Falling d-Ideals in d-Algebras

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Volume 2011, Article ID 516418,14pages doi:10.1155/2011/516418

Research Article

Falling d-Ideals in d-Algebras

Young Bae Jun,

¹

Sun Shin Ahn,

²

and Kyoung Ja Lee

³

1Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea

2Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

3Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea

Correspondence should be addressed to Sun Shin Ahn,[email protected] Received 24 August 2011; Accepted 25 October 2011

Academic Editor: Bo Yang

Copyrightq2011 Young Bae Jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the theory of a falling shadow which was first formulated by Wang1985, a theoretical approach of the ideal structure in d-algebras is established. The notions of a falling d-subalgebra, a falling d-ideal, a falling BCK-ideal, and a fallingd-ideal of a d-algebra are introduced. Some fundamental properties are investigated. Relations among a falling d-subalgebra, a falling d-ideal, a falling BCK-ideal, and a fallingd-ideal are stated. Characterizations of falling d-ideals and falling d-ideals are discussed. A relation between a fuzzy d-subalgebra and a falling d-subalgebra is provided.

1. Introduction

Is´eki and Tanaka introduced two classes of abstract algebrasBCK-algebras andBCI-algebras 1,2. It is known that the class ofBCK-algebras is a proper subclass of the class of BCI- algebras. BCK-algebras have several connections with other areas of investigation, such as: lattice ordered groups, MV-algebras, Wajsberg algebras, and implicative commutative semigroups. Font et al.3have discussed Wajsberg algebras which are term-equivalent to MV-algebras. Mundici4proved thatMV-algebras are categorically equivalent to bounded commutativeBCK-algebras. Meng5proved that implicative commutative semigroups are equivalent to a class of BCK-algebras. Neggers and Kim6 introduced the notion of d- algebras which is another useful generalization ofBCK-algebras. They investigated several relations betweend-algebras andBCK-algebras as well as several other relations between d-algebras and oriented digraphs. After that, some further aspects were studied in 7, 8.

Neggers et al.9introduced the concept ofd-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 onto0,1along with some consequences.

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In the study of a unified treatment of uncertainty modelled by means of combining probability and fuzzy set theory, Goodman 10 pointed out the equivalence of a fuzzy set and a class of random sets. Wang and Sanchez 11 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 in12. Tan et al.13,14established a theoretical approach to define a fuzzy inference relation and fuzzy set operations based on the theory of falling shadows. Jun and Kang15established a theoretical approach to define a fuzzy positive implicative ideal in a BCK-algebra based on the theory of falling shadows. They provided relations between falling fuzzy positive implicative ideals and falling fuzzy ideals. They also considered relations between fuzzy positive implicative ideals and falling fuzzy positive implicative ideals. Jun and Kang16considered the fuzzification of generalized Tarski filters of generalized Tarski algebras and investigated related properties.

They established characterizations of a fuzzy-generalized Tarski filter and introduced the notion of falling fuzzy-generalized Tarski filters in generalized Tarski algebras based on the theory of falling shadows. They provided relations between fuzzy-generalized Tarski filters and falling fuzzy-generalized Tarski filters and established a characterization of a falling fuzzy-generalized Tarski filter.

In this paper, we establish a theoretical approach to define a fallingd-subalgebra, a fallingd-ideal, a fallingBCK-ideal, and a fallingd-ideal ind-algebras based on the theory of falling shadows which was first formulated by Wang12. We provide relations among a fallingd-subalgebra, a fallingd-ideal, a fallingBCK-ideal, and a fallingd-ideal. We consider characterizations of falling d-ideals and falling d-ideals and discuss a relation between a fuzzyd-subalgebra and a fallingd-subalgebra.

2. Preliminaries

Ad-algebra is a nonempty setXwith a constant 0 and a binary operation “∗” satisfying the following axioms:

ix∗x0, ii0∗x0,

iiix∗y0 andy∗x0 implyxy, for allx, y∈X.

ABCK-algebra is ad-algebraX,∗,0satisfying the following additional axioms:

iv x∗y∗x∗z∗z∗y 0, v x∗x∗y∗y0,

for allx, y, z∈X.

