Based on the theory of falling shadows and fuzzy sets, the notion of a falling
fuzzy implicative ideal of a BCK-algebra is introduced. Relations among
falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative
ideals, and falling fuzzy commutative ideals are given. Relations
between fuzzy implicative ideals and falling fuzzy implicative ideals are provided.

1. Introduction and Preliminaries1.1. 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. Falling shadow representation theory shows us the way of selection relaid on the joint degrees distributions. It is reasonable and convenient approach for the theoretical development and the practical applications of fuzzy sets and fuzzy logics. 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. Jun and Park [6] discussed the notion of a falling fuzzy subalgebra/ideal of a BCK/BCI-algebra. Jun and Kang [7, 8] also considered falling fuzzy positive implicative ideals and falling fuzzy commutative ideals. In this paper, we establish a theoretical approach to define a fuzzy implicative ideal in a BCK-algebra based on the theory of falling shadows. We consider relations between fuzzy implicative ideals and falling fuzzy implicative ideals. We provide relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals.

1.2. Basic Results on BCK-Algebras and Fuzzy Aspects

A BCK/BCI-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.

An algebra (X;*,0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions:

(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0),

(∀x,y∈X)((x*(x*y))*y=0),

(∀x∈X)(x*x=0),

(∀x,y∈X)(x*y=0,y*x=0⇒x=y).

(∀x∈X)(0*x=0),

(∀x∈X)(x*0=x),

(∀x,y,z∈X)(x≤y⇒x*z≤y*z,z*y≤z*x),

(∀x,y,z∈X)((x*y)*z=(x*z)*y),

0∈I,

(∀x∈X)(∀y∈I)(x*y∈I⇒x∈I).

A subset I of a BCK-algebra X is called a positive implicative ideal of X if it satisfies (b1) and

(b3)

(∀x,y,z∈X)((x*y)*z∈I,y*z∈I⇒x*z∈I).

A subset I of a BCK-algebra X is called a commutative ideal of X if it satisfies (b1) and

(b4)

(∀x,y,z∈X)((x*y)*z∈I,z∈I⇒x*(y*(y*x))∈I).

A subset I of a BCK-algebra X is called an implicative ideal of X if it satisfies (b1) and

(b5)

(∀x,y,z∈X)((x*(y*x))*z∈I,z∈I⇒x∈I).

We refer the reader to the paper [9] and book [10] for further information regarding BCK-algebras.

A fuzzy set μ in a BCK-algebra X is called a fuzzy ideal of X (see [11]) if it satisfies the following:

(c1)

(∀x∈X)(μ(0)≥μ(x)),

(c2)

(∀x,y∈X)(μ(x)≥min{μ(x*y),μ(y)}).

A fuzzy set μ in a BCK-algebra X is called a fuzzy positive implicative ideal of X (see [12]) if it satisfies (c1) and

(c3)

(∀x,y,z∈X)(μ(x*z)≥min{μ((x*y)*z),μ(y*z)}).

A fuzzy set μ in a BCK-algebra X is called a fuzzy commutative ideal of X (see [13]) if it satisfies (c1) and

(c4)

(∀x,y,z∈X)(μ(x*(y*(y*x)))≥min{μ((x*y)*z),μ(z)}).

A fuzzy set μ in a BCK-algebra X is called a fuzzy implicative ideal of X (see [14]) if it satisfies (c1) and

(c5)

(∀x,y,z∈X)(μ(x)≥min{μ((x*(y*x))*z),μ(z)}).

Proposition 1.1 (see [<xref ref-type="bibr" rid="B11">11</xref>, <xref ref-type="bibr" rid="B14">14</xref>]).

Let μ be a fuzzy set in a BCK-algebra X. Then μ is a fuzzy (implicative) ideal of X if and only if
(1.2)(∀t∈[0,1])(μt≠∅⟹μtisan(implicative)idealofX),
where μt:={x∈X∣μ(x)≥t}.

1.3. The Theory of Falling Shadows

We first display the basic theory on falling shadows. We refer the reader to the papers [1–5] for further information regarding falling shadows.

Given a universe of discourse U, let 𝒫(U) denote the power set of U. For each u∈U, let
(1.3)u̇:={E∣u∈E,E⊆U},
and for each E∈𝒫(U), let
(1.4)Ė:={u̇∣u∈E}.
An ordered pair (𝒫(U),ℬ) is said to be a hyper-measurable structure on U if ℬ is a σ-field in 𝒫(U) and U̇⊆ℬ. Given a probability space (Ω,𝒜,P) and a hyper-measurable 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,
(1.5)(∀C∈B)(ξ-1(C)={ω∣ω∈Ω,ξ(ω)∈C}∈A).
Suppose that ξ is a random set on U. Let
(1.6)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 H̃t:={u∈U∣H̃(u)≥t} be a t-cut of H̃. Then
(1.7)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]).

