# On Fuzzy Derivation of Ideals of BF-Algebras

Authors: Gerima Tefera Dejen; Tsige Gebrekidan
Journal: Journal of Mathematics
 (2022)

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## Abstract

The concept of fuzzy derivation of ideals of BF algebra is introduced. In addition, fuzzy left derivation of ideals of BF-algebras and fuzzy right derivation of ideals of BF- algebra are discussed. We also investigated fuzzy left (right) derivation of Cartesian product of ideals of BF-algebra. We initiated the idea of level cut in fuzzy derivation of ideals of BF-algebra and showed that the intersection of any set of left (right) fuzzy derivations of ideals of BF-algebra is also fuzzy left (right) derivation of ideals of BF-algebra.


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## Body

## 1. Introduction

Neggers and Hee Sik Kim [1] introduced B-algebra which is related to several classes of algebra such as BCH/BCK-algebra. The notion of BF-algebras as a generalization of B-algebra initiated by Andrzes [2] and also introduced ideals and normal ideal in BF-algebra. Nora, O., and Al-Shehrie in [3] introduced the notion of left-right (right-left) derivation of B-algebra and some related properties. Zadeh in [4] discussed on fuzzy set and Gerima, and Fasil [5] discussed on the derivation of BF-sub algebra, derivations of ideals of BF-algebra.Saied and Rezvani in [6] investigated on fuzzy BF-algebra with some properties, and Gerima and Tigist [7] initiated the idea of fuzzy dot BF ideals of BF-algebra with different characterizations. Abujabal and Al-Shehri [8] introduced left-right-derivation of BCI-algebra and Mostofa et al. in [9] discussed about properties of derivation of BF-algebras. S.R. in [10] explained about Doubt fuzzy ideals of BF-algebra and Mostafa et al. introduced properties of derivations of KH-algebra with different properties in [9]. Consequently, Mostafa et al. in [11] explained on fuzzy derivations of KU ideals on KU algebra, and Walendziak in [12] studied about note on normal sub algebra in B-algebras. A new kind of derivation was introduced on X called (T, Γ, Υ)-derivation such that F is monotone, as a generalization of derivation of c-subtraction and characterized some of its relevant properties in [13].In this paper, fuzzy derivation of ideals of BF-algebra which is fuzzification of derivation of ideals of BF-algebra with different properties are investigated. The Cartesian product of fuzzy left (right) derivation of ideals of BF-algebra is elaborated with some related properties. The theories of the properties of derivations are important to apply in the notion of rings and near-ring theory of BF-algebra and some related concepts.


## 2. Preliminaries

Definition 1 (see [2]).
A BF-algebra is a nonempty setX with a constant 0 and a binary operation ∗ satisfying the following axioms:(i)
x∗x=0(ii)
x∗0=x(iii)
0∗x∗y=y∗x∀x,y∈X.
Ifx≤y, then x∗y=0,∀x,y∈X.Definition 2 (see [5]).
A nonempty subsetS of a BF-algebra X is called sub algebra of X if x∗y∈S for any x,y∈S.Definition 3 (see [4]).
A fuzzy setμ of a BF-algebra X is called a fuzzy ideal of X if it satisfies the following conditions:(i)
μ0≥μx(ii)
μx≥minμx∗y,μyDefinition 4 (see [6]).
A fuzzy subsetμ of X is called a fuzzy sub algebra of a BF-algebra X if μx∗y≥minμx,μy for all x,y∈X.Proposition 1 (see [4]).
Ifμ is a fuzzy ideal of BF-algebra, then μ is a fuzzy sub algebra of a BF-algebra X.Definition 5 (see [2]).
Letd:X⟶X be a self-map on BF-algebra. Then, d is said to be regular if d0=0.Lemma 1 (see [4]).
IfA=<x,μAx>,x∈X is a fuzzy BF-sub algebra of X, and μ is a fuzzy sub algebra of X, then for all μA0≥μAx,∀x∈A.
This lemma is the consequence of Definitions1 and 4.Theorem 1 (see [6]).
LetA1 and A2 be fuzzy BF-sub algebras of X. Then, A1∩A2 is a fuzzy BF-sub algebra of X.Definition 6 (see [2]).
A nonempty subsetI of a BF-algebra X is said to be an ideal of X if(i)
0∈I(ii)
x∗y∈I and y∈I implies x∈I,∀x,y∈IDefinition 7 (see [10]).
IfX,∗,0 is a BF-algebra, then we have the following: A left-right derivation of X is a self-map d:X⟶X satisfying the identity dx∗y=dx∗y∩x∗dy,∀x,y∈X.(i)
A right-left derivation ofX is a self-map d:X⟶X satisfying the identitydx∗y=x∗dy∩dx∗y,∀x,y∈X(ii)
Ifd satisfying both a left-right and a right-left derivation, then d is called a derivation of XRemark 11.
IfX,∗,0 is a BF- algebra, then x∩y=y∗y∗x for all x,y in XProposition 2 (see [4]).
Letd be a left-right derivation of BF-algebra X. Then,(i)
d0=dx∗x, for all x in X(ii)
d is one -to- one(iii)
Ifd is regular, then it is the identity map(iv)
If there is an elementx in X such that dx=x, then d is the identity map(v)
If there is an elementx in X such that dy∗x=0 or x∗dy=0 for all y in X, then dy=x, for all y in X. That is d is constantDefinition 8 (see [10]).
LetX be a BF-algebra, and let A and B be two fuzzy sets on X. Then, A×Bx,y=minAx,By.