AnyBCK-algebraX,∗,0satisfies the following conditions:

a1 for allx, y∈X x∗y∗x0,

a2 for allx, y, z∈X x∗z∗y∗z∗x∗y 0.

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A subsetIof aBCK-algebraXis called aBCK-ideal ofXif it satisfies b10∈I,

b2 for allx∈X for ally∈I x∗y∈I ⇒ x∈I.

We now display the basic theory on falling shadows. We refer the reader to the papers 10–14for further information regarding the theory of falling shadows.

Given a universe of discourseU, letPUdenote the power set ofU. For eachu∈U, let

˙

u:{E|u∈EandE⊆U}, 2.1

and for eachE∈ PU, let

E˙ :{u˙ |u∈E}. 2.2

An ordered pairPU,Bis said to be a hypermeasurable structure onUifBis a σ-field inPUand ˙U ⊆ B. Given a probability spaceΩ,A, Pand a hypermeasurable structure PU,BonU, a random set onUis defined to be a mappingξ :Ω → PUwhich isA-B measurable, that is,

∀C∈ B

ξ⁻¹C {ω|ω∈Ωandξω∈C} ∈ A

. 2.3

Suppose thatξis a random set onU. Let

Hu :Pω|u∈ξωfor eachu∈U. 2.4

ThenHis a kind of fuzzy set inU. We callHa falling shadow of the random setξ, andξis called a cloud ofH.

For example,Ω,A, P 0,1,A, m, whereAis a Borel field on0,1andmis the usual Lebesgue measure. LetHbe a fuzzy set inUand letH_t:{u∈U|Hu ≥t}be at-cut ofH. Then

ξ:0,1−→ PU, t −→Ht 2.5

is a random set andξis a cloud ofH. We will call ξdefined above as the cut-cloud ofHsee 10.

3. Falling d-Subalgebras/Ideals

In what follows letXdenote ad-algebra unless otherwise specified.

A nonempty subsetSofXis called ad-subalgebra ofXsee8ifx∗y∈Swhenever x∈Sandy∈S.

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A subsetI ofX is called aBCK-ideal ofX see8if it satisfies conditionsb1and b2.

A subsetIofXis called ad-ideal ofX see8if it satisfies conditionsb2andb3 for allx, y∈Xx∈I ⇒ x∗y∈I.

Definition 3.1. LetΩ,A, Pbe a probability space, and let

ξ:Ω−→ PX 3.1

be a random set. Ifξωis ad-subalgebraresp.,BCK-ideal andd-idealofXfor anyω∈Ω withξω/∅, then the falling shadowHof the random setξ, that is,

Hx Pω|x∈ξω 3.2

is called a fallingd-subalgebraresp., fallingBCK-ideal and fallingd-idealofX.

Example 3.2. LetΩ,A, Pbe a probability space and let

FX:

f |f :Ω−→X is a mapping

. 3.3

Define an operationonFXby

∀ω∈Ω fg

ω fω∗gω

3.4

for allf, g ∈FX. Letθ ∈FXbe defined byθω 0 for allω∈Ω. It is routine to check thatFX;, θis ad-algebra. For anyd-subalgebraresp.,BCK-ideal andd-idealAofX andf∈FX, let

A_f :

ω∈Ω|fω∈A , ξ:Ω−→ PFX, ω −→

f∈FX|fω∈A

. 3.5

ThenA_f ∈ Aandξω {f ∈ FX | fω ∈ A}is ad-subalgebraresp.,BCK-ideal and d-idealofFX. Since

ξ⁻¹f˙

ω∈Ω|f∈ξω

ω∈Ω|fω∈A

A_f ∈ A, 3.6

ξis a random set ofFX. Hence the falling shadowHf Pω |fω∈ AonFXis a fallingd-subalgebraresp., fallingBCK-ideal and fallingd-idealofFX.

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Example 3.3. LetX:{0, a, b, c}be ad-algebra which is not aBCK-algebra with the following Cayley table:

a

a a

a

b

b b

b

c

c c c 0

0 0 0 0

0 0 0 0 0 0

∗

3.7

LetΩ,A, P 0,1,A, mand define a random setξ:0,1 → PXas follows:

ξt:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∅, ift∈0,0.2, {0, a, c}, ift∈0.2,0.6, X, ift∈0.6,1.

