新推：型-2模糊关系与区间型-2模糊集的运算与性质探讨

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本文探讨了"类型-2模糊关系与区间值类型-2模糊集"这一主题，作者通过对模糊真值代数中的新型运算——扩展上确界（extended supremum）和扩展下确界（extended infimum）的引入，对类型-2模糊集之间的这些扩展操作进行了深入研究。这两种运算作为类型-2模糊集合中最大值和最小值操作的推广，为理解和处理类型-2模糊系统中的复杂决策提供了理论基础。在讨论了类型-2模糊关系的基本性质后，文章重点分析了它们的复合性，这是构建多级模糊逻辑系统的关键环节。通过复合，模糊关系能够表示更为复杂的依赖关系和不确定性，这对于处理模糊推理、机器学习和控制问题具有重要意义。作者提出并定义了区间值类型-2模糊集（interval-valued type-2 fuzzy sets），这是一种将区间概念融入类型-2模糊集的新形式，它不仅保留了类型-2模糊集的模糊特性，还引入了额外的度量和量化层次，使得决策过程更具稳健性和精确度。区间值类型-2模糊关系（interval-valued type-2 fuzzy relations）在此基础上进一步发展，它们可以捕捉和表达更加精确和灵活的关联模式。本文探讨了这些新的结构在处理数据不确定性、模糊决策和系统建模中的应用，如模糊决策支持系统、模糊控制系统以及模糊优化问题。通过这些新颖的操作和概念，本文的研究有助于拓展模糊理论的应用范围，并为设计更高效、准确的模糊算法提供理论支持。这篇文章对于理解模糊逻辑的高级形式，提升模糊系统的性能和精确度具有重要的学术价值。感兴趣的读者可参考ScienceDirect和Elsevier的《Fuzzy Sets and Systems》期刊，发表日期为2014年，获取全文细节。

6 B.Q. Hu, C.Y. Wang / Fuzzy Sets and Systems 236 (2014) 1–32

In Proposition 2.6, Mizumoto and Tanaka [37,38] added conditions that J is a ﬁnite subset of I and A

is a closed

interval on I . Proposition 2.5 tells us that these conditions are unnecessary. Proposition 2.6 appeared in [46], where J

is bounded.

3. Extended supremum and extended inﬁmum

Only operations combining a ﬁnite number of fuzzy truth values are considered in Deﬁnition 2.3. In this section,

we propose two operations combining an inﬁnite number of fuzzy truth values.

Deﬁnition 3.1. Let A

∈ Map(J, I ) for all i ∈ Λ. Then we deﬁne



sup

i∈Λ



(x) = sup

x=sup

i∈Λ



inf

i∈Λ

)







inf

i∈Λ



(x) = sup

x=inf

i∈Λ



inf

i∈Λ

)



It is clear that for crisp sets A

⊆ J for all i ∈ Λ,

sup

i∈Λ



x ∈ J : x = sup

i∈Λ

∈ A

for all i ∈ Λ





inf

i∈Λ



x ∈ J : x = inf

i∈Λ

∈ A

for all i ∈ Λ



. (8)

Let A

∈ Map(J, I ) for all i ∈ Λ and Λ be ﬁnite. We have

sup

i∈Λ





i∈Λ

and



inf

i∈Λ





i∈Λ

. (9)

Notice that sup and



inf need not satisfy idempotency property, while Λ is inﬁnite. In other words, sup

i∈Λ

A = A and



inf

i∈Λ

A =A need not hold for arbitrary inﬁnite index set Λ.

Example 3.1. Consider J = I and A

= (0, 1) for all i ∈ N. Then by Eq. (8), the following hold

sup

i∈N

= (0, 1]=(0, 1) and



inf

i∈N

=[0, 1) =(0, 1).

In the sequel, some basic properties of the extended supremum and extended inﬁmum are presented.

Lemma 3.1. Let A

∈ Map(J, I ) for all i ∈ Λ. Then the following hold

(1) ( sup

i∈Λ

)





inf

i∈Λ

)



(2) sup

i∈Λ

⊆ sup

i∈Λ



inf

i∈Λ

⊆



inf

i∈Λ

,ifA

⊆ B

for all i ∈ Λ.

(3) sup

i∈Λ

∪B

) ⊇( sup

i∈Λ

) ∪ ( sup

i∈Λ



inf

i∈Λ

∪B

) ⊇(



inf

i∈Λ

) ∪ (



inf

i∈Λ

(4) sup

i∈Λ

∩B

) ⊆( sup

i∈Λ

) ∩ ( sup

i∈Λ



inf

i∈Λ

∩B

) ⊆(



inf

i∈Λ

) ∩ (



inf

i∈Λ

Proof. It

follows immediately from Deﬁnition 3.1. 2

Proposition 3.1. Let A

∈ Map(J, I ) for all i ∈Λ. Then the following hold

(1) ( sup

i∈Λ

)

⊆ sup

i∈Λ

)

, (



inf

i∈Λ

)

⊆



inf

i∈Λ

)

( sup

i∈Λ

)

⊆ sup

i∈Λ

)

, (



inf

i∈Λ

)

⊆



inf

i∈Λ

)

(2) ( sup

i∈Λ

)

⊇ sup

i∈Λ

)

, (



inf

i∈Λ

)

⊇



inf

i∈Λ

)

(3) ( sup

i∈Λ

)

⊆ sup

i∈Λ

)

, (



inf

i∈Λ

)

⊆



inf

i∈Λ

)

If Λ is ﬁnite, then identities in items (1) and (3) hold.