### 2.1. Fuzzy Derivation of Ideals of BF-Algebras

Definition 9.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subsetμ:dX⟶0,1 is said to be a fuzzy right derivation of ideals of BF-algebras. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμx∗dy,μdy,∀x,y∈XExample 1.
LetX=0,1,2,3,4 be a BF-algebra, in which the operation ∗ is defined as follows:
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,4
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4
To showμ is a fuzzy right derivation of ideals of BF-algebra
we assume.t0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t1=μd2=μd4=μx,∀x∉0,1,3.Hence,μ0≥μx,∀x≠0,1,3.(2)
Letx=1,y=4. Then,(1)μd1=μ0=t0≥t1,=mint1,t1,
=minμ1∗d4,μd4, where μ1∗d4=μ1∗4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμ1∗d4,μd4.
Henceμd1≥minμ1∗d4,μd4.
Letx=3,y=4. Then,(2)μd3=μ0=t0≥t1,=mint1,t1,
=minμ3∗d4μd4, where μ3∗d4=μ3∗4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμ3∗d4,μd4.
Hence,μd3≥minμ3∗d4,μd4.
Letx=3,y=2. Then,(3)μd3=μ0=t0≥t1,=mint1,t1,
=minμ3∗d2,μd2, where μ3∗d2=μ3∗4=μ4=t1,μd2=μ4=t1.
Since.t0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμ3∗d2,μd2.
Hence,μd3≥minμ3∗d2,μd2.
Therefore,μdx≥minμx∗dy,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy right derivation of ideals of BF-algebra.
We assume.t0≤t1
Ifx=0,y=2,(4)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.
Hence,t0≤t1 contradicts. μ0≥μx,∀x∈X.Definition 10.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subset μ:dX⟶0,1 is said to be a fuzzy left derivation of ideals of BF-algebras. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμdx∗y,μdy,∀x,y∈XExample 2.
LetX=0,1,2,3,4. From example 15, μ satisfies a fuzzy left derivation of ideals of BF-algebra.
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,49.
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4.
To showμ is a fuzzy right derivation of ideals of BF-algebra,
We assume.t0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t=μzd2=μd4=μx,∀x∉0,1,3Hence,μ0≥μx,∀x≠0,1,3(2)
Letx=1,y=4. Then,(5)μd1=μ0=t0≥t1,=mint1,t1,
=minμd1∗4,μd4, where μd1∗4=μ0∗4=μ2=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμd1∗4,μd4.
Hence,μd1≥minμd1∗4,μd4.
Letx=3,y=4. Then,(6)μd3=μ0=t0≥t1,=mint1,t1,
=minμd3∗4,μd4, where μd3∗4=μ0∗4=μ2=t1,μd4=μ4=t1.
Sincet0≥t1
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗4,μd4.
Hence,μd3≥minμd3∗4,μd4.
Letx=3,y=2. Then,(7)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗2,μd2, where μd3∗2=μ0∗2=μ4=t1,μd2=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗2,μd2.
Hence,μd3≥minμd3∗2,μd2.
Therefore,μdx≥miμdx∗y,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy left derivation of ideals of BF-algebra.
We assumet0≤t1,
Ifx=0,y=2,(8)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.


