# P-Fuzzy Ideals and P-Fuzzy Filters in P-Algebras

Authors: Wondwosen Zemene Norahun
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

In this paper, we introduce the concept of p-fuzzy ideals and p-fuzzy filters in a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filter.


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## Body

## 1. Introduction

The concept of fuzzy sets was firstly introduced by Zadeh [1]. In 1971, Rosenfeld used the notion of a fuzzy subset of a set to introduce the concept of a fuzzy subgroup of a group [2]. Rosenfeld’s paper inspired the development of fuzzy abstract algebra. Since then, several authors have developed interesting results on fuzzy theory (see [3–14]). In this paper, we introduce the concept of p-fuzzy ideals and p-fuzzy filters in p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. Moreover, we prove that, for any fuzzy ideal of L, there is the smallest p-fuzzy ideal containing it. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we prove that the image and inverse image of a p-fuzzy ideal is a p-fuzzy ideal under a ∗-epimorphism mapping. Finally, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filters.


## 2. Preliminaries

In this section, we recall some definitions and basic results on p-algebra and fuzzy theory.Definition 1 (see [15]).
An algebraL=L;∧,∨,∗,0,1 of type 2,2,1,0,0 is a p-algebra if the following conditions hold:(1)
L;∧,∨,0,1 is a bounded lattice(2)
For alla,b∈L,a∧b=0⇔a∧b∗=aTheorem 1 (see [15]).
For any two elementsa,b of a p-algebra, we have the following:(1)
0∗∗=0(2)
a∧a∗=0(3)
a≤b⇒b∗≤a∗(4)
a≤a∗∗(5)
a∗∗∗=a∗(6)
a∨b∗=a∗∧b∗(7)
a∧b∗∗=a∗∗∧b∗∗
An elementx of a p-algebra is called closed, if x=x∗∗.Definition 2 (see [15]).
A nonempty subsetI of L is called an ideal of L if for any x,y∈I,x∨y∈I and x∈I,y∈L,x∧y∈I.Definition 3 (see [15]).
A nonempty subsetF of L is called a filter of L if for any x,y∈F,x∧y∈F and x∈F,y∈L,x∨y∈I.Definition 4 (see [16]).
An idealI of L is called a p-ideal if any x∈I, x∗∗∈I.Definition 5 (see [16]).
A filterF of L is called a p-filter if for any x∗∗∈F, x∈F.Definition 6 (see [1]).
LetX be any nonempty set. A mapping μ:X⟶0,1 is called a fuzzy subset of X.
The unit interval0,1 together with the operations min and max forms a complete lattice satisfying the infinite meet distributive law; i.e.,(1)α∧∨β∈Mβ=∨β∈Mα∧β,for all α∈0,1 and any M⊆0,1.
