.2, a fuzzy subset λ:H→0,1 as λ0=0.9,λw=λz=0.7, λx=λy=λ1=0.4 and μ:H→0,1 as μ0=0.9,μw=μz=μx=μy=μ1=0.3 are fuzzy ideals of H.Theorem 9.
Letξ be a fuzzy subset of H. Then ξ is a fuzzy ideal if and only if for any γ∈0,1, ∅≠ξγ is an ideal of H.Proof.
The proof is straight forward.Corollary 1.
I is an ideal of H iff χI is a fuzzy ideal.Definition 9.
LetH be a bounded pseudo-hoop and ξ be a fuzzy subset of H. The fuzzy ideal generated by ξ means the smallest fuzzy ideal inluding ξ, which is denoted by ξ.Theorem 10.
Letξi:i∈Δ is a fuzzy ideal of pseuo-hoop algebra H. Then ∩i∈Δξi is a fuzzy ieal of H.Theorem 11.
Letξ is a fuzzy subset of H. Then ξ=∩λ:ξ⊆λ,λis any fuzzy ideal ofH.Proof.
The proof is clear.Theorem 12.
LetA be any subset of H. Then χA=χA.Theorem 13.
Letξ be a nonconstant fuzzy subset of a good pseudo-hoop algebra H such that ξt≤ξ0 for any t∈H. Then the following conditions are equivalent:(1)
ξ is a fuzzy ideal of H,(2)
ξu≥ξt∧ξt−⊙u for every t,u∈H,(3)
ξu≥ξt∧ξu⊙t− for every t,u∈H.Proof.
1⇒2 Assume that ξ is a fuzzy ideal of H. Since t−⊙u≤t−⊙u, then we have u≤t−⇝t−⊙u=t−⊙u⊲t. Now, ξt−⊙u∧ξt≤ξt−⊙u⊲t≤ξu . This implies 1⇒2 holds.
2⇒1 Assume that 1 holds. Let t,u∈H such that t≤u. Then we have u−≤t− and so u−⊙t≤t−⊙t=0.
ξu=ξu∧ξ0=ξu∧ξt−⊙t≤ξt by (2) and hypothesis.
ASu−⊙t⊲u=u−⊙u−⇝t≤t, then ξt∧ξu≤ξu−⊙t⊲u∧ξu≤ξt⊲u. Since H good pseudo-hoop, t∼−=t−∼ and u−⊙t⊳u=u−⊙t∼→u≤t∼−. Also we know that t−⊙t∼−=0ξt∧ξu=ξt∧ξ0∧ξu=ξt∧ξu−⊙t∼−∧ξu≤ξt∼−∧ξu≤ξt∼−∧ξu≤ξu−⊙t⊳u∧ξu≤ξt⊳u.
Thusξ is fuzzy ideal of H. Hence 1⇔2 holds. Similarly, we show that 1⇔3. □Theorem 14.
Letξ be a nonconstant fuzzy subset of a good pseudo-hoop algebra H such that ξt≤ξ0 for any t∈H. Then the following conditions are equivalent:(1)
ξ is a fuzzy ideal of H,(2)
ξu≥ξt∧ξt−→u−∼ for every t,u∈H,(3)
ξu≥ξt∧ξt∼→u∼− for every t,u∈H.Proof.
1⇔2 Assume that ξ is a fuzzy ideal of H. Since t−⊙u−∼≤t−⊙u−∼−∼=t−→u−∼∼=t−→u−∼ and u≤u−∼.
ξt∧ξt−→u−∼≤ξt∧ξt−⊙u−∼≤ξu−∼≤ξu by Theorem 11 (2) and Definition 7 (1) and (3).
1⇔2 Suppose that 2 holds. Since t−→t−∼−∼=t−→t−∼=0.
ξt=ξt∧ξ0=ξt∧ξt−→t−∼≤ξt−→t−∼−∼≤ξt−∼ by (2).