### 2.2.t0≤t1 Contradicting μ0≥μx,∀x∈X

Definition 11.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subset μ:dX⟶0,1 is said to be a fuzzy derivation of ideals of BF-algebra. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμdx∗y,μdy,∀x,y∈XExample 3.
LetX=0,1,2,3,4. From example 15, μ satisfies a fuzzy derivation of ideals of BF-algebra.
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,4.
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4.
To showμ is a fuzzy right derivation of ideals of BF-algebra.
We assumet0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t1=μd2=μd4=μx,∀x∉0,1,3Hence,μ0≥μx,∀x≠0,1,3(2)
Letx=1,y=4. Then,(9)μd1=μ0=t0≥t1,=mint1,t1,=minμd1∗4,μd4, where μd1∗4=μd4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμd1∗4,μd4.
Hence,μd1≥minμd1∗4,μd4.
Letx=3,y=4. Then,(10)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗4,μd4, where μd3∗4=μd4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗4,μd4.
Hence,μd3≥minμd3∗4,μd4.
Letx=3,y=2. Then,(11)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗2,μd2, where μd3∗2=μd4=μ4=t1,μd2=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗2,μd2.
Hence,μd3≥minμd3∗2,μd2.
Therefore,μdx≥minμdx∗y,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy derivation of ideals of BF-algebra.
We assumet0≤t1.
Ifx=0,y=2,(12)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.
Hence,t0≤t1 contradicts μ0≥μx,∀x∈X.Lemma 2.
Ifμ is a fuzzy right derivation of ideals of BF-algebra X and x≤dy, then μdx≥μdy.Proof.
Ifx≤dy, then x∗dy=0 since dy≤y and μ0≥μx (by definition), we get
μdx≥minμx∗dy,μdy, since μ is a fuzzy right derivation of ideal of BF-algebras.
=minμ0,μdy, since x∗dy=0.(13)=μdy.
Therefore,μdx≥μdy.Lemma 3.
Ifμ is a fuzzy left derivation of ideals of a BF-algebra X and x≤y, then μdx≥μdy.Proof.
Ifx≤y, then dx∗y≤x∗y=0.
μdx≥minμdx∗y,μdy since μ is a fuzzy left derivation of ideals of a BF-algebra.
Sincedx∗y≤x∗y implying dx∗y≤x∗y=0.
Impliesdx∗y=0.=minμ0,μdy since dx∗y=0.(14)=μdy.
Therefore,μdx≥μdy.Proposition 3.
Letd be a regular self-map of X and let μ be a fuzzy ideal of a BF-algebra X. Then, μd0≥μdx.Proposition 4.
The intersection of any set of fuzzy left derivations of ideals of BF-algebraX is also fuzzy left derivations of BF-algebra.Proof.
Letμi,i∈I be a family of fuzzy left derivations of ideals of BF-algebra X, then for any i∈I.(1)