We often write∧ for minimum or infimum and ∨ for maximum or supremum. That is, for all α,β∈0,1, we have α∧β=minα,β and α∨β=maxα,β.
The characteristics function of any setA is defined as(2)χAx=1,ifx∈A,0,ifx∉A.Definition 7 (see [2]).
Letμ and θ be fuzzy subsets of a set A. Define the fuzzy subsets μ∪θ and μ∩θ of A as follows: for each x∈A, μ∪θx=μx∨θx and μ∩θx=μx∧θx.
Then,μ∪θ and μ∩θ are called the union and intersection of μ and θ, respectively.
For any collection,μi:i∈I of fuzzy subsets of X, where I is a nonempty index set, the least upper bound ∪i∈Iμi and the greatest lower bound ∩i∈Iμi of the μi’s are given for each x∈X, ∪i∈Iμix=∨i∈Iμix and ∩i∈Iμix=∧i∈Iμix, respectively.
For eacht∈0,1, the set(3)μt=x∈A:μx≥tis called the level subset of μ at t [1].Definition 8 (see [2]).
Letf be a function from X into Y, μ be a fuzzy subset of X, and θ be a fuzzy subset of Y.(1)
The image ofμ under f, denoted by fμ, is a fuzzy subset of Y defined by the following: for each y∈Y, fμy=Supμx:x∈f−1y,iff−1y≠ϕ,0,otherwise(2)
The preimage ofθ under f, denoted by f−1θ, is a fuzzy subset of X defined by for each x∈X, f−1θx=θfxDefinition 9 (see [17]).
A fuzzy subsetμ of a bounded lattice L is called a fuzzy ideal of L, if for all x,y∈L the following conditions are satisfied:(1)
μ0=1(2)
μx∨y≥μx∧μy(3)
μx∧y≥μx∨μyDefinition 10 (see [17]).
A fuzzy subsetμ of a bounded lattice L is called a fuzzy filter of L, if for all x,y∈L the following conditions are satisfied:(1)
μ1=1(2)
μx∨y≥μx∨μy(3)
μx∧y≥μx∧μy
We define the binary operations “+” and “⋅” on the set of all fuzzy subsets of L as(4)μ+θx=Supμy∧θz:y,z∈L,y∨z=x,μ⋅θx=Supμy∧θz:y,z∈L,y∧z=x.
Ifμ and θ are fuzzy ideals of L, then μ⋅θ=μ∧θ=μ∩θ and μ+θ=μ∨θ is a fuzzy ideal generated by μ∪θ.
Ifμ and θ are fuzzy filters of L, then μ+θ=μ∧θ (the pointwise infimum of μ and θ) and μ⋅θ=μ∨θ (the supremum of μ and θ).Theorem 2 (see [18]).
LetL be a lattice, x∈L and α∈0,1. Define a fuzzy subset αx of L as(5)αxy=1,ify≤x,α,ify≰x,which is a fuzzy ideal of L.Remark 1 (see [18]).
αx is called the α-level principal fuzzy ideal corresponding to x.
Similarly, a fuzzy subsetαx of L defined as(6)αxy=1,ifx≤y,α,ifx≰y,is the α-level principal fuzzy filter corresponding to x.Remark 2.
LetL be a lattice with 0. Then, L is called 0-distributive if for any a,b,c∈L with a∧b=0=a∧c implying a∧b∨c=0.
Throughout the rest of this paper,L stands for the 0-distributive p-algebra unless otherwise mentioned.