Now,ξt∧ξt−⊙u≤ξt∧ξt−⊙u−∼≤ξt∧ξt−→u→∼≤ξu By (2). Hence ξ is a fuzzy ideal of H by (2).
Hence1⇔2 holds. Similarly, we show that 1⇔3.Theorem 15.
LetH be a bounded and normal pseudo-hoop algebra and ξ be a non constant fuzzy subset of H. Then(1)ξa=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊲s2⊲…⊲sn,=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊳s2⊳…⊳sn.Proof.
Putλa=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊲s2⊲…⊲sn. We prove that λ is the smallest ideal containing ξ. Clearly ξ⊆λ.(2)λa∧λa−⊙b=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊲s2⊲…⊲sn∧∨ξt1∧ξt2∧….∧ξtn:a−⊙b≤s1⊲s2⊲…⊲sn≤∨ξs1∧ξs2∧….∧ξsn∧ξt1∧ξt2∧….∧ξtn:b≤a−⊙b⊲a≤t1⊲t2⊲…⊲tn⊲s1⊲s2⊲…⊲sn≤∨ξb∧ξb∧….∧ξb:b≤a−⊙b⊲a≤t1⊲t2⊲…⊲tn⊲s1⊲s2⊲…⊲sn=λb.
This implies by Theorem 15,λ is fuzzy ideals of H.
Suppose thatμ is any fuzzy ideal of H containing ξ. Now,(3)λa=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊲s2⊲…⊲sn≤∨μs1∧μs2∧….∧μsn:a≤s1⊲s2⊲…⊲sn,=μa,=μa.
Henceξa=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊲s2⊲…⊲sn. With similar procider, we have ξa=∨ξs1∧ξs2∧….∧ξsn:a≤s1⊳s2⊳…⊳sn.


## 4. Fuzzy Ideals and Fuzzy Congruences Relations

In this subsection, we introuce fuzzy congruences relation on pseudo-hoop algebras induced by fuzzy ideals. We construct the fuzzy quotient pseudo-hoop algebras via fuzzy ideals and prove that there is a one-to-one correspondence between the set of all normal fuzzy ideals of a pseudo-hoop algebraH with condition pDN and the set of all fuzzy congruences relation on H.In pseudo-hoop algebra’s, pre image of (normal) fuzzy ideals is (normal) fuzzy ideals under homomorphism, while image is (normal) fuzzy ideals under isomorphism.Definition 10.
LetH,⊙,→,⇝ be a pseudo-hoop algebra and Θ be a fuzzy equivalence relation on H. Then(1)
The fuzzy equivalence relationΘ is Known as a left fuzzy congruence relation if Θt,u≤Θa⊙t,b⊙u∧Θa→t,a→u∧Θa⇝t,a⇝u for every t,u,a∈H.(2)
The fuzzy equivalence relationΘ is Known as a right fuzzy congruence relation if Θt,u≤Θx⊙a,u⊙a∧Θx→a,u→a∧Θt⇝a,u⇝a for every t,u,a∈H.(3)
The fuzzy equivalence relationΘ is known as a fuzzy congruence relation if Θt1,u1∧Θt2,u2≤Θt1⊙t2,u1⊙u2∧Θt1→t2,u1→u2∧Θt1⇝t2,u1⇝u2 for every t1,t2u1,u2∈H.Example 2.
LetH be a hoop algebra of eExample 4. Define a fuzzy relation Θ on H as Θ0,0=Θw,w=Θx,x=Θy,y=Θz,z=Θ1,1=0.9,Θ0,w=Θ0,z=Θw,0=Θw,z=Θz,0=Θz,w=Θx,y=Θx,1=Θy,x=Θy,1=Θ1,x=Θ1,y=0.6,Θ0,y=Θy,0=Θ0,1=Θ1,0=Θw,x=Θx,w=Θw,y=Θy,w=Θ1,w=Θw,1=Θx,z=Θz,x=Θy,z=Θz,y=Θ1,z=Θz,1=Θ0,x=Θx,0=0.3. It is easily check that Θ is a fuzzy congruence relation.Theorem 16.