3.8

Then the falling shadowHofξis a fallingd-subalgebra ofX.

Example 3.4. LetX : {0, a, b, c}be ad-algebra which is not aBCK-algebra with the Cayley table as follows:

a

c

a b

a

b

b c

b

c

c c c 0

0 0 0 0

0 0 0 0 0 0

∗

3.9

ξt:

⎧⎨

⎩

{0, a, b}, ift∈0,0.9,

X, ift∈0.9,1. 3.10

Then the falling shadowHofξis a fallingBCK-ideal ofX.

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Example 3.5. LetX:{0, a, b, c, d}be ad-algebra which is not aBCK-algebra with the Cayley table as follows:

a

b

a a a

a

b

b b

b

c

c c c

a a

d

c c

c c 0

0 0

0 0 0 0 0

∗

3.11

ξt:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

{0, a}, if t∈0,0.3, X, if t∈0.3,0.8,

∅, if t∈0.8,1.

3.12

Then the falling shadowHofξis a fallingd-ideal ofX.

Note that the falling shadowHofξinExample 3.4is not a fallingd-subalgebra ofX because if we taket∈0,0.9, thenξt {0, a, b}is not ad-subalgebra ofX. This shows that, in ad-algebra, a fallingBCK-ideal need not be a fallingd-subalgebra.

The following example shows that a fallingd-subalgebra need not be a fallingBCK- ideal ind-algebras.

Example 3.6. Consider the d-algebra X which is given in Example 3.4. Let Ω,A, P 0,1,A, mand define a random set

ξ:0,1−→ PX, t −→

⎧⎨

⎩

{0, c}, ift∈0,0.4,

X, ift∈0.4,1. 3.13

Then the falling shadowHofξis a fallingd-subalgebra ofX, but it is not a fallingBCK-ideal ofXsinceξt {0, c}is not aBCK-ideal ofXfort∈0,0.4.

Theorem 3.7. Every fallingd-ideal is a fallingd-subalgebra.

Proof. It is clear, and we omit the proof.

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The following example shows that the converse ofTheorem 3.7is not true.

Example 3.8. LetX : {0, a, b, c}be ad-algebra which is not aBCK-algebra with the Cayley table as follows:

a

c

a b

a

b

b b

b

c

c c c 0

0 0 0 0

0 0 0 0 0 0

∗

3.14

LetΩ,A, P 0,1,A, mand define a random set

ξ:0,1−→ PX, t −→

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∅, if t∈0,0.2, {0, a}, if t∈0.2,0.5, X, if t∈0.5,1.

3.15

Then the falling shadowHofξis a fallingd-subalgebra ofX, but not a fallingd-ideal ofX, sinceξt {0, a}is not ad-ideal ofXfort∈0.2,0.5.

LetΩ,A, Pbe a probability space andHa falling shadow of a random setξ:Ω → PX. For anyx∈X, let

Ωx;ξ:{ω∈Ω|x∈ξω}. 3.16

ThenΩx;ξ∈ A.

Lemma 3.9. IfHis a fallingd-subalgebra ofX, then

∀x∈X Ωx;ξ⊆Ω0;ξ. 3.17

Proof. If Ωx;ξ ∅, then it is clear. Assume that Ωx;ξ/∅ and letω ∈ Ω be such that ω∈Ωx;ξ. Thenx∈ξω, and so 0x∗x∈ξωsinceξωis ad-subalgebra ofX. Hence ω∈Ω0;ξ, and thereforeΩx;ξ⊆Ω0;ξfor allx∈X.

CombiningTheorem 3.7andLemma 3.9, we have the following corollary.

Corollary 3.10. IfHis a fallingd-ideal ofX, then3.17is valid.

We provide a characterization of a fallingd-ideal.

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Theorem 3.11. LetHbe a falling shadow of a random setξonX. ThenHis a fallingd-ideal ofXif and only if the following conditions are valid:

a for allx, y∈X Ωx∗y;ξ∩Ωy;ξ⊆Ωx;ξ, b for allx, y∈X Ωx;ξ⊆Ωx∗y;ξ.