## 3. P-Fuzzy Ideals

In this section, we study the concept of p-fuzzy ideals of a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. Moreover, we prove that, for any fuzzy ideal ofL, there is the smallest p-fuzzy ideal containing it. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Finally, we show that the image and inverse image of a p-fuzzy ideal is a p-fuzzy ideal under a ∗-epimorphism mapping.Definition 11.
A fuzzy idealμ of L is called a p-fuzzy ideal if μx=μx∗∗ for each x∈L.Lemma 1.
A fuzzy idealμ of L is a p-fuzzy ideal if and only if μa∗∧b∗∗≥μa∧μb for all a,b∈L.Lemma 2.
For anyx∈L, αx∗ is a p-fuzzy ideal.Proof.
Leta∈L. If αx∗a=1, then a≤x∗ and a∗∗≤x∗. Thus, αx∗a∗∗=1. If αx∗a=α, then a≰x∗ and a∗∗≰x∗. Thus, αx∗a∗∗=α. Hence, αx∗ is a p-fuzzy ideal.Theorem 3.
αx is a p-fuzzy ideal if and only if x is a closed element.Proof.
Letαx be a p-fuzzy ideal. Then, αxx∗∗=1 and x∗∗≤x. Since x≤x∗∗, we have x=x∗∗. Thus, x is closed.
Conversely, suppose thatx is a closed element. Let a∈L. If αxa=1, then a≤x and a∗∗≤x. Thus, αxa∗∗=1. If αxa=α, then a≰x and a∗∗≰x. Thus, αxa∗∗=α. Hence αx is a p-fuzzy ideal.Corollary 1.
The following conditions onL are equivalent:(1)
Every fuzzy ideal is a p-fuzzy ideal(2)
Every level principal fuzzy ideal is a p-fuzzy ideal(3)
L is Boolean algebraProof.
The proofs of1⇒2 and 3⇒1 are straightforward. To show that 2⇒3, suppose that every level principal fuzzy ideal is a p-fuzzy ideal. Then, by the above theorem, every element of L is closed. For any x,y∈L, the supremum is given by x⊻y=x∗∧y∗∗.
To show thatL is distributive, it suffices to prove that(7)x∧y⊻z≤x∧y⊻x∧z,∀x,y,z∈L.
For this purpose, lett=x∧y∨x∧z. Then, x∧y≤t=t∗∗ gives x∧y∧t∗=0 and x∧t∗≤y∗. Similarly, x∧t∗≤z∗ and therefore x∧t∗≤y∗∧z∗=y∗∧z∗∗∗. It follows from this that x∧t∗∧y∗∧z∗∗=0 and hence that x∧y⊻z=x∧y∗∧z∗∗≤t∗∗=t=x∧y⊻x∧z.
To see thatL is also complemented, observe that 1=0∗ and 0=1∗. Since every x∈L, we have x∧x∗=0 and x∨x∗=x∗∧x∗∗∗=0∗=1. We see that the complement of x is x∗. Thus, L is complemented. Hence L is a Boolean algebra.Theorem 4.
A fuzzy subsetμ of L is a p-fuzzy ideal if and only if every level subset of μ is a p-ideal of L.Corollary 2.