B.Q. Hu, C.Y. Wang / Fuzzy Sets and Systems 236 (2014) 1–32 7

Proof. (1) For all x ∈ J , we get



sup

i∈Λ



(x) = sup

yx



sup

i∈Λ



(y) = sup

yx

sup

y=sup

i∈Λ

inf

i∈Λ

) = sup

xsup

i∈Λ

inf

i∈Λ



sup

i∈Λ

)



(x) = sup

x=sup

i∈Λ

inf

i∈Λ

)

) = sup

x=sup

i∈Λ

inf

i∈Λ

sup

x

)

 sup

xsup

i∈Λ

inf

i∈Λ

) =



sup

i∈Λ



(x).

The others are similarly proven.

If Λ is

ﬁnite, it follows immediately from Proposition 6 in [46].

(2) Let α ∈ I .I

f sup

i∈Λ

)

=∅, then we have ( sup

i∈Λ

)

⊇ sup

i∈Λ

)

. Otherwise, let x ∈ sup

i∈Λ

)

Then there exists an x

∈ (A

)

for each i ∈ Λ such that x = sup

i∈Λ

and ( sup

i∈Λ

)(x)  inf

i∈Λ

)  α. Thus

( sup

i∈Λ

)

⊇ sup

i∈Λ

)

. Similarly, we have (



inf

i∈Λ

)

⊇



inf

i∈Λ

)

(3) Let α ∈[0, 1).If( sup

i∈Λ

)

=∅, then we get ( sup

i∈Λ

)

⊆ sup

i∈Λ

)

. Otherwise, let x ∈ ( sup

i∈Λ

)

Then there exists an x

∈ J for each i ∈ Λ such that x = sup

i∈Λ

and inf

i∈Λ

)>α. Thus ( sup

i∈Λ

)

⊆

sup

i∈Λ

)

. Similarly, we obtain (



inf

i∈Λ

)

⊆



inf

i∈Λ

)

If Λ is ﬁnite, it follows immediately from Proposition 2.1(7). 2

When α = 0o

rα = 1, we have the following corollary.

Corollary 3.1. Let A

∈ Map(J, I ) for all i ∈Λ. Then the following hold

(1) Ker( sup

i∈Λ

) ⊇ sup

i∈Λ

Ker A

, Ker(



inf

i∈Λ

) ⊇



inf

i∈Λ

Ker A

(2) Supp( sup

i∈Λ

) ⊆ sup

i∈Λ

Supp A

, Supp(



inf

i∈Λ

) ⊆



inf

i∈Λ

Supp A

If Λ is

ﬁnite, then identities in item (2) hold.

A mapping H : I → 2

is called a nest of sets on X, if it satisﬁes that α

 α

implies H(α

) ⊇ H(α

) for all

,α

∈ I , where 2

denotes the powerset of X. The set of all nests of sets on X is denoted as N (X).Itiswell

known that if H ∈ N (X), A ∈Map(X, I ) and A

⊆ H(α)⊆ A

for all α ∈ I , then A =



α∈I

αH(α).

Deﬁnition 3.1 can be equivalently represented as follows.

Corollary 3.2. Let A

∈ Map(J, I ), H

∈ N (J ) and (A

)

⊆ H

(α) ⊆ (A

)

for all i ∈ Λ and α ∈ I . Then

sup

i∈Λ



α∈I

αH



(α) and



inf

i∈Λ



α∈I

αH



(α) with H



(α) = sup

i∈Λ

(α) and H



(α) =



inf

i∈Λ

(α).

Proof. It follows immediately from Proposition 3.1(2) and (3). 2

By the previous corollary, we obtain that extended supremum and extended inﬁmum on the algebra of fuzzy

numbers

had been proposed and investigated in [5] with nests of closed intervals, where J is the unit interval.

The distributive laws between t-norm extended operations and extended supremum (resp. extended inﬁmum) are

vestigated as follows.

Proposition 3.2. Let A,

∈ Map(I, I ) for all i ∈ Λ and  be a continuous t-norm. Then the following hold









sup

i∈Λ



⊆ sup

i∈Λ







), A











inf

i∈Λ



⊆



inf

i∈Λ







);



⊥





sup

i∈Λ



⊆ sup

i∈Λ



⊥



), A



⊥







inf

i∈Λ



⊆



inf

i∈Λ



⊥



Proof. By Deﬁnition 3.1 and Lemma 2.1,wehave

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