A fuzzy relation on a pseudo-hoop algebraH,⊙,→,⇝ is a fuzzy congruence relation iff it is a left and a right fuzzy congruence relation.Proof.
The proof of it is clear.
Letξ be a fuzzy ideal of a bounded pseudo-hoop algebra H. Then we expres a fuzzy relation Θξ on H as follows:(4)Θξt,u=ξt−⊙u∧ξu−⊙t∧ξt∼⊙u∧ξu∼⊙t.for every t,u∈H.Theorem 17.
LetH be a bounded pseudo-hoop algebra and ξ a fuzzy ideal of H. Then Θξ is a fuzzy equivalence relation on H.Proof.
Sincet⊙t−=0 and t⊙t∼=0, Θξt,t=ξt⊙t−∧ξt⊙t∼=ξ0∧ξ0≥ξu−⊙v∧ξu−⊙u∧ξu∼⊙v∧ξv∼⊙u=Θξu,v for any t,u,v∈H. Hence Θξ is fuzzy reflexive.
ClearlyΘξ is fuzzy symmetric. Next, we show that Θξ is fuzzy transitive.
Sincev−⊙u−⊙v−⊙t=v−→u−⊙v−⊙t≤u−⊙t ,(5)t−⊙u−⊙t−⊙v=t−→u−⊙t−⊙v≤u−⊙z,t⊙v∼⊙u⊙v∼∼=t⊙v∼⊙v∼⇝v∼≤t⊙u∼ and(6)v⊙t∼⊙v⊙t∼∼=v⊙t∼⊙t∼⇝u∼≤v⊙u∼,Θξt,u∧Θξu,v=ξt−⊙u⊙ξu−⊙t∧ξu−⊙v∧ξv∼⊙yξx⊙u∼⊙ξu⊙t∼∧ξu⊙v∼∧ξv⊙u∼≤ξt−⊙u⊙ξv−⊙u−⊙v−⊙t∧ξt−⊙u−⊙t−⊙v∧ξv∼⊙u∧ξt⊙v∼⊙u⊙v∼∼⊙ξu⊙t∼∧ξu⊙v∼∧ξv⊙t∼⊙u⊙t∼∼≤ξt−⊙v∧ξv−⊙t∧ξt⊙v∼∧ξv⊙t∼by Theorem3.8=Θξt,v.
This impliesΘξ is transitive. Hence Θξ is fuzzy equivalence relation on H. □Theorem 18.
LetH be a good pseudo-hoop algebra and ξ a fuzzy normal ideal of H. Then Θξ is a fuzzy congruence relation on H.Proof.
Θξ is an equvalence relation, by Theorem 15.
Ast⊙a⊙u⊙a∼=t⊙a⊙a⇝u∼≤t⊙u∼,
u⊙a⊙t⊙a∼=u⊙a⊙a⇝t∼≤u⊙t∼ an ξ is a fuzzy normal. Then we have(7)Θξt,u=ξt−⊙u∧ξu−⊙t∧ξt⊙u∼∧ξu⊙t∼≤ξt⊙u∼∧ξu⊙t∼≤ξt⊙a⊙u⊙a∼∧ξu⊙a⊙t⊙a∼=ξt⊙a⊙u⊙a∼∧ξu⊙a⊙t⊙a∼ξu⊙a−⊙t⊙a∧ξt⊙a−⊙u⊙a=Θξt⊙a,u⊙a.
Similarly,θξt,u≤Θξa⊙t,a⊙u.