A nonempty subsetI of L is a p-ideal if and only if χI is a p-fuzzy ideal.In the following result, we prove that, for any fuzzy ideal ofL, there is the smallest p-fuzzy ideal containing it.Theorem 5.
Letμ be a fuzzy ideal of L. Define(8)pμx=Supμa:x≤a∗∗:a∈L.Then,pμ is the smallest p-fuzzy ideal containing μ and hence μ is a p-fuzzy ideal if and only if μ=pμ.Proof.
Letμ be a fuzzy ideal of L. Then clearly pμ is a fuzzy ideal of L. To prove pμ is a p-fuzzy ideal, let x∈L. Clearly pμx≥pμx∗∗. On the other hand, pμx=Supμa:x≤a∗∗≤Supμa:x∗∗≤a∗∗=pμx∗∗. Thus, pμ is a p-fuzzy ideal containing μ.
Now we proceed to show thatpμ is the smallest p-fuzzy ideal containing μ. Let θ be a p-fuzzy ideal containing μ. Let x,a∈L such that x≤a∗∗. Then, θa∗∗≤θx. Since θ is a p-fuzzy ideal and μ⊆θ, we get that θa≤θx and μa≤θx. This shows that θx is an upper bound of μa:x≤a∗∗,a∈L. This implies Supμa:x≤a∗∗,a∈L≤θx. Thus, pμ⊆θ. So pμ is the smallest p-fuzzy ideal containing μ.Lemma 3.
Ifμ and θ are fuzzy ideals of L, then μ⊆θ implies pμ⊆pθ.Lemma 4.
For any two fuzzy idealsμ and θ of L, pμ∩θ=pμ∩pθ.Proof.
Letμ and θ be two fuzzy ideals of L. Clearly pμ∩θ⊆pμ∩pθ. To show the other inclusion, let x∈L. Then(9)pμ∩pθx=Supμa:x≤a∗∗,a∈L∧Supθb:x≤b∗∗,b∈L≤Supμa∧b∧θa∧b:x≤a∗∗∧b∗∗≤Supμ∩θa∧b:x≤a∧b∗∗≤Supμ∩θc:x≤c∗∗,c∈L=pμ∩θx.
Thus,pμ∩θ=pμ∩pθ.Lemma 5.
For any fuzzy idealμ of L, the map μ⟶pμ is a closure operator on FIL. That is,(1)
μ⊆pμ(2)
ppμ=pμ(3)
μ⊆θ⇒pμ⊆pθ, for any two fuzzy ideals μ,θ of L
p-fuzzy ideals are simply the closed elements ofFIL with respect to the closure operator.Every p-fuzzy ideal is a fuzzy ideal but the converse may not be true. For this, we have the following example.Example 1.
Consider the p-algebraL=0,a,b,c,1 whose Hasse diagram is given below.
Define fuzzy subsetsμ and θ of L as follows:(10)μ0=1=μa,μb=μc=μ1=0,θ0=θb=1,θa=θc=θ1=0.
Then, it can be easily verified thatμ and θ are p-fuzzy ideals of L. Moreover, we observe that the fuzzy ideal μ∨θ of L is not a p-fuzzy ideal.
The set of all p-fuzzy ideals ofL is denoted by FIpL. We now prove that FIpL is a lattice.Theorem 6.
Ifλ,η∈FIpL, then the supremum of λ and η is given by(11)λ⊻ηx=Supλa∧ηb:x≤a∗∧b∗∗,a,b∈L.Proof.
Putγ=λ⊻η. Clearly γ0=1. For any x,y∈L,(12)γx∧γy=Supλa1∧ηb1:x≤a1∗∧b1∗∗,a1,b1∈L∧Supλa2∧ηb2:y≤a2∗∧b2∗∗,a2,b2∈L=Supλa1∧λa2∧ηb1∧ηb2:x≤a1∗∧b1∗∗,y≤a2∗∧b2∗∗=Supλa1∨a2∧ηb1∨b2:x≤a1∗∧b1∗∗,y≤a2∗∧b2∗∗.