Proof. Assume thatHis a fallingd-ideal ofX. For anyx, y∈X, if ω∈Ω

x∗y;ξ

∩Ω y;ξ

, 3.18

thenx∗y∈ξωandy∈ξω. Sinceξωis ad-ideal ofX, it follows fromb2thatx∈ξω so thatω ∈Ωx;ξ. HenceΩx∗y;ξ∩Ωy;ξ ⊆Ωx;ξfor allx, y ∈X. Now letx, y ∈ X and ω ∈ Ω be such that ω ∈ Ωx;ξ. Then x ∈ ξω and so x∗y ∈ ξωbyb3. Thus ω∈Ωx∗y;ξ, and thereforeΩx;ξ⊆Ωx∗y;ξfor allx, y∈X.

Conversely, suppose that two conditionsaandbare valid. Letx, y∈Xandω∈Ω be such thatx∗y∈ξωandy∈ξω. Thenω∈Ωx∗y;ξandω∈Ωy;ξ. It follows from athatω ∈Ωx∗y;ξ∩Ωy;ξ⊆ Ωx;ξso thatx∈ξω. Now, assume thatx∈ξωfor everyx∈X andω ∈ Ω. Thenω ∈ Ωx;ξ⊆ Ωx∗y;ξfor ally ∈X, and sox∗y ∈ ξω.

Thereforeξωis ad-ideal ofXfor allω∈Ω. HenceHis a fallingd-ideal ofX.

Proposition 3.12. For a falling shadowHof a random setξonX, ifHis a fallingBCK-ideal ofX, then

a for allx, y∈X x∗y0⇒Ωy;ξ⊆Ωx;ξ, b for allx, y∈X Ωx∗y;ξ∩Ωy;ξ⊆Ωx;ξ, c for allx∈X Ωx;ξ⊆Ω0;ξ.

Proof. aLetx, y ∈ X andω ∈ Ωbe such thatx∗y 0 andω ∈ Ωy;ξ. Then y ∈ ξω andx∗y 0 ∈ ξωbyb1. It follows fromb2thatx ∈ξωso thatω ∈ Ωx;ξ. Hence Ωy;ξ⊆Ωx;ξfor allx, y∈Xwithx∗y0.

bLetx, y ∈Xandω ∈Ωbe such thatω ∈Ωx∗y;ξ∩Ωy;ξ. Thenx∗y∈ ξω and y ∈ ξω. Sinceξω is aBCK-ideal ofX, it follows from b2that x ∈ ξω so that ω∈Ωx;ξ. HenceΩx∗y;ξ∩Ωy;ξ⊆Ωx;ξfor allx, y∈X.

cIt follows fromiianda.

We give conditions for a falling shadow to be a fallingBCK-ideal.

Theorem 3.13. For a falling shadowHof a random setξonX, assume that the following conditions are satisfied:

a Ω Ω0;ξ,

b for allx, y∈X Ωx∗y;ξ∩Ωy;ξ⊆Ωx;ξ.

ThenHis a fallingBCK-ideal ofX.

Proof. Using a, we have 0 ∈ ξω for allω ∈ Ω. Letx, y ∈ X and ω ∈ Ωbe such that x∗y ∈ξωandy ∈ξω. Thenω ∈Ωx∗y;ξ∩Ωy;ξ⊆Ωx;ξbyb, and sox∈ξω.

Thereforeξωis aBCK-ideal ofXfor allω∈Ω. HenceHis a fallingBCK-ideal ofX.

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Proposition 3.14. IfHis a fallingd-ideal ofX, then

∀x, y∈X y∗x0⇒Ωx;ξ⊆Ω y;ξ

. 3.19

Proof. Letx, y∈Xbe such thaty∗x0. Letω ∈Ωx;ξ. Thenx∈ξωandω∈Ω0;ξby Corollary 3.10. Hencey∗x0 ∈ξω. Sinceξωis ad-ideal ofX, it follows fromb2that y∈ξω. Therefore3.19holds.