Sincet−⊙u−⊙t−⊙u−∼=t−→u−⊙t−⊙u−∼≤u−⊙u−∼=0 and u−⊙t−⊙u−⊙t−∼=u−→t−⊙u−⊙t−∼≤t−⊙t−∼=0.(8)Θξt,u=ξt−⊙u∧ξu−⊙t∧ξt⊙u∼∧ξu⊙t∼=ξt−⊙u∧ξu−⊙t∧ξt⊙u∼∧ξu⊙t∼∧ξ0∧ξ0≤ξt−⊙u∧ξu−⊙t∧ξt⊙u∼∧ξu⊙t∼∧ξt−⊙u−⊙t−⊙u−∼∧ξu−⊙t−⊙u−⊙t−∼≤ξt−⊙u−∼∧ξu−⊙t−∼by Theorem3.10=ξt−⊙u−∼∧ξu−⊙t−∼∧ξu−−⊙t−∧ξt−−⊙u−,ξis fuzzy normal ideal=Θξt−,u−.
Similarly,Θξt,u≤Θξt∼,u∼.
For anyt,u,a∈H,(9)Θξt,u≤Θξt−∼,u−∼≤Θξt−∼,u−∼≤Θξt−∼⊙a∼−,u−∼⊙a∼−=Θξt−∼→a−∼,u−∼→a−∼≤Θξt−∼→a−∼,u−∼→a−∼=Θξt→a−∼,u→a−∼His good pseudo−hoop=Θξt→a,u→a.
Similarly,Θξt,u≤Θξa→t,a→u, Θξt,u≤Θξt⇝a,u⇝a and Θξt,u≤Θξa⇝t,a⇝u. Hence Θξ is a fuzzy congruence relation on H.
LetH be a good pseudo-hoop algebra and ξ be a fuzzy normal ideal of H. We define H/ξ=tΘξ:t∈H, where tΘξu=Θξt,u. For every t,u∈H, we expres the operation ⊙,→,⇝ on H/ξ by tΘξ⊙uΘξ=t⊙uΘξ,tΘξ→uΘξ=t→uΘξ,tΘξ⇝uΘξ=t⇝uΘξ.Theorem 19.
LetH be a good pseudo-hoop algebra and ξ be a fuzzy normal ideal of H. Then H/ξ,⊙,→,⇝,1Θξ is a bounded pseudo hoop algebra with condition PDN.Theorem 20.
.LetH be a good pseudo-hoop algebra and λ is a fuzzy subset of H. Then(1)
LetΘ be a fuzzy congruence relation on H. Then λt=Θλt,0 is a normal fuzzy ideal of H and Θλ is a fuzzy congruence relation on H. Let H satisfies the condition PDN. Then Θλ concide with Θ,(2)
Letξ be a fuzzy normal ideal of H. Then Θξ is a fuzzy congruence relation on H. Additionaly, 0Θξt=Θξ0,t is a fuzzy normal ideal of H and coincides with ξ,(3)
LetH satisfies the condition PDN. Then there is a one-to-one correspondence between the set of fuzzy congruence relations on H and the set of fuzzy normal ideals of H.Proof.
(1)
Firest, we want to show thatλ is a normal fuzzy ideal of H. For every t,u∈H,(10)λt∧λu=Θt,0∧Θu,0≤Θt,0∧Θu−,0−≤Θt,0∧Θu−→t,0−→t=Θt,0∧Θt⊲u,t≤Θt⊲u,0by fuzzy transtive=λt⊳u.Similarly,λt∧λu≤λt⊳u.Also, lett,u∈H such that t≤u. Now,(11)λu=Θu,0≤Θu∼,0∼≤Θt⊙u∼,x⊙0∼=Θ0,t,sincet⊙u∼=0andt⊙0∼=t=λt.Thusλ is a fuzzy ideal of H.SinceH is a good pseudo-hoop, we have u∧t−∼⊙t∼=t−∼→⊙t−∼⊙t∼=t−∼→⊙t−∼⊙t∼=0.For everyt,u∈H,(12)λt−⊙u=Θt−⊙u,0≤Θt−⊙u∼,0∼≤Θu⊙u⇝t−∼,u⊙0∼,t−⊙u∼=u⇝t−∼=Θu∧t−∼,u≤Θu∧t−∼⊙t∼,u⊙t∼=Θ0,u⊙t∼=λu⊙t∼.Similarly,λu⊙t∼≤λt−⊙u. Hence