Let0∈X. Then, ∩μi,i∈I0=infμi,i∈I0≥infμi,i∈Ix=∩μi,i∈Ix(2)
Letx,y∈X. Then, ∩μii∈Idx=infμi,i∈Idx≥infminμi,i∈Idx∗y,μi,i∈Idy
Sinceμi,i∈I is fuzzy left derivations of ideals of BF-algebra.(15)=mininfμi,i∈Idx∗y,infμi,i∈Idy,=min∩μi,i∈Idx∗y,∩μi,i∈Idy,Proposition 5.
The intersection of any set of fuzzy right derivations of ideals of BF-algebraX is also fuzzy right derivations of BF-algebra.Definition 12.
Letμ be a fuzzy derivation of ideal of a BF-algebra X.
Then, the set∪μ,α=x∈X|μdx≥α, where α∈0,1 is called upper-level cut
ofμ (level subset of μ).Theorem 2.
μ is a fuzzy sub algebra of a BF-algebra X if and only if ∪μ,α is a sub algebra of X, where α∈0,1.Proof.
Supposeμ be a fuzzy sub algebra of a BF-algebra X.
We want to show∪μ,α is sub algebra of X.
Letx∈S⊆X. Then, put μx=α,α∈0,1.(16)μ0≥μx≥α.
Sinceμ0≥α implies 0∈∪μ,α.
Hence,∪μ,α≠ϕ.
Letx,y∈∪μ,α. Then, μx≥α,μy≥α.(17)α=minα,α≤minμx,μy≤μx,y.
Hence,μx∗y≥α.
This∪μ,α is a sub algebra of X.
Conversely,∪μ,α be a sub algebra of X.
Letx,y∈∪μ,α. Then, x∗y∈∪μ,α.(18)μx≥α,μy≥α.
It impliesμx≥α,μy≥α and μx∗y≥α.(19)μx∗y≥α≥minα,α=minμx,μy.
Hence,μx∗y≥minμx,μy.
Therefore,μ is a fuzzy sub algebra of a BF-algebra X.Theorem 3.
μ is a fuzzy left derivation of ideals of a BF-algebras X if and only if∪μ,α is an ideal of X.Proof.
Supposeμ be a fuzzy left derivation of ideals of a BF-algebra X.
We want show∪μ,α is an ideal of X.
Letx∈X. Then, μx=α,α∈0,1.
Sinceμ0≥μx=α implies 0∈∪μ,α.
Hence,∪μ,α≠ϕ.
Letx∗y∈∪μ,α and y∈∪μ,α. Then, μx∗y≥α and μy≥α.
Sinceμ is a fuzzy left derivation of ideals of a BF-algebra.
We haveμdx≥minμdx∗y,μdy.≥minμx∗y,μy, since dx∗y≤x∗y. implies μdx∗y≥μx∗y.≥minα,α=α
It is implied thatμdx≥α. But μdx≥μx.
Sinceμx≥α. Therefore, x∈∪μ,α.
Hence,∪μ,α is an ideal of X.
Conversely,∪μ,α be an ideal of X.
We need to proveμ is a fuzzy left derivation of ideals of a BF-algebra.(1)
Since∪μ,α is an ideal of X(20)0∈∪μ,α,μ0≥α.Letx∈∪μ,α. Then, put μx=αso thatμ0≥μx,∀x∈X(2)
Letx∗y∈∪μ,α and y∈∪μ,α. Then, μx∗y≥α and μy≥α.
We getx∈∪μ,α. implying μx≥α.
Sincedx≤x and dy≤y. It is implied that μdx≥μx≥α and μdy≥μy≥α.(21)μdx∗y≥μx∗yμdx∗y≥1−α
Putμdx∗y=1−α and μdy=α.
Then,μdx≥min1−α,α(22)=minμdx∗y,μdy.
Hence,μdx≥minμdx∗y,μdy.
Therefore,μ is a fuzzy left derivation of ideals of a BF-algebra X.Theorem 4.
μ is a fuzzy right derivation of ideals of a BF-algebra X if and only if ∪μ,α is an ideal of X.