Ad-ideal I of X is called a d-ideal ofX see8if, for arbitrary x, y, z ∈ X,b4 x∗z∈Iwheneverx∗y∈Iandy∗z∈I.

Definition 3.15. LetΩ,A, Pbe a probability space, and let

ξ:Ω−→ PX 3.20

be a random set. Ifξωis ad-ideal ofXfor anyω∈Ωwithξω/∅, then the falling shadow Hof the random setξis called a fallingd-ideal ofX.

Example 3.16. LetXbe ad-algebra as inExample 3.8. LetΩ,A, P 0,1,A, mand define a random set

ξ:Ω−→ PX, ω −→

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

{0, a, b}, ifω∈0,0.3, X, ifω∈0.3,0.8,

∅, ifω∈0.8,1.

3.21

Then the falling shadowHofξis a fallingd-ideal ofX, and it is represented as follows:

Hx

⎧⎨

⎩

0.8, ifx∈ {0, a, b},

0.5, ifxc. 3.22

Theorem 3.17. Every fallingd-ideal is a fallingd-ideal.

Proof. Straightforward.

We provide an example to show that the converse ofTheorem 3.17is not true.

Example 3.18. Consider the falling d-ideal Hof X which is given in Example 3.5. For t ∈ 0,0.3, ξt {0, a} is not a d-ideal of X since b∗d 0 ∈ ξt,d∗c a ∈ ξt, but b∗cc /∈ξt. HenceHis not a fallingd-ideal ofX.

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In the above discussion, we can see the following relations:

Falling d-ideal

Fallingd-ideal

Falling d-subalgebra Falling BCK-ideal

3.23

In this diagram, the reverse implications are not true, and we need additional conditions for considering the reverse implications.

Ad-algebraXis called ad^∗-algebrasee8if it satisfies the identityx∗y∗x0 for allx, y∈X.

Theorem 3.19. In ad^∗-algebra, every fallingBCK-ideal is a fallingd-ideal.

Proof. LetHbe a fallingBCK-ideal of ad^∗-algebraX. ThenΩx∗y;ξ∩Ωy;ξ⊆Ωx;ξfor allx, y ∈ X byProposition 3.12. Letx, y ∈X andω ∈ Ωx;ξ. Thenx ∈ ξω. SinceX is a d^∗-algebra, we havex∗y∗x0∈ξωand sox∗y∈ξωbyb2. Henceω∈Ωx∗y;ξ, which shows thatΩx;ξ⊆Ωx∗y;ξfor allx, y∈X. UsingTheorem 3.11, we conclude that His a fallingd-ideal ofX.

Corollary 3.20. In ad^∗-algebra, every fallingBCK-ideal is a fallingd-subalgebra.

Proof. It follows from Theorems3.7and3.19.

The following example shows that, in ad^∗-algebra, any fallingd-subalgebra is neither a fallingBCK-ideal nor a fallingd-ideal.

Example 3.21. Let X : {0, a, b, c} be a d^∗-algebra which is not a BCK-algebra with the following Cayley table:

a

a a

a

b

b b

b

c

c c c 0

0 0 0 0 0

0 0 0 0 0 0

∗

3.24

ξt:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∅, ift∈0,0.3, {0, a, c}, ift∈0.3,0.7, X, ift∈0.7,1.

3.25

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Then the falling shadowHofξis a fallingd-subalgebra ofX, but it is neither fallingBCK- ideal nor a fallingd-ideal ofXsinceξt {0, a, c}is neither aBCK-ideal nor ad-ideal ofX fort∈0.3,0.7.

Hence, in ad^∗-algebra, we have the following relations among fallingd-ideals, falling d-subalgebras, and fallingBCK-ideals:

Fallingd-ideal

Fallingd-subalgebra Falling BCK-ideal

3.26

We now establish a characterization of a fallingd-ideal.

