# The Theory of Falling Shadows Applied tod-Ideals in d-Algebras

Authors: Young Bae Jun; Sun Shin Ahn
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

On the basis of the theory of a falling shadow which was first formulated by Wang (1985), the notion of fallingd∗-ideals in d-algebras is introduced, and related properties are investigated. Characterizations of a falling d∗-ideal are established. Relations among falling d∗-ideals, falling d-ideals, falling d#-ideals, falling d-subalgebras, and falling BCK-ideals are discussed.


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## Body

## 1. Introduction

In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman [1] pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez [2] introduced the theory of falling shadows which directly relates probability concepts with the membership function of fuzzy sets. The mathematical structure of the theory of falling shadows is formulated in [3]. Tan et al. [4, 5] established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Yuan and Lee [6] considered a fuzzy subgroup (subring, ideal) as the falling shadow of the cloud of the subgroup (subring, ideal). Iséki and Tanaka introduced two classes of abstract algebras: BCK-algebras and BCI-algebras ([7, 8]). It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras; that is, a BCI-algebra is a generalization of a BCK-algebra. As another useful generalization of BCK-algebras, Neggers and Kim [9] introduced the notion of d-algebras. They investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. After that, some further aspects were studied in [10, 11]. Neggers et al. [12] introduced the concept of d-fuzzy function which generalizes the concept of fuzzy subalgebra to a much larger class of functions in a natural way. In addition, they discussed a method of fuzzification of a wide class of algebraic systems onto [0,1] along with some consequences. Jun et al. [13] discussed implicative ideals of BCK-algebras based on the fuzzy sets and the theory of falling shadows. Also, Jun et al. [14] used the theory of a falling shadow for considering falling d-subalgebras, falling d-ideals, falling d#-ideals, and falling BCK-ideals in d-algebras.In this paper, we introduce the notion of fallingd*-ideals in d-algebras, and investigate several properties. We establish characterizations of falling d*-ideals, and we use these characterizations for considering relations among falling d*-ideals, falling d-ideals, falling d#-ideals, falling d-subalgebras and falling BCK-ideals.


## 2. Preliminaries

Ad-algebra is a nonempty set X with a constant 0 and a binary operation “*" satisfying the following axioms:(I)
x*x=0,(II)
0*x=0,(III)
x*y=0 and y*x=0 imply x=yfor allx,y∈X.ABCK-algebra is a d-algebra (X,*,0) satisfying the following additional axioms:(IV)
((x*y)*(x*z))*(z*y)=0,(V)
(x*(x*y))*y=0for allx,y,z∈X.AnyBCK-algebra (X,*,0) satisfies the following conditions:(a1)
(∀x,y∈X)((x*y)*x=0),(a2)
(∀x,y,z∈X)((x*z)*(y*z))*(x*y)=0).A subsetI of a BCK-algebra X is called a BCK-ideal of X if it satisfies,(b1)
0∈I.(b2)
(∀x∈X)(∀y∈I)(x*y∈I⇒x∈I).We now display the basic theory on falling shadows. We refer the reader to the papers [1–5] for further information regarding the theory of falling shadows.Given a universe of discourseU, let 𝒫(U) denote the power set of U. For each u∈U, let(2.1)u̇:={E∣u∈EandE⊆U},
and for each E∈𝒫(U), let(2.2)Ė:={u̇∣u∈E}.
An ordered pair (𝒫(U),ℬ) is said to be a hypermeasurable structure on U if ℬ is a σ-field in 𝒫(U) and U̇⊆ℬ. Given a probability space (Ω,𝒜,P) and a hypermeasurable structure (𝒫(U),ℬ) on U, a random set on U is defined to be a mapping ξ:Ω→𝒫(U) which is 𝒜-ℬ measurable, that is,
(2.3)(∀C∈B)(ξ-1(C)={ω∣ω∈Ωandξ(ω)∈C}∈A).
Suppose that ξ is a random set on U. Let(2.4)H̃(u):=P(ω∣u∈ξ(ω))foreachu∈U.
Then H̃ is a kind of fuzzy set in U. We call H̃ a falling shadow of the random set ξ, and ξ is called a cloud of H̃.For example,(Ω,𝒜,P)=([0,1],𝒜,m), where 𝒜 is a Borel field on [0,1] and m is the usual Lebesgue measure. Let H̃ be a fuzzy set in U, and let H̃t:={u∈U∣H̃(u)≥t} be a t-cut of H̃. Then
(2.5)ξ:[0,1]⟶P(U),t⟼H̃t
is a random set and ξ is a cloud of H̃. We will call ξ defined above as the cut-cloud of H̃ (see [1]).