Ifx≤a1∗∧b1∗∗ and y≤a2∗∧b2∗∗, then x∨y≤a1∨a2∗∧b1∨b2∗∗. Thus(13)γx∧γy≤Supλa1∨a2∧ηb1∨b2:x∨y≤a1∨a2∗∧b1∨b2∗∗≤Supλc1∧ηc2:x∨y≤c1∗∧c2∗∗=γx∨y.
Thus,γx∨y≥γx∧γy.
On the other hand,(14)γx≤Supλa∧ηb:x≤a∗∧b∗∗≤Supλa∧ηb:x∧y≤a∗∧b∗∗=γx∧y.
Thus,γx∧y≥γx∨γy.
To showγ is a p-fuzzy ideal, let x∈L. Then clearly γx∗∗≤γx.(15)γx=Supλa∧ηb:x≤a∗∧b∗∗≤Supλa∧ηb:x∗∗≤a∗∧b∗∗=γx∗∗.
Thus,γ is a p-fuzzy ideal of L.
Now we proceed to show thatγ is the smallest p-fuzzy ideal containing λ and η. Clearly λ⊆γ and η⊆γ. Let θ be any p-fuzzy ideal containing λ and η. For any x∈L,(16)γx=Supλa∧ηb:x≤a∗∧b∗∗≤Supθa∧θb:x≤a∗∧b∗∗=Supθa∨b:x≤a∨b∗∗≤Supθc:x≤c∗∗=pθx.
This shows thatpθ=θ. Thus, γ⊆θ. So γ is the smallest p-fuzzy ideal of L containing λ and η.Theorem 7.
The setFIpL of p-fuzzy ideal of L forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.Proof.
ClearlyFIpL,⊆ is a partially ordered set. For λ,γ∈FIpL, the infimum and the supremum of λ and γ are λ∧γ=λ∩γ and λ⊻γ, respectively. Thus, FIpL,∧,⊻ is a lattice.
To show the distributivity, it suffices to showλ∩γ⊻η⊆λ∩γ⊻λ∩η for all λ,γ,ηFIpL. For any x∈L,(17)λ∩γ⊻ηx=Supλx∧γa∧ηb:x≤a∗∧b∗∗,a,b∈L=Supλx∧γa∧λx∧ηb:x≤a∗∧b∗∗=Supλx∧γa∗∗∧λx∧ηb∗∗:x≤a∗∧b∗∗≤Supλx∧a∗∗∧γx∧a∗∗∧λx∧b∗∗∧ηx∧b∗∗:x≤a∗∧b∗∗=Supλ∩γx∧a∗∗∧λ∩ηx∧b∗∗:x≤a∗∧b∗∗.
Ifx≤a∗∧b∗∗, then x∧a∗∧b∗=0. Since L is a p-algebra, a∧b∗=a∧a∧b∗ for all a,b∈L, and we get that x≤x∧a∗∗∗∧x∧b∗∗∗∗. Thus(18)λ∩γ⊻ηx≤Supλ∩γx∧a∗∗∧λ∩ηx∧b∗∗:x≤x∧a∗∗∗∧x∧b∗∗∗∗≤Supλ∩γc∧λ∩ηd:x≤c∗∧d∗∗,c,d∈L≤λ∩γ⊻λ∩ηx.
Henceλ∩γ⊻η=λ∩γ⊻λ∩η. So FIpL is distributive.
Now we show the completeness. SinceFIp,⊆ is a poset and χL and χ0 are the greatest and least elements of FIp, respectively, then FIpL is a complete distributive lattice.Theorem 8.
Letγ be a p-fuzzy ideal of L and η be a fuzzy filter of L such that γ∩η≤β, α∈0,1. Then, there exists a prime p-fuzzy ideal θ of L such that γ⊆θ and θ∩η≤α.Proof.
PutΓ=λ∈FIpL:γ⊆λandλ∩η≤β. Since γ∈Γ, Γ is nonempty and it forms a poset together with the inclusion ordering of fuzzy sets. Let A=μii∈I be any chain in Γ. Then clearly ∪i∈Iμi is a p-fuzzy ideal. Since μi∩η≤α for each i∈I, then ∪i∈Iμi∩η≤α. Thus, ∪i∈Iμi∈Γ. By applying Zorn’s lemma, we get a maximal element. Let us say θ∈Γ; that is, θ is a p-fuzzy ideal of L such that γ⊆θ and θ∩η≤α.