2. Falling Fuzzy Implicative Ideals

In what follows let X denote a BCK-algebra unless otherwise.

Definition 2.1 (see [<xref ref-type="bibr" rid="B6">6</xref>–<xref ref-type="bibr" rid="B8">8</xref>]).

Let (Ω,𝒜,P) be a probability space, and let
(2.1)ξ:Ω⟶P(X)
be a random set. If ξ(ω) is an ideal (resp., positive implicative ideal and commutative ideal) of X for any ω∈Ω, then the falling shadow H̃ of the random set ξ, that is,
(2.2)H̃(x)=P(ω∣x∈ξ(ω))
is called a falling fuzzy ideal (resp., falling fuzzy positive implicative ideal and falling fuzzy commutative ideal) of X.

Let (Ω,𝒜,P) be a probability space and let
(2.3)F(X):={f∣f:Ω⟶Xisamapping},
where X is a BCK-algebra. Define an operation ⊛ on F(X) by
(2.4)(∀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 BCK-algebra (see [6]).

Definition 2.2.

Let (Ω,𝒜,P) be a probability space and let
(2.5)ξ:Ω⟶P(X)
be a random set. If ξ(ω) is an implicative ideal of X for any ω∈Ω, then the falling shadow H̃ of the random set ξ, that is,
(2.6)H̃(x)=P(ω∣x∈ξ(ω))
is called a falling fuzzy implicative ideal of X.

For any subset A of X and f∈F(X), let
(2.7)Af:={ω∈Ω∣f(ω)∈A},ξ:Ω⟶P(F(X)),ω⟼{f∈F(X)∣f(ω)∈A}.
Then Af∈𝒜.

Theorem 2.3.

If A is an implicative ideal of X, then
(2.8)ξ(ω)={f∈F(X)∣f(ω)∈A}
is an implicative ideal of F(X).

Proof.

Assume that A is an implicative ideal of X and let ω∈Ω. Since θ(ω)=0∈A, we see that θ∈ξ(ω). Let f,g,h∈F(X) be such that (f⊛(g⊛f))⊛h∈ξ(ω) and h∈ξ(ω). Then
(2.9)(f(ω)*(g(ω)*f(ω)))*h(ω)=((f⊛(g⊛f))⊛h)(ω)∈A
and h(ω)∈A. Since A is an implicative ideal of X, it follows from (b5) that f(ω)∈A and so f∈ξ(ω). Hence ξ(ω) is an implicative ideal of F(X).

Since
(2.10)ξ-1(ḟ)={ω∈Ω∣f∈ξ(ω)}={ω∈Ω∣f(ω)∈A}=Af∈A,
we see that ξ is a random set on F(X). Let
(2.11)H̃(f)=P(ω∣f(ω)∈A).
Then H̃ is a falling fuzzy implicative ideal of F(X).

Example 2.4.

Consider a BCK-algebra X={0,a,b,c,d} with a Cayley table which is given by Table 1 (see [10, page 274]). Let (Ω,𝒜,P)=([0,1],𝒜,m) and let ξ:[0,1]→𝒫(X) be defined by
(2.12)ξ(t):={{0,a}ift∈[0,0.25),{0,b}ift∈[0.25,0.55),{0,b,d}ift∈[0.55,0.7),{0,a,b,c}ift∈[0.7,1].
Then ξ(t) is an implicative ideal of X for all t∈[0,1]. Hence H̃, which is given by H̃(x)=P(t∣x∈ξ(t)), is a falling fuzzy implicative ideal of X, and it is represented as follows:
(2.13)H̃(x)={1ifx=0,0.55ifx=a,0.75ifx=b,0.3ifx=c,0.15ifx=d.
Then
(2.14)H̃t={{0}ift∈(0.75,1],{0,a}ift∈(0.55,0.75],{0,a,b}ift∈(0.3,0.55],{0,a,b,c}ift∈(0.15,0.3],Xift∈[0,0.15].
If t∈(0.3,0.55], then H̃t={0,a,b} is not an implicative ideal of X since (c*(b*c))*a=(c*0)*a=c*a=b∈H̃t and a∈H̃t, but c∉H̃t. It follows from Proposition 1.1 that H̃ is not a fuzzy implicative ideal of X.