## 3. Fallingd*-Ideals

In what follows letX denote a d-algebra unless otherwise specified.A nonempty subsetS of X is called a d-subalgebra of X (see [11]) if x*y∈S whenever x∈S and y∈S.A subsetI of X is called a BCK-ideal of X (see [11]) if it satisfies conditions (b1) and (b2).A subsetI of X is called a d-ideal of X (see [11]) if it satisfies condition (b2) and (b3)
(∀x,y∈X)(x∈I⇒x*y∈I).Ad-ideal I of X is called a d#-ideal of X (see [11]) if, for arbitrary x,y,z∈X,(b4)
x*z∈I whenever x*y∈I and y*z∈I.Definition 3.1 (see [14]).
Let(Ω,𝒜,P) be a probability space, and let
(3.1)ξ:Ω⟶P(X)
be a random set. If ξ(ω) is a d-subalgebra (BCK-ideal, d-ideal and d#-ideal, resp.) of X for any ω∈Ω with ξ(ω)≠∅, then the falling shadow H̃ of the random set ξ, that is,
(3.2)H̃(x)=P(ω∣x∈ξ(ω))
is called a fallingd-subalgebra (fallingBCK-ideal,  fallingd-ideal and fallingd♯-ideal,resp.) of X.Lemma 3.2 (see [14]).
LetH̃ be a falling shadow of a random set ξ on X. Then H̃ is a falling d-ideal of X if and only if the following conditions are valid:(a)
(∀x,y∈X)(Ω(x*y;ξ)∩Ω(y;ξ)⊆Ω(x;ξ)),(b)
(∀x,y∈X)(Ω(x;ξ)⊆Ω(x*y;ξ)).Lemma 3.3 (see [14]).
IfH̃ is a falling d-ideal of X, then
(3.3)(∀x,y∈X)(y*x=0⟹Ω(x;ξ)⊆Ω(y;ξ)).Proposition 3.4.
For a falling shadowH̃ of a random set ξ on X, if H̃ is a falling d-ideal of X, then
(3.4)(∀x,y,z∈X)((x*y)*z=0⟹Ω(y;ξ)∩Ω(z;ξ)⊆Ω(x;ξ)).Proof.
Letx,y,z∈X be such that (x*y)*z=0. Using Lemma 3.3, we have Ω(z;ξ)⊆Ω(x*y;ξ). It follows from Lemma 3.2(a) that
(3.5)Ω(y;ξ)∩Ω(z;ξ)⊆Ω(y;ξ)∩Ω(x*y;ξ)⊆Ω(x;ξ).
This completes the proof.A fuzzy setμ on X is called a fuzzyd-ideal of X (see [10]) if it satisfies(i)
(∀x,y∈X)(μ(x)≥min{μ(x*y),μ(y)}),(ii)
(∀x,y∈X)(μ(x*y)≥μ(x)).Lemma 3.5 (see [10]).
A fuzzy setμ on X is a fuzzy d-ideal of X if and only if, for every λ∈[0,1],μλ:={x∈X∣μ(x)≥λ} is a d-ideal of X when it is nonempty.Theorem 3.6.
If we take the probability space(Ω,𝒜,P)=([0,1],𝒜,m), where 𝒜 is a Borel field on [0,1] and m is the usual Lebesgue measure, then every fuzzy d-ideal of X is a falling d-ideal of X.Proof.
Letμ be a fuzzy d-ideal of X. Then μλ(≠∅) is a d-ideal of X for all λ∈[0,1] by Lemma 3.5. Let
(3.6)ξ:Ω⟶P(X)