### 2.3. Cartesian Product of Fuzzy Derivation of Ideals of BF-Algebras

Definition 13.
LetX be a BF-algebra, and let μ and λ be a fuzzy derivation of ideals of a BF-algebra. Then, μ×λx,y=minμx,λy,∀x,y∈X.
μ×λ:X×X⟶0,1, where μ:X⟶0,1λ:X⟶0,1.Theorem 5.
Letμ and λ be a fuzzy left derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy left derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy left derivation of ideals of a BF-algebra X.
We want to showμ×λ is a fuzzy left derivation of ideals of BF-algebra X.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(23)≥minμx,λy=μ×λx,y, by definition of fuzzy left derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.Sinceμ and λ are fuzzy derivation of ideals of a BF-algebra,(24)≥minminμdx1∗x2,μdx2,minλdy1∗y2,λdy2=minminμdx1∗x2,λdy1∗y2,minμdx2,λdy2=minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2=minμ×λdx1,dy1∗x2,y2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λdx1,dy1∗x2,y2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy left derivation of ideals of BF-algebra X.Theorem 6.
Letμ and λ be a fuzzy right derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy right derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy right derivation of ideals of a BF-algebra X.
We want to show thatμ×λ is a fuzzy right derivation of ideals of BF-algebraX.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(25)≥minμx,λy=μ×λx,y, by definition of fuzzy right derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.
Sinceμ and λ are fuzzy derivation of ideals of a BF-algebra.(26)≥minminμx1∗dx2,μdx2,minλy1∗dy2,λdy2=minminμx1∗dx2,λy1∗dy2,minμdx2,λdy2=minμ×λx1∗dx2,y1∗dy2,μ×λdx2,dy2=minμ×λx1,y1∗dx2,dy2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λx1,y1∗dx2,dy2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy right derivation of ideals of BF-algebra X.Theorem 7.
Letμ and λ be a fuzzy derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy derivation of ideals of a BF-algebra X.
We want to show thatμ×λ is a fuzzy derivation of ideals of BF-algebra X.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(27)≥minμx,λy=μ×λx,y, by definition of fuzzy derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.
Sinceμ and λ are fuzzy derivation of ideals of a BF-algebras.(28)≥minminμdx1∗x2,μdx2,minλdy1∗y2,λdy2=minminμdx1∗x2,λdy1∗y2,minμdx2,λdy2=minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy derivation of ideals of BF-algebra X.


## 2.1. Fuzzy Derivation of Ideals of BF-Algebras

Definition 9.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subsetμ:dX⟶0,1 is said to be a fuzzy right derivation of ideals of BF-algebras. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμx∗dy,μdy,∀x,y∈XExample 1.
LetX=0,1,2,3,4 be a BF-algebra, in which the operation ∗ is defined as follows:
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,4
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4
To showμ is a fuzzy right derivation of ideals of BF-algebra
we assume.t0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t1=μd2=μd4=μx,∀x∉0,1,3.Hence,μ0≥μx,∀x≠0,1,3.(2)
Letx=1,y=4. Then,(1)μd1=μ0=t0≥t1,=mint1,t1,
=minμ1∗d4,μd4, where μ1∗d4=μ1∗4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμ1∗d4,μd4.
Henceμd1≥minμ1∗d4,μd4.
Letx=3,y=4. Then,(2)μd3=μ0=t0≥t1,=mint1,t1,
=minμ3∗d4μd4, where μ3∗d4=μ3∗4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμ3∗d4,μd4.
Hence,μd3≥minμ3∗d4,μd4.
Letx=3,y=2. Then,(3)μd3=μ0=t0≥t1,=mint1,t1,
=minμ3∗d2,μd2, where μ3∗d2=μ3∗4=μ4=t1,μd2=μ4=t1.
Since.t0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμ3∗d2,μd2.
Hence,μd3≥minμ3∗d2,μd2.
Therefore,μdx≥minμx∗dy,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy right derivation of ideals of BF-algebra.
We assume.t0≤t1
Ifx=0,y=2,(4)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.
Hence,t0≤t1 contradicts. μ0≥μx,∀x∈X.Definition 10.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subset μ:dX⟶0,1 is said to be a fuzzy left derivation of ideals of BF-algebras. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμdx∗y,μdy,∀x,y∈XExample 2.
LetX=0,1,2,3,4. From example 15, μ satisfies a fuzzy left derivation of ideals of BF-algebra.
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,49.
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4.
To showμ is a fuzzy right derivation of ideals of BF-algebra,
We assume.t0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t=μzd2=μd4=μx,∀x∉0,1,3Hence,μ0≥μx,∀x≠0,1,3(2)
Letx=1,y=4. Then,(5)μd1=μ0=t0≥t1,=mint1,t1,
=minμd1∗4,μd4, where μd1∗4=μ0∗4=μ2=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμd1∗4,μd4.
Hence,μd1≥minμd1∗4,μd4.
Letx=3,y=4. Then,(6)μd3=μ0=t0≥t1,=mint1,t1,
=minμd3∗4,μd4, where μd3∗4=μ0∗4=μ2=t1,μd4=μ4=t1.
Sincet0≥t1
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗4,μd4.
Hence,μd3≥minμd3∗4,μd4.
Letx=3,y=2. Then,(7)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗2,μd2, where μd3∗2=μ0∗2=μ4=t1,μd2=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗2,μd2.
Hence,μd3≥minμd3∗2,μd2.
Therefore,μdx≥miμdx∗y,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy left derivation of ideals of BF-algebra.
We assumet0≤t1,
Ifx=0,y=2,(8)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.