Now we proceed to showθ is a prime fuzzy ideal. Assume that θ is not prime fuzzy ideal. Let μ1,μ2∈FIL and μ1∩μ2⊆θ such that μ1⊈θ and μ2⊈θ. If we put θ1=pμ1∨θ and θ2=pμ2∨θ, then both θ1 and θ2 are p-fuzzy ideals of L properly containing θ. Since θ is maximal in Γ, we get θ1∉Γ and θ2∉Γ. Thus, θ1∩η≰α and θ2∩η≰α. This implies there exist x,y∈L such that θ1∩ηx>α and θ2∩ηy>α. So θ1∩θ2∩ηx∧y>α.(19)⇒pθ∨μ1∧μ2x∧y∧ηx∧y>α⇒pθx∧y∧ηx∧y>α⇒θx∧y∧ηx∧y>α⇒θ∩ηx∧y>α.
This is a contradiction. Henceθ is prime p-fuzzy ideal of L.
LetL and M be p-algebras. Then, a lattice morphism f:L⟶M is said to be a ∗-morphism if fx∗=fx∗ for all x,y∈L.Theorem 9.
Letf:L⟶M be a ∗-epimorphism. If μ is a p-fuzzy ideal of L, then fμ is a p-fuzzy ideal of M.Proof.
Letμ be a p-fuzzy ideal of L. Then, fμ is a fuzzy ideal of M. To show fμ is a p-fuzzy ideal, let y∈M. Since y≤y∗∗, we have fμy∗∗≤fμy. On the other hand,(20)fμy=Supμx:x∈f−1y.
Sincex∈f−1y and f is a ∗-morphism, we have x∗∗∈f−1y∗∗. Thus(21)fμy≤Supμx∗∗:x∗∗∈f−1y∗∗≤Supμa:a∈f−1y∗∗=fμy∗∗.
Thus,fμy≤fμy∗∗. So fμ is a p-fuzzy ideal of M.Theorem 10.
Letf:L⟶M be a ∗-epimorphism. If θ is a p-fuzzy ideal of M, then f−1θ is a p-fuzzy ideal of L.Theorem 11.
Letf:L⟶M be a ∗-epimorphism. Then, the map g:FIp⟶FMp defined by μ↦fμ is a lattice epimorphism.Proof.
Letμ,θ∈FIpL. Then, μ∩θ and μ⊻θ are p-fuzzy ideals of L. Thus, by Theorem 9, fμ∩θ and fμ⊻θ are p-fuzzy ideals of M. Clearly fμ∩θ⊆fθ∩fμ. For any y∈M,(22)fμ∩fθy=Supμa:a∈f−1y,a∈L∧Supθb:b∈f−1y,b∈L.
Ifa∈f−1y and b∈f−1y, then a∧b∈f−1y. Thus,(23)fμ∩fθy≤Supμa∧b∧θa∧b:a∧b∈f−1y=Supμ∩θa∧b:a∧b∈f−1y≤Supμ∩θc:c∈f−1y=fμ∩θy.
Sofμ∩fθ=fμ∩θ.
Again clearly,fμ⊻fθ⊆fμ⊻θ. For any x∈M,(24)fμ⊻fθx=Supfμx1∧fθx2:x≤x1∗∧x2∗∗=SupSupμb1:b1∈f−1x1∧Supμb2:b2∈f−1x2x≤x1∗∧x2∗∗,fμ⊻θx=Supμ⊻θa:a∈f−1x,a∈L=SupSupμa1∧θa2:a≤a1∗∧a2∗∗:a∈f−1x.
Iffa=x and a≤a1∗∧a2∗∗, then x=fa≤fa1∗∧fa2∗∗. Put fa1=y1 and fa2=y2. Then, a1∈f−1y1,a2∈f−1y2 and x≤y1∗∧y2∗∗. Based on this fact, we have(25)fμ⊻θx≤SupSupμa1:a1∈f−1y1∧Supθa2:a2∈f−1y2:≤y1∗∧y2∗∗=Supfμy1∧fθy2:x≤y1∗∧y2∗∗=fμ⊻fθx.
Thus,fμ⊻θ=fμ⊻fθ. So g is a homomorphism. Now we proceed to show g is an epimorphism. Let μ∈FIpM. Then, by Theorem 10, f−1μ∈FIpL. Since f is onto, we have ff−1μ=μ. Thus, g is onto. So g is a lattice epimorphism.