be a random set and ξ(λ)=μλ for every λ∈Ω. Then μ is a falling d-ideal of X.We provide an example to show that the converse of Theorem3.6 is not true.Example 3.7.
LetX:={0,a,b,c} be a d-algebra which is not a BCK-algebra with the Cayley table as follows:
(3.7)*0abc00000aa0aabbb00ccca0
Let (Ω,𝒜,P)=([0,1],𝒜,m) and define a random set
(3.8)ξ:Ω⟶P(X),ω⟼{{0}ifω∈[0,0.2),∅ifω∈[0.2,0.3),{0,a}ifω∈[0.3,0.6),{0,b}ifω∈[0.6,0.85),Xifω∈[0.85,1].
Then the falling shadow H̃ of ξ is a falling d-ideal of X, and it is represented as follows:
(3.9)H̃(x)={0.9ifx=0,0.45ifx=a,0.4ifx=b,0.15ifx=c.
We know that H̃ is not a fuzzy d-ideal of X since
(3.10)H̃(c)=0.15≱0.4=min⁡⁡{H̃(c*b),H̃(b)}.Let(Ω,𝒜,P) be a probability space and let
(3.11)F(X):={f∣f:Ω⟶Xisamapping}.
Define an operation ⊛ on F(X) by
(3.12)(∀ω∈Ω)((f⊛g)(ω)=f(ω)*g(ω))
for all f,g∈F(X). Let θ∈F(X) be defined by θ(ω)=0 for all ω∈Ω. Then (F(X);⊛,θ) is a d-algebra [14]. For any subset A of X and f∈F(X), let(3.13)Af:={ω∈Ω∣f(ω)∈A},ξ:Ω⟶P(F(X)),ω⟼{f∈F(X)∣f(ω)∈A}.
Then Af∈𝒜.Theorem 3.8.
IfA is a d-ideal of X, then
(3.14)ξ(ω)={f∈F(X)∣f(ω)∈A}
is a d-ideal of F(X).Proof.
Assume thatA is a d-ideal of X, and let ω∈Ω. Let f,g∈F(X) be such that g∈ξ(ω) and f⊛g∈ξ(ω). Then g(ω)∈A and f(ω)*g(ω)=(f⊛g)(ω)∈A. Since A is a d-ideal of X, it follows from (b2) that f(ω)∈A so that f∈ξ(ω). For any f∈F(X), if f∈ξ(ω) then f(ω)∈A. It follows that from (b3) that (f⊛g)(ω)=f(ω)*g(ω)∈A for all g∈F(X). Hence f⊛g∈ξ(ω) for all g∈F(X). Therefore ξ(ω) is a d-ideal of F(X).Theorem 3.9.
IfH̃ is a falling d-ideal of X, then(a)
(∀x,y∈X)(H̃(x*y)≥H̃(x)),(b)
(∀x,y∈X)(H̃(x)≥Tm(H̃(x*y),H̃(y))),where Tm(s,t)=max{s+t-1,0} for any s,t∈[0,1].Proof.
(a) It is clear.
(b) By Definition3.1, ξ(ω) is a d-ideal of X for any ω∈Ω with ξ(ω)≠∅. Hence
(3.15){ω∈Ω∣x*y∈ξ(ω)}∩{ω∈Ω∣y∈ξ(ω)}⊆{ω∈Ω∣x∈ξ(ω)},
and thus
(3.16)H̃(x)=P(ω∣x∈ξ(ω))≥P({ω∣x*y∈ξ(ω)}∩{ω∣y∈ξ(ω)})≥P(ω∣x*y∈ξ(ω))+P(ω∣y∈ξ(ω))-P(ω∣x*y∈ξ(ω)ory∈ξ(ω))≥H̃(x*y)+H̃(y)-1.
Hence
(3.17)H̃(x)≥max⁡⁡{H̃(x*y)+H̃(y)-1,0}=Tm(H̃(x*y),H̃(y)).
This completes the proof.Ad-algebra X is called a d*-algebra (see [11]) if it satisfies the identity (x*y)*x=0 for all x,y∈X.If ad#-ideal I of X satisfies(b5)