λu⊙t∼=λt−⊙u and λ is normal fuzzy ideal.By Theorem 16,Θλ is a fuzzy congruence relation on H.Assume thatH satisfies PDN condition.Next, we want to show thatΘ=Θλ. For any t,u∈H, Θt,u≤Θt−,u−≤Θt−⊙u,u−⊙u=Θt−⊙u,0=λt−⊙u. With similar procider, Θt,u≤Θt,u≤λu−⊙t, Θt,u≤Θt,u≤λt⊙u∼ and Θt,u≤Θt,u≤λu⊙t∼. This implies Θt,u≤λu−⊙t∧λu−⊙t∧λt⊙u∼∧λu⊙t∼=Θλt,u.Conversely,(13)Θλt,u=λu−⊙t∧λu−⊙t∧λt⊙u∼∧λu⊙t∼≤λu⊙t∼≤Θu⊙t∼,0≤Θu⊙t∼−⊙u,0−⊙u=Θu∧t∼−,u=Θu∧t,u,SinceHis satisfiesPDNcondition.SimilarlyΘλt,u≤Θu∧t,t. This implies Θλt,u≤Θu∧t,t∧Θu∧t,u≤Θλt,u.HenceΘλ=Θ.(2)
By Theorem 20Θλ is a fuzzy congruence relation on H. By (1), 0Θξ is normal fuzzy ideal of H. Next, we want to show that 0Θξ=ξ.Sincet−⊙0=0,0−⊙t=t,t⊙0∼=t,0⊙t∼=0. Then we have ξt=ξt−⊙0∧ξ0−⊙t∧ξt⊙0∼∧ξ0⊙t∼=Θξt,0=0Θξt. Hence 0Θξ=ξ.(3)
It is clear by (1) and (2).Theorem 21.
LetH,J be two pseudo-hoop algebras with boundes and f:H→J a bounded pseudo-hoop homomorphism. Then we have the following results:(1)
Letξ be a (normal) fuzzy ideal of J. Then f−1ξ is a (normal) fuzzy ideal of H,(2)
Letf be a isomorphism and λ be a fuzzy (normal) ideal of H. Then fλ is a fuzzy (normal) ideal of J.Proof.
(1)
Assume thatξ is a fuzzy ideal of J such that t≤u. Then we have that t→u=1. As f is a bounded pseudo-hoop homomorphism, ft→u=ft→fu=f1=1. Hence ft≤fu. Now, f−1ξu=ξfu≤ξft=f−1ξt.Next, for anyt,u∈H, f−1ξt∧f−1ξu=ξft∧ξfu≤ξft⊲fu=ξft⊲u=f−1ξt⊲u. With similarly, f−1ξt∧f−1ξu≤f−1ξt⊳u. This implies f−1ξ is a fuzzy ideal of H.Assume thatξ is a normal fuzzy ideal of J. Let t,u∈H, f−1ξt−⊙u=ξft−⊙u=ξft−⊙fu=ξft⊙fu∼=f−1t⊙u∼. This implies f−1ξ is a normal fuzzy ideal of H.(2)
Assume thatλ is a fuzzy ideal of H. We want to show that fλ is a fuzzy ideal of J. Let t,u∈J such that t≤u . As f is bounded pseudo-homomorphism, there exist a,b∈H such that fa=t and fb=u.
This implies1=t→u=fa→fb=fa→b⇒a→b=1 and also a≤b.
Now,fλu=∨λb:fb=u≤∨λa:fa=t=λt. Additionaly, fλt∧fλu=∨λa:fa=t∧∨λb:fb=u=∨λa∧λb:fb=u,fa=t≤∨λa⊲b:fb=u,fa⊲b=t⊲u=fλt⊲u. With similar procidure, fλt∧fλu≤fλt⊳u.
This impliesfλ is a fuzzy ideal of H.