## 4. P-Fuzzy Filters

In this section, we study the concept of p-fuzzy filter of a p-algebra. We prove that the class of p-fuzzy filters forms a complete distributive lattice.Definition 12.
A fuzzy filterμ of L is called a p-fuzzy filter of L if μx=μx∗∗ for any x∈L.Theorem 12.
A fuzzy subsetμ of L is a p-fuzzy filter if and only if every level subset of μ is a p-filter of L.Corollary 3.
A nonempty subsetI of L is a p-filter if and only if χI is a p-fuzzy filter.
The set of all dense elements ofL is denoted by D.Corollary 4.
χD is the smallest p-fuzzy filter of L.Proof.
Since the set of all dense elements ofL is a p-filter of L, by the above corollary χD is a p-fuzzy filter of L. To show χD is the smallest p-fuzzy filter, let μ be a p-fuzzy filter of L. Then clearly χDx≤μx,∀x∉D. If x∈D, then x∗∗∈D. Since μ is a p-fuzzy filter, we get that μx=μx∗∗=1. This implies χD⊆μ. Thus, χD is the smallest p-fuzzy filter.Corollary 5.
χ1 is p-fuzzy filter of L if and only if χD=χ1.Proof.
Supposeχ1 is p-fuzzy filter of L. Then, by the above corollary χ1=χD. Conversely, let χD=χ1. Since χD is a p-fuzzy filer, we get that χ1 is a p-fuzzy filter of L.Theorem 13.
A proper fuzzy filterμ of L is a p-fuzzy filter if and only if for any x,y∈L such that x∗=y∗ and μx=μy.The set of all p-fuzzy filters of a p-algebraL is denoted by FFpL. The following result shows that FFpL forms a lattice.Theorem 14.
Ifλ,η∈FFpL, then the supremum of λ and η is given by(26)λ⊻ηx=Supλa∧ηb:x≤a∧b∗,a,b∈L.Proof.
Putγ=λ⊻η. Clearly γ1=1. For any x,y∈L,(27)γx∧γy=Supλa1∧ηb1:x∗≤a1∧b1∗,a1,b1∈L∧Supλa2∧ηb2:y∗≤a2∧b2∗,a2,b2∈L=Supλa1∧λa2∧ηb1∧ηb2:x∗≤a1∧b1∗,y∗≤a2∧b2∗=Supλa1∧a2∧ηb1∧b2:x∗≤a1∧b1∗,y∗≤a2∧b2∗.
Ifx∗≤a1∧b1∗ and y∗≤a2∧b2∗, then x∧y∗≤a1∧a2∧b1∧b2∗. Thus(28)γx∧γy≤Supλa1∧a2∧ηb1∧b2:x∧y∗≤a1∧a2∧b1∧b2∗≤Supλc1∧ηc2:x∧y∗≤c1∧c2∗=γx∧y.
Thus,γx∧y≥γx∧ηy.
On the other hand,(29)γx≤Supλa∧ηb:x∗≤a∧b∗≤Supλa∧ηb:x∗∧y∗≤a∧b∗=Supλa∧ηb:x∨y∗≤a∧b∗=γx∨y.
Thus,γx∨y≥γx∨γy.
To showγ is a p-fuzzy filter of L, let x∈L.(30)γx∗∗=Supλa∧ηb:x∗∗∗≤a∧b∗=Supλa∧ηb:x∗≤a∧b∗=γx.
Thus,γ is a p-fuzzy filter of L. We now show that γ is the smallest p-fuzzy filter containing λ and η. Let θ be a p-fuzzy filter containing λ and η. For any x∈L,(31)γx≤Supλa∧ηb:x∗≤a∧b∗≤Supθa∧θb:x∗≤a∧b∗=θ⊻θx=θx.
Thus,γ is the smallest p-fuzzy filter containing λ and η.Theorem 15.
The setFFpL of p-fuzzy filter of L forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.Proof.
ClearlyFFpL,⊆ is a partially ordered set. For λ,γ∈FFpL, the infimum and the supremum of λ and γ are λ∧γ=λ∩γ and λ⊻γ, respectively. Thus, FFpL,∧,⊻ is a lattice.
To show the distributivity, it suffices to showλ∩γ⊻η⊆λ∩γ⊻λ∩η for all λ,γ,ηFIpL. For any x∈L,(32)λ∩γ⊻ηx=Supλx∧γa∧ηb:x∗≤a∧b∗,a,b∈L=Supλx∧γa∧λx∧ηb:x∗≤a∧b∗≤Supλx∨a∧γx∨a∧λx∨b∧ηx∨b:x∗≤a∧b∗=Supλx∨a∗∗∧γx∨a∗∗∧λx∨b∗∗∧ηx∨b∗∗:x∗≤a∧b∗=Supλ∩γx∨a∗∗∧λ∩ηx∨b∗∗:x∗≤a∧b∗.
Ifx∗≤a∧b∗, then x∗∧a∧b∗∗=0. We can easily verify that x∗≤x∨a∗∗∧x∨b∗∗∗. Thus,(33)λ∩γ⊻ηx≤Supλ∩γx∨a∗∗∧λ∩ηx∨b∗∗:x∗≤x∨a∗∗∧x∨b∗∗∗≤Supλ∩γc∧λ∩ηd:x∗≤c∧d∗,c,d∈L≤λ∩γ⊻λ∩ηx.
Thus,λ∩γ⊻η=λ∩γ⊻λ∩η. So FFpL is distributive.