x*y∈I and y*x∈I imply (x*z)*(y*z)∈I and (z*x)*(z*y)∈I for all x,y,z∈X, then we say that I is a d*-ideal of X (see [11]).Definition 3.10.
For a a probability space(Ω,𝒜,P) and a random set ξ on X, if ξ(ω) is a d*-ideal of X for any ω∈Ω with ξ(ω)≠∅, then the falling shadow H̃ of the random set ξ is called a fallingd*-ideal of X.Example 3.11.
LetX:={0,a,b,c} be a d-algebra which is not a BCK-algebra with the following Cayley table:
(3.18)*0abc00000aa00abbb00ccca0
Let (Ω,𝒜,P)=([0,1],𝒜,m) and define a random set
(3.19)ξ:Ω⟶P(X),ω⟼{{0,a}ifω∈[0,0.6),∅ifω∈[0.6,0.7),Xifω∈[0.7,1].
Then the falling shadow H̃ of ξ is a falling d*-ideal of X.Obviously, every fallingd*-ideal is a falling d#-ideal, but the converse does not hold in general.Example 3.12.
LetX:={0,a,b,c} be a d-algebra which is not a BCK-algebra with the Cayley table as follows:
(3.20)*0abc00000aa00abcb0cccbb0
For a probability space (Ω,𝒜,P)=([0,1],𝒜,m), define a random set
(3.21)ξ:Ω⟶P(X),ω⟼{{0,a}ifω∈[0,0.3),Xifω∈[0.3,0.8),∅ift∈[0.8,1].
Then the falling shadow H̃ of ξ is a falling d#-ideal of X, but not a falling d*-ideal of X because if ω∈[0,0.3) then ξ(ω)={0,a} is not a d*-ideal of X.A characterization of a fallingd#-ideal is established as follows.Lemma 3.13 (see [14]).
For a falling shadowH̃ of a random set ξ on X, the following are equivalent:(a)
H̃ is a falling d#-ideal of X,(b)
H̃ is a falling d-ideal of X that satisfies the following inclusion:
(3.22)(∀x,y,z∈X)(Ω(x*y;ξ)∩Ω(y*z;ξ)⊆Ω(x*z;ξ)).We provide characterizations of a fallingd*-ideal.Theorem 3.14.
For a falling shadowH̃ of a random set ξ on X,H̃ is a falling d*-ideal of X if and only if the following conditions are valid for every x,y,z∈X:(a)
Ω(x*y;ξ)∩Ω(y;ξ)⊆Ω(x;ξ),(b)
Ω(x;ξ)⊆Ω(x*y;ξ),(c)
Ω(x*y;ξ)∩Ω(y*z;ξ)⊆Ω(x*z;ξ),(d)
Ω(x*y;ξ)∩Ω(y*x;ξ)⊆Ω((x*z)*(y*z);ξ)∩Ω((z*x)*(z*y);ξ).Proof.
Assume thatH̃ is a falling d*-ideal of X. Then H̃ is a falling d#-ideal of X, and so conditions (a), (b), and (c) are valid by Lemmas 3.2 and 3.13. Let x,y,z∈X and ω∈Ω. If ω∈Ω(x*y;ξ)∩Ω(y*x;ξ), then x*y∈ξ(ω) and y*x∈ξ(ω). Since ξ(ω) is a d*-ideal of X, it follows from (b5) that (x*z)*(y*z)∈ξ(ω) and (z*x)*(z*y)∈ξ(ω) so that
(3.23)ω∈Ω((x*z)*(y*z);ξ)∩Ω((z*x)*(z*y);ξ)