Assume thatλ is a normal fuzzy ideal of H. Let t,u∈J, fλt−⊙u=fλfa−⊙fb=∨λa−⊙b:fa−⊙b=t−⊙u=∨λa⊙b∼:fa⊙b∼=t⊙u∼=fλt⊙u∼. Thus implies fλ normal fuzzy ideal of J.


## 5. Fuzzy Ideals and Fuzzy Filters

The set of complement elements of pseudo-hoop algebras is taken into consideration in this part as we discuse the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. In addition, we investigate the relationship between fuzzy prime ideals and fuzzy -prime ideals.Letλ be a nonempty fuzzy subset of a bounded pseudo-hoop algebra H. We represents λ−t−=λt and λ∼t∼=λt.Definition 11.
LetH,⊙,→,⇝,0,1 be a pseudo-hoop algebra with boundes and ξ be a fuzzy subset of H. The fuzzy sets Mξt=ξt− and Nξt=ξt∼.Example 3.
LetH=0,r,s,t,u,1. We define ⊙,→ and ⇝ on H as follows:
ThenH,⊙,→,⇝,1 is a bouneded pseudo-hoop algebra. Define a fuzzy subset ξ:H→0,1 as ξ1=ξu=ξv=ξw=0.8,ξ0=ξr=ξs=ξt=0.4. Then Mξ0=ξ1=0.4,Mξr=ξu=0.8,Mξs=ξu=0.8,Mξt=ξu=0.8,Mξu=ξt=0.4,Mξv=ξ0=0.4,Mξw=ξ0=0.4,Mξ0=ξ1=0.8. Also, Nξ0=ξ1=0.4,Nξr=ξu=0.8,Nξs=ξu=0.8,Nξt=ξu=0.8,Nξu=ξt=0.4,Nξv=ξ0=0.4,Nξw=ξ0=0.4,Nξ0=ξ1=0.8.
After routine calculation show thatξ is a fuzzy filter of H.Theorem 22.
Letξ be a fuzzy filter of a good pseudo-hoop algebra H. Then Mξ is a fuzzy ideal generated by ξ∼ and Nξ is a fuzzy ideal generated by ξ−.Proof.
For anyt,u∈H,(14)Mξt−⊙u∧Mξt=λt−⊙u−∧λt−=λu−→u−∧λt−≤λu−asλis a fuzzy filter ofH=Mλu.
This impliesMλ is a fuzzy ideal of H by Theorem 12.
Next, we show that,λ∼⊆Mλ.
For anyt∈H, there exist u∈H such that t=u∼ and also clearly u≤u∼−=t−.λ∼t=λ∼u∼=λu≤λt−=Mλt. This implies λ∼⊆Mλ.
Suppose thatη is a fuzzy ideal H such that λ∼⊆η.
Sincet≤t−∼Mλt=λt−=λ∼t−∼≤ηt−∼≤ηt.
This impliesMλ⊆η. Hence Mλ is a fuzzy ideal generated by λ∼. Similarly, Nξ is a fuzzy ideal generated by ξ−.Theorem 23.
LetH be a bounded pseudo-hoop algebra and λ a fuzzy ideal of H. If H is good, then Mλ and Nλ are fuzzy filters of H such that λ∼⊆Mλ amd λ−⊆Nλ.Proof.
For anyt,u∈H, if t≤u, then u−≤t− and t⊙u−=x→u−=t−∼→u−=t−⊙u−.(15)Mλt=λt−≤λu−=Mλu.
Also,Mλt∧Mλu=λt−∧λu−=λt−⊳u−=λt⊙u−=Mλt⊙u. This implies Mλ is a fuzzy filter of H. For any t∈H, there exists u∈H such that t=u∼. Now, λ∼t=λ∼u∼=λu=λu∼− as by Lemma 2. Thus λu∼−=λt−=Mλt. This implies λ∼⊆Mλ. Similarly, we can prove that Nλ is fuzzy filter of H and λ−⊆Nλ.Theorem 24.