## 5. Relation betweenFIpL and FFpL

In this section, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filters.Lemma 6.
Letμ be a p-fuzzy ideal. Define(34)μ†x=μx∗.
Then,μ† is a p-fuzzy filter.Proof.
Letμ be a p-fuzzy ideal of L. Since 1∗=0, we get that μ†1=1. For any x,y∈L, μ†x∧μ†y=μx∗∧μy∗=μx∗∨y∗=μx∗∨y∗∗∗=μx∧y∗=μ†x∧y. Thus, μ† is a fuzzy filter. To show μ† is a p-fuzzy ideal, let x∈L. Then, μ†x∗∗=μx∗=μ†x. Hence μ† is a p-fuzzy filter of L.Lemma 7.
Letθ be a fuzzy filter. Define(35)θ‡x=θx∗.Then,θ‡ is a p-fuzzy ideal.Proof.
Letθ be a fuzzy filter. Since 1=0∗, we get θ‡0=1. For any x,y∈L,(36)θ‡x∨y=θx∗∧y∗=θx∗∧θy∗=θ‡x∧θ‡y.
Thus,θ‡ is a fuzzy ideal of L. To show θ‡ is a p-fuzzy ideal, let x∈L. Then, θ‡x∗∗=θx∗=θ‡x. Thus, θ‡ is a p-fuzzy ideal of L.Theorem 16.
Letμ be a fuzzy filter of L. Then, μ‡†=μ if and only if μ is a p-fuzzy filter.Proof.
Letμ be a fuzzy filter of L and μ‡†=μ. Then, by Lemmas 6 and 7, μ is a p-fuzzy filter. Conversely, let μ be a p-fuzzy filter and x∈L. Then, μ‡†x=μ‡x∗=μx∗∗=μx. Thus, μ‡†=μ.Theorem 17.
InL, FIpL≅FFpL.Proof.
Definef:FIpL⟶FFpL by fμ=μ†. Since μ∈FIpL, by Lemma 6, fμ∈FFpL. Let μ,θ∈FIpL such that fμ=fθ. Then, μ†=θ†. Since μ† and θ† are p-fuzzy filters of L, then μ†‡ and θ†‡ are p-fuzzy ideals of L and μ†‡=θ†‡. To show μ=θ, let x∈L. Then, μ†‡x=θ†‡x. This implies μx∗∗=θx∗∗. Since μ and θ are p-fuzzy ideal, we get that μx=θx. Thus, f is one to one.
Letθ∈FFpL. Then, by Theorem 16, θ=θ‡†. Since θ is a fuzzy filter, then θ‡ is a p-fuzzy ideal of L and fθ‡=θ. So f is onto.
Letμ,θ∈FIpL and x∈L. Then(37)fμ⊻θx=Supμa∧θb:x∗≤a∗∧b∗∗,a,b∈L=Supμa∗∗∧θb∗∗:x∗≤a∗∧b∗∗=Supμ†a∗∧θ†b∗:x∗≤a∗∧b∗∗≤Supμ†c∧θ†d:x∗≤c∧d∗,c,d∈L=μ†⊻θ†x=fμ⊻fθx.
Thus,fμ⊻θ⊆fμ⊻fθ. On the other hand,(38)fμ⊻fθx=Supμ†a∧θ†b:x∗≤a∧b∗,a,b∈L=Supμa∗∧θb∗:x∗≤a∧b∗=Supμa∗∧θb∗:x∗≤a∗∗∧b∗∗∗≤Supμc∧θd:x∗≤c∗∧d∗∗,c,d∈L=μ⊻θx∗=μ⊻θ†x=fμ⊻θx.
Thus,fμ⊻θ=fμ⊻fθ. Similarly, fμ∩θ=fμ∩fθ. Hence FIpL≅FFpL.


## 6. Conclusion

In this paper, we introduced the concept of p-fuzzy ideals and p-fuzzy filters in a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. We also studied the image and inverse image of p-fuzzy ideals under a∗-epimorphism mapping. Moreover, we prove that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filter. Our future work will focus on σ-fuzzy ideals in a (0, 1) distributive lattice.



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