## 2.2.t0≤t1 Contradicting μ0≥μx,∀x∈X

Definition 11.
LetX be a BF-algebra and let d:X⟶X be a self-map. Then, a fuzzy subset μ:dX⟶0,1 is said to be a fuzzy derivation of ideals of BF-algebra. If the following condition holds:(1)
μ0≥μx,∀x∈X(2)
μdx≥minμdx∗y,μdy,∀x,y∈XExample 3.
LetX=0,1,2,3,4. From example 15, μ satisfies a fuzzy derivation of ideals of BF-algebra.
It is defined byμ:dX⟶0,1 by. dx=0ifx=0,1,34ifx=2,4.
It is defined byμdx=t0ifx=0,1,3t1ifx=2,4 and. μx=t0x∈0,1,3t1x∈2,4.
To showμ is a fuzzy right derivation of ideals of BF-algebra.
We assumet0≥t1.
Ifx=0,y=2,(1)
μd0=μ0=t0≥t1=μd2=μd4=μx,∀x∉0,1,3Hence,μ0≥μx,∀x≠0,1,3(2)
Letx=1,y=4. Then,(9)μd1=μ0=t0≥t1,=mint1,t1,=minμd1∗4,μd4, where μd1∗4=μd4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd1=t0≥t1=mint1,t1=minμd1∗4,μd4.
Hence,μd1≥minμd1∗4,μd4.
Letx=3,y=4. Then,(10)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗4,μd4, where μd3∗4=μd4=μ4=t1,μd4=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗4,μd4.
Hence,μd3≥minμd3∗4,μd4.
Letx=3,y=2. Then,(11)μd3=μ0=t0≥t1,=mint1,t1,=minμd3∗2,μd2, where μd3∗2=μd4=μ4=t1,μd2=μ4=t1.
Sincet0≥t1.
Therefore,μd3=t0≥t1=mint1,t1=minμd3∗2,μd2.
Hence,μd3≥minμd3∗2,μd2.
Therefore,μdx≥minμdx∗y,μdy,∀x∈0,1,3 and y∈2,4.
Hence,μ is a fuzzy derivation of ideals of BF-algebra.
We assumet0≤t1.
Ifx=0,y=2,(12)μd0=μ0=t0≤t1=μd2=μd4=μx,∀x∉0,1,3.
Therefore,μ0≤μx,∀x≠0,1,3.
Hence,t0≤t1 contradicts μ0≥μx,∀x∈X.Lemma 2.
Ifμ is a fuzzy right derivation of ideals of BF-algebra X and x≤dy, then μdx≥μdy.Proof.
Ifx≤dy, then x∗dy=0 since dy≤y and μ0≥μx (by definition), we get
μdx≥minμx∗dy,μdy, since μ is a fuzzy right derivation of ideal of BF-algebras.
=minμ0,μdy, since x∗dy=0.(13)=μdy.
Therefore,μdx≥μdy.Lemma 3.
Ifμ is a fuzzy left derivation of ideals of a BF-algebra X and x≤y, then μdx≥μdy.Proof.
Ifx≤y, then dx∗y≤x∗y=0.
μdx≥minμdx∗y,μdy since μ is a fuzzy left derivation of ideals of a BF-algebra.
Sincedx∗y≤x∗y implying dx∗y≤x∗y=0.
Impliesdx∗y=0.=minμ0,μdy since dx∗y=0.(14)=μdy.
Therefore,μdx≥μdy.Proposition 3.
Letd be a regular self-map of X and let μ be a fuzzy ideal of a BF-algebra X. Then, μd0≥μdx.Proposition 4.
The intersection of any set of fuzzy left derivations of ideals of BF-algebraX is also fuzzy left derivations of BF-algebra.Proof.
Letμi,i∈I be a family of fuzzy left derivations of ideals of BF-algebra X, then for any i∈I.(1)