Letλ be a fuzzy ideal of a bounded pseudo-hoop algebra H. Then λ=MNλ=NMλ.Proof.
For anyt∈H, λt=λt−∼=Nλt−=MNλt. Hence λ=MNλ. Similarly λ=NMλ.Theorem 25.
Letξ be a fuzzy filter of a bounded pseudo-hoop algebra H. Then ξ⊆MNξ and ξ⊆NMξ.Proof.
Sincet≤t−∼, Now, ξt≤ξt−∼=Nξt−=MNξt. Hence ξ⊆MNξ. Similarly, ξ⊆NMξ.Theorem 26.
LetH,⊙,→,⇝ be a bounded ξ is known asa fuzzy prime ideal if ξ is ξt∧u=ξt∨ξu for every t,u∈H.Example 4.
In example 4,ξ and λ are fuzzy prime ideal of H. As μw∧y=μ0=0.9 and μw∨μy=0.3∨0.3=0.3. This show that μw∨μy=0.3≠0.9=μw∧y. Therefore μ is not fuzzy prime ideal of H.Theorem 27.
LetH,J be two bounded pseudo-hoop algebras and f:H→J be a bounded pseudo-hoop homomorphism. Then, the following facts are correct:(1)
Ifξ is a fuzzy prime ideal of J, then f−1ξ is a fuzzy prime ideal of H.(2)
Ifλ is a fuzzy prime ideal of H and f:H→J is a bounded pseudo-hoop isomorphism, then fλ is a fuzzy prime ideal of J.Proof.
(1)
f−1ξ is a fuzzy ideal of X by Theorem 21. Next, we want to show that f−1ξ is a fuzzy prime ideal of H. Let t,u∈H, f−1ξt∧u=ξft∧u=ξft∨ξfu=f−1ξt∨f−1ξu. Therefore f−1ξ is a fuzzy prime ideal of H.(2)
Also,fξ is a fuzzy ideal of J, by Theorem 22 and also easily proved that fξ is a fuzzy prime ideal of J.Definition 12.
LetH,⊙,→,⇝,1 be a pseudo-hoop algebra with boundes and λ a fuzzy ideal of H. Then λ is known as a fuzzy ⊙-prime ideal of H if λt⊙u=λt∨λu for every t,u∈H.Example 5.
In example 4, a fuzzy subsetλ:H→0,1 as λ0=λw=λz=0.6, λx=λy=λ1=0.2 is a fuzzy ⊙-prime ideal of H.Theorem 28.
LetH be a pseudo-hoop algebra with bounds. Then every fuzzy ⊙-prime ideal of H is a fuzzy prime ideal of H, but the converse ot it may not be true.Proof.
Assume thatξ is fuzzy ⊙-prime ideal of H. We prove that ξ is fuzzy prime ideal. Since t⊙u≤t∧u≤t,u. Now, ξt∧u≤ξt⊙u=ξt∨ξu. Conversely, ξt∨ξu≤ξt∧u. Hence ξt∨ξu=ξt∧u. This implies every fuzzy ⊙ -prime ideal of H is a fuzzy prime ideal of H.
In example 4, a fuzzy subsetξ:H→0,1 as ξ0=0.6=ξy, ξw=ξy=ξz=ξ1=0.2. It is simple to demonstrate that ξ is the fuzzy ⊙ -prime ideal of H, but not the fuzzy prime ideal.


## 6. Conclusions

In this study, fuzzy ideals in pseudo-hoop algebras are discused. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebraH with conditions pDN and the set of all fuzzy congruences relation on H. For a pseudo-hoop algebra, the pre image of the (normal) fuzzy ideals is the (normal) fuzzy ideals under homomorphism, and the image is the (normal) fuzzy ideals under isomorphism.Lastly, by taking into account the set of complement elements of pseudo-hoop algebras, we investigate the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. Additionally, we investigate the relationship between fuzzy prime ideals and⊙ -prime ideals.



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