for all x,y,z∈X. Therefore (d) holds.
Conversely, suppose that conditions (a), (b), (c), and (d) are valid. Three conditions (a), (b), and (c) imply thatH̃ is a falling d#-ideal of X by Lemmas 3.2 and 3.13. Finally, let x,y,z∈X and ω∈Ω be such that x*y∈ξ(ω) and y*x∈ξ(ω). Using the condition (d), we have
(3.24)ω∈Ω(x*y;ξ)∩Ω(y*x;ξ)⊆Ω((x*z)*(y*z);ξ)∩Ω((z*x)*(z*y);ξ),
which implies that (x*z)*(y*z)∈ξ(ω) and (z*x)*(z*y)∈ξ(ω). Therefore H̃ is a falling d*-ideal of X.The following relation is described in [14]Combining this relation and the fact that every fallingd*-ideal is a falling d#-ideal, we have the following relation:In this diagram, the reverse implications are not true, and we need additional conditions for considering the reverse implications. Jun et al. [14] showed that the following relation holds in d*-algebras:Lemma 3.15 (see [14]).
For a falling shadowH̃ of a random set ξ on X, if H̃ is a falling BCK-ideal of X, then(a)
(∀x,y∈X)(x*y=0⇒Ω(y;ξ)⊆Ω(x;ξ)),(b)
(∀x,y∈X)(Ω(x*y;ξ)∩Ω(y;ξ)⊆Ω(x;ξ)).Theorem 3.16.
IfX is a BCK-algebra, then every falling BCK-ideal of X is a falling d*-ideal of X.Proof.
LetH̃ be a falling BCK-ideal of a BCK-algebra X. Then
(3.25)Ω(x*y;ξ)∩Ω(y;ξ)⊆Ω(x;ξ)
for all x,y∈X by Lemma 3.15(b). Using (a1), we have (x*y)*x=0 for all x,y∈X. Hence, by Lemma 3.15(a), we get Ω(x;ξ)⊆Ω(x*y;ξ) for all x,y∈X. If ω∈Ω(x*y;ξ)∩Ω(y*z;ξ), then x*y∈ξ(ω) and y*z∈ξ(ω). Note that ((x*z)*(y*z))*(x*y)=0∈ξ(ω). Since ξ(ω) is a BCK-ideal of X, it follows from (b2) that x*z∈ξ(ω) so that ω∈Ω(x*z;ξ). Thus Ω(x*y;ξ)∩Ω(y*z;ξ)⊆Ω(x*z;ξ). Let ω∈Ω(x*y;ξ)∩Ω(y*x;ξ). Then x*y∈ξ(ω) and y*x∈ξ(ω). By (IV) and (a2), we have ((z*x)*(z*y))*(y*x)=0∈ξ(ω) and ((x*z)*(y*z))*(x*y)=0∈ξ(ω). It follows from (b2) that (z*x)*(z*y)∈ξ(ω) and (x*z)*(y*z)∈ξ(ω) so that ω∈Ω((x*z)*(y*z);ξ)∩Ω((z*x)*(z*y);ξ). Hence
(3.26)Ω(x*y;ξ)∩Ω(y*x;ξ)⊆Ω((x*z)*(y*z);ξ)∩Ω((z*x)*(z*y);ξ).
Using Theorem 3.14, we conclude that H̃ is a falling d*-ideal of X.Note that everyBCK-algebra is a d*-algebra (see [11]). Therefore, the above diagrams together with Theorem 3.16 induce the following diagram in BCK-algebras:



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