Theorem 3.22. For a falling shadowHof a random setξonX, the followings are equivalent.

aHis a fallingd-ideal ofX.

bHis a fallingd-ideal ofXthat satisfies the following inclusion:

∀x, y, z∈X Ω

x∗y;ξ

∩Ω

y∗z;ξ

⊆Ωx∗z;ξ

. 3.27

Proof. Assume thatHis a fallingd-ideal ofX. ThenHis a fallingd-ideal ofX. Letx, y, z∈X andω∈Ωbe such thatω∈Ωx∗y;ξ∩Ωy∗z;ξ. Thenx∗y∈ξωandy∗z∈ξω, and sox∗z∈ξωsinceξωis ad-ideal ofX. Henceω∈Ωx∗z;ξ, and thereforeΩx∗y;ξ∩ Ωy∗z;ξ⊆Ωx∗z;ξfor allx, y, z∈X.

Conversely, letHbe a fallingd-ideal ofXsatisfying the condition3.27. Thenξωis ad-ideal ofX. Letx, y, z∈ X andω ∈Ωbe such thatx∗y ∈ξωandy∗z ∈ξω. Then ω∈Ωx∗y;ξ∩Ωy∗z;ξ⊆Ωx∗z;ξby3.27, and thusx∗z∈ξω. HenceHis a falling d-ideal ofX.

We now discuss relations between a fallingd-subalgebra and a fuzzyd-subalgebra. As a result, we can make a statement that the notion of a fallingd-subalgebra is a generalization of the notion of a fuzzyd-subalgebra.

A fuzzy set μ on X is called a fuzzy d-subalgebra of X see 7 if μx ∗ y ≥ min{μx, μy}for allx, y∈X.

Lemma 3.23 see7. A fuzzy setμ ofX is a fuzzyd-subalgebra ofX if and only if, for every λ∈0,1, μλ:{x∈X|μx≥λ}is ad-subalgebra ofXwhen it is nonempty.

Theorem 3.24. If one takes the probability space Ω,A, P 0,1,A, m, whereA is a Borel field on0,1andmis the usual Lebesgue measure, then every fuzzyd-subalgebra ofX is a falling d-subalgebra ofX.

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Proof. Letμbe a fuzzyd-subalgebra ofX. Thenμλis ad-subalgebra ofXfor allλ∈0,1by Lemma 3.23. Let

ξ:0,1−→ PX 3.28

be a random set andξλ μ_λfor everyλ∈0,1. Thenμis a fallingd-subalgebra ofX.

We provide an example to show that the converse ofTheorem 3.24is not true.

Example 3.25. LetXbe ad-algebra as inExample 3.4. LetΩ,A, P 0,1,A, mand define a random set

ξ:Ω−→ PX, ω −→

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

{0, c}, if t∈0,0.2,

∅, if t∈0.2,0.3, {0, b}, if t∈0.3,0.6, {0, a}, if t∈0.6,0.85, X, if t∈0.85,1.

3.29

Then the falling shadowHofξis a fallingd-subalgebra ofX, and it is represented as follows:

Hx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0.9, ifx0, 0.4, ifxa, 0.45, ifxb, 0.35, ifxc.

3.30

We know thatHis not a fuzzyd-subalgebra ofXsince

Hb ∗a Hc 0.35/≥0.4min

Hb, Ha

. 3.31

Theorem 3.26. Every fallingd-subalgebra ofX is aTm-fuzzyd-subalgebra ofX; that is, ifHis a fallingd-subalgebra ofX, then

∀x, y∈X H x∗y

≥T_m

Hx, H

y

, 3.32

whereT_ms, t max{st−1,0}for anys, t∈0,1.

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Proof. ByDefinition 3.1,ξωis ad-subalgebra ofXfor anyω∈Ωwithξω/∅. Hence {ω∈Ω|x∈ξω} ∩

ω∈Ω|y∈ξω

⊆

ω∈Ω|x∗y∈ξω

, 3.33

which implies that H

x∗y P

ω|x∗y∈ξω

≥P

{ω|x∈ξω} ∩

ω|y∈ξω

≥Pω|x∈ξω P

ω|y∈ξω

−P

ω|x∈ξωorω|y∈ξω

≥Hx H y

−1.

3.34

Hence

H x∗y

≥max

Hx H y

−1,0 Tm

Hx, H y

. 3.35

This completes the proof.

Acknowledgments

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

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