Let0∈X. Then, ∩μi,i∈I0=infμi,i∈I0≥infμi,i∈Ix=∩μi,i∈Ix(2)
Letx,y∈X. Then, ∩μii∈Idx=infμi,i∈Idx≥infminμi,i∈Idx∗y,μi,i∈Idy
Sinceμi,i∈I is fuzzy left derivations of ideals of BF-algebra.(15)=mininfμi,i∈Idx∗y,infμi,i∈Idy,=min∩μi,i∈Idx∗y,∩μi,i∈Idy,Proposition 5.
The intersection of any set of fuzzy right derivations of ideals of BF-algebraX is also fuzzy right derivations of BF-algebra.Definition 12.
Letμ be a fuzzy derivation of ideal of a BF-algebra X.
Then, the set∪μ,α=x∈X|μdx≥α, where α∈0,1 is called upper-level cut
ofμ (level subset of μ).Theorem 2.
μ is a fuzzy sub algebra of a BF-algebra X if and only if ∪μ,α is a sub algebra of X, where α∈0,1.Proof.
Supposeμ be a fuzzy sub algebra of a BF-algebra X.
We want to show∪μ,α is sub algebra of X.
Letx∈S⊆X. Then, put μx=α,α∈0,1.(16)μ0≥μx≥α.
Sinceμ0≥α implies 0∈∪μ,α.
Hence,∪μ,α≠ϕ.
Letx,y∈∪μ,α. Then, μx≥α,μy≥α.(17)α=minα,α≤minμx,μy≤μx,y.
Hence,μx∗y≥α.
This∪μ,α is a sub algebra of X.
Conversely,∪μ,α be a sub algebra of X.
Letx,y∈∪μ,α. Then, x∗y∈∪μ,α.(18)μx≥α,μy≥α.
It impliesμx≥α,μy≥α and μx∗y≥α.(19)μx∗y≥α≥minα,α=minμx,μy.
Hence,μx∗y≥minμx,μy.
Therefore,μ is a fuzzy sub algebra of a BF-algebra X.Theorem 3.
μ is a fuzzy left derivation of ideals of a BF-algebras X if and only if∪μ,α is an ideal of X.Proof.
Supposeμ be a fuzzy left derivation of ideals of a BF-algebra X.
We want show∪μ,α is an ideal of X.
Letx∈X. Then, μx=α,α∈0,1.
Sinceμ0≥μx=α implies 0∈∪μ,α.
Hence,∪μ,α≠ϕ.
Letx∗y∈∪μ,α and y∈∪μ,α. Then, μx∗y≥α and μy≥α.
Sinceμ is a fuzzy left derivation of ideals of a BF-algebra.
We haveμdx≥minμdx∗y,μdy.≥minμx∗y,μy, since dx∗y≤x∗y. implies μdx∗y≥μx∗y.≥minα,α=α
It is implied thatμdx≥α. But μdx≥μx.
Sinceμx≥α. Therefore, x∈∪μ,α.
Hence,∪μ,α is an ideal of X.
Conversely,∪μ,α be an ideal of X.
We need to proveμ is a fuzzy left derivation of ideals of a BF-algebra.(1)
Since∪μ,α is an ideal of X(20)0∈∪μ,α,μ0≥α.Letx∈∪μ,α. Then, put μx=αso thatμ0≥μx,∀x∈X(2)
Letx∗y∈∪μ,α and y∈∪μ,α. Then, μx∗y≥α and μy≥α.
We getx∈∪μ,α. implying μx≥α.
Sincedx≤x and dy≤y. It is implied that μdx≥μx≥α and μdy≥μy≥α.(21)μdx∗y≥μx∗yμdx∗y≥1−α
Putμdx∗y=1−α and μdy=α.
Then,μdx≥min1−α,α(22)=minμdx∗y,μdy.
Hence,μdx≥minμdx∗y,μdy.
Therefore,μ is a fuzzy left derivation of ideals of a BF-algebra X.Theorem 4.
μ is a fuzzy right derivation of ideals of a BF-algebra X if and only if ∪μ,α is an ideal of X.