Table 1

Cayley table.

*

0

a

b

c

d

0

0

0

0

0

0

a

a

0

a

0

a

b

b

b

0

0

b

c

c

b

a

0

c

d

d

d

d

d

0

Theorem 2.5.

Every fuzzy implicative ideal of X is a falling fuzzy implicative ideal of X.

Proof.

Consider the probability space (Ω,𝒜,P)=([0,1],𝒜,m), where 𝒜 is a Borel field on [0,1] and m is the usual Lebesque measure. Let μ be a fuzzy implicative ideal of X. Then μt is an implicative ideal of X for all t∈[0,1] by Proposition 1.1. Let
(2.15)ξ:[0,1]⟶P(X)
be a random set and ξ(t)=μt for every t∈[0,1]. Then μ is a falling fuzzy implicative ideal of X.

Example 2.4 shows that the converse of Theorem 2.5 is not valid.

Theorem 2.6.

Every falling fuzzy implicative ideal is a falling fuzzy ideal.

Proof.

Let H̃ be a falling fuzzy implicative ideal of X. Then ξ(ω) is an implicative ideal of X, and hence it is an ideal of X. Thus H̃ is a falling fuzzy ideal of X.

The converse of Theorem 2.6 is not true in general as shown by the following example.

Example 2.7.

Consider a BCK-algebra X={0,a,b,c,d} with a Cayley table which is given by Table 2 (see [10, page 260]). Let (Ω,𝒜,P)=([0,1],𝒜,m) and let ξ:[0,1]→𝒫(X) be defined by
(2.16)ξ(t):={{0,c}ift∈[0,0.3),{0,a,b,d}ift∈[0.3,1].
Then ξ(t) is an ideal of X for all t∈[0,1]. Hence H̃(x)=P(t∣x∈ξ(t)) is a falling fuzzy ideal of X, and
(2.17)H̃(x)={0.3ifx=c,0.7ifx∈{a,b,d},1ifx=0.
In this case, we can easily check that H̃ is a fuzzy ideal of X (see [6]). If t∈[0,0.3), then ξ(t)={0,c} is not an implicative ideal of X since (a*(b*a))*c∈ξ(t) and c∈ξ(t), but a∉ξ(t). Therefore H̃ is not a falling fuzzy implicative ideal of X.

Table 2

Cayley table.

*

0

a

b

c

d

0

0

0

0

0

0

a

a

0

0

a

0

b

b

a

0

b

0

c

c

c

c

0

c

d

d

d

d

d

0

Theorem 2.8.

Every falling fuzzy implicative ideal is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.

Proof.

Let H̃ be a falling fuzzy implicative ideal of X. Then ξ(ω) is an implicative ideal of X, and hence it is both a positive implicative ideal and a commutative ideal of X. Thus H̃ is both a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal.

The following example shows that the converse of Theorem 2.8 may not be true.

Example 2.9.

(1) Consider a BCK-algebra X={0,a,b,c,d} with a Cayley table which is given by Table 3 (see [10, page 263]). Let (Ω,𝒜,P)=([0,1],𝒜,m) and let ξ:[0,1]→𝒫(X) be defined by
(2.18)ξ(t):={{0,c}ift∈[0,0.15),{0,d}ift∈[0.15,0.45),{0,a,b}ift∈[0.45,0.75),{0,c,d}ift∈[0.75,1].
Then ξ(t) is a commutative ideal of X for all t∈[0,1]. Hence H̃, which is given by H̃(x)=P(t∣x∈ξ(t)), is a falling fuzzy commutative ideal of X (see [8]). But it is not a falling fuzzy implicative ideal of X because if t∈[0.75,1] then ξ(t)={0,c,d} is not an implicative ideal of X since (a*(b*a))*c=(a*a)*c=0*c=0∈ξ(t) and c∈ξ(t), but a∉ξ(t).

(2) Let X={0,a,b,c} be a BCK-algebra in which the *-multiplication is defined by Table 4. Let (Ω,𝒜,P)=([0,1],𝒜,m) and let
(2.19)ξ:[0,1]⟶P(X),t⟼{{0,a}ift∈[0,0.2),{0,b}ift∈[0.2,0.5),{0,a,b}ift∈[0.5,0.9),Xift∈[0.9,1].
Then ξ(t) is a positive implicative ideal of X for all t∈[0,1]. Thus H̃, which is given by H̃(x)=P(t∣x∈ξ(t)), is a falling fuzzy positive implicative ideal of X. But it is not a falling fuzzy implicative ideal of X because if t∈[0.2,0.5) then ξ(t)={0,b} is not an implicative ideal of X since (a*(c*a))*b=(a*c)*b=0*b=0∈ξ(t) and b∈ξ(t), but a∉ξ(t).