## 2.3. Cartesian Product of Fuzzy Derivation of Ideals of BF-Algebras

Definition 13.
LetX be a BF-algebra, and let μ and λ be a fuzzy derivation of ideals of a BF-algebra. Then, μ×λx,y=minμx,λy,∀x,y∈X.
μ×λ:X×X⟶0,1, where μ:X⟶0,1λ:X⟶0,1.Theorem 5.
Letμ and λ be a fuzzy left derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy left derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy left derivation of ideals of a BF-algebra X.
We want to showμ×λ is a fuzzy left derivation of ideals of BF-algebra X.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(23)≥minμx,λy=μ×λx,y, by definition of fuzzy left derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.Sinceμ and λ are fuzzy derivation of ideals of a BF-algebra,(24)≥minminμdx1∗x2,μdx2,minλdy1∗y2,λdy2=minminμdx1∗x2,λdy1∗y2,minμdx2,λdy2=minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2=minμ×λdx1,dy1∗x2,y2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λdx1,dy1∗x2,y2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy left derivation of ideals of BF-algebra X.Theorem 6.
Letμ and λ be a fuzzy right derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy right derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy right derivation of ideals of a BF-algebra X.
We want to show thatμ×λ is a fuzzy right derivation of ideals of BF-algebraX.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(25)≥minμx,λy=μ×λx,y, by definition of fuzzy right derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.
Sinceμ and λ are fuzzy derivation of ideals of a BF-algebra.(26)≥minminμx1∗dx2,μdx2,minλy1∗dy2,λdy2=minminμx1∗dx2,λy1∗dy2,minμdx2,λdy2=minμ×λx1∗dx2,y1∗dy2,μ×λdx2,dy2=minμ×λx1,y1∗dx2,dy2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λx1,y1∗dx2,dy2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy right derivation of ideals of BF-algebra X.Theorem 7.
Letμ and λ be a fuzzy derivation of ideals of a BF-algebra X. Then, μ×λ is a fuzzy derivation of ideals of BF-algebra X.Proof.
Letμ and λ be fuzzy derivation of ideals of a BF-algebra X.
We want to show thatμ×λ is a fuzzy derivation of ideals of BF-algebra X.(1)
Let0,0∈X×X. Then, μ×λ0,0=minμ0,λ0.(27)≥minμx,λy=μ×λx,y, by definition of fuzzy derivation of ideals of a BF-algebra X.Hence,μ×λ0,0≥μ×λx,y,∀x,y∈X.(2)
Letx1,y1 and x2,y2. Then, μ×λdx1,dy1=minμdx1,λdy1.
Sinceμ and λ are fuzzy derivation of ideals of a BF-algebras.(28)≥minminμdx1∗x2,μdx2,minλdy1∗y2,λdy2=minminμdx1∗x2,λdy1∗y2,minμdx2,λdy2=minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2
Therefore,μ×λdx1,dy1≥minμ×λdx1∗x2,dy1∗y2,μ×λdx2,dy2.
Hence,μ×λ is a fuzzy derivation of ideals of BF-algebra X.


## 3. Conclusions

In this paper, we introduced the concepts of fuzzy derivation of ideals of BF-algebra, the fuzzy left derivation of ideals of BF-algebra, and fuzzy right derivation of ideals of BF-algebra. In addition, Cartesian product of fuzzy derivation of ideals of BF-algebra is investigated. In general, this theory can be applied to the notion of rings and near-ring theory of BF-algebra and some related concepts in the future.



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