The notions of a falling fuzzy positive implicative ideal and a falling fuzzy commutative ideal are independent, that is, a falling fuzzy commutative ideal need not be a falling fuzzy positive implicative ideal, and vice versa. In fact, the falling fuzzy commutative ideal H̃ in Example 2.9(1) is not a falling fuzzy positive implicative ideal. Also the falling fuzzy positive implicative ideal H̃ in Example 2.9(2) is not a falling fuzzy commutative ideal.

Let (Ω,𝒜,P) be a probability space and H̃ a falling shadow of a random set ξ:Ω→𝒫(X). For any x∈X, let
(2.20)Ω(x;ξ):={ω∈Ω∣x∈ξ(ω)}.
Then Ω(x;ξ)∈𝒜.

Table 3

Cayley table.

*

0

a

b

c

d

0

0

0

0

0

0

a

a

0

0

a

a

b

b

a

0

b

b

c

c

c

c

0

c

d

d

d

d

d

0

Table 4

*-Multiplication.

*

0

a

b

c

0

0

0

0

0

a

a

0

a

0

b

b

b

0

0

c

c

c

c

0

Proposition 2.10.

If a falling shadow H̃ of a random set ξ:Ω→𝒫(X) is a falling fuzzy implicative ideal of X, then (1)

(∀x,y,z∈X)(Ω((x*(y*x))*z;ξ)∩Ω(z;ξ)⊆Ω(x;ξ)),

(2)

(∀x,y,z∈X)(Ω(x;ξ)⊆Ω((x*(y*x))*z;ξ)).

Proof.

(1) Let ω∈Ω((x*(y*x))*z;ξ)∩Ω(z;ξ). Then (x*(y*x))*z∈ξ(ω) and z∈ξ(ω). Since ξ(ω) is an implicative ideal of X, it follows from (b5) that x∈ξ(ω) so that ω∈Ω(x;ξ). Therefore
(2.21)Ω((x*(y*x))*z;ξ)∩Ω(z;ξ)⊆Ω(x;ξ),
for all x,y,z∈X.

(2) If ω∈Ω(x;ξ), then x∈ξ(ω). Since
(2.22)((x*(y*x))*z)*x=((x*(y*x))*x)*z=((x*x)*(y*x))*z=(0*(y*x))*z=0*z=0,
we have ((x*(y*x))*z)*x=0∈ξ(ω). Since ξ(ω) is an implicative ideal and hence an ideal of X, it follows that (x*(y*x))*z∈ξ(ω) and so ω∈Ω((x*(y*x))*z;ξ). Hence Ω(x;ξ)⊆Ω((x*(y*x))*z;ξ) for all x,y,z∈X.

Theorem 2.11.

If H̃ is a falling fuzzy implicative ideal of X, then
(2.23)(∀x,y,z∈X)(H̃(x)≥Tm(H̃((x*(y*x))*z),H̃(z))),
where Tm(s,t)=max{s+t-1,0} for any s,t∈[0,1].

Proof.

By Definition 2.2, ξ(ω) is an implicative ideal of X for any ω∈Ω. Hence
(2.24){ω∈Ω∣(x*(y*x))*z∈ξ(ω)}∩{ω∈Ω∣z∈ξ(ω)}⊆{ω∈Ω∣x∈ξ(ω)},
and thus
(2.25)H̃(x)=P(ω∣x∈ξ(ω))≥P({ω∣(x*(y*x))*z∈ξ(ω)}∩{ω∣z∈ξ(ω)})≥P(ω∣(x*(y*x))*z∈ξ(ω))+P(ω∣z∈ξ(ω))-P(ω∣(x*(y*x))*z∈ξ(ω)orz∈ξ(ω))≥H̃((x*(y*x))*z)+H̃(z)-1.
Therefore
(2.26)H̃(x)≥max{H̃((x*(y*x))*z)+H̃(z)-1,0}=Tm(H̃((x*(y*x))*z),H̃(z)).
This completes the proof.

Theorem 2.11 means that every falling fuzzy implicative ideal of X is a Tm-fuzzy implicative ideal of X.

Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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