Study of Fuzzy Multisets and Their Algebra

Study of Fuzzy Multisets and Their Algebra

Chapter One

 Aim and Objectives

The aim of this Dissertation is to study Fuzzy Sets, Multisets, Fuzzy Multisets and develop some algebraic structures.

The objectives are as follows:

To comprehensively study the origin, structure and development of fuzzy set theory.

To critically study the fundamentals of fuzzy sets, multisets, and fuzzy multi- sets.

To extend the properties of α– cuts for fuzzy sets to fuzzy multisets.

To strengthen properties of inverse α-cuts for fuzzy sets and extend them to fuzzy multisets.

To formulate certain monoids of multiset partitions and that of fuzzy partitions.

To study fuzzy groups, multigroups, and fuzzy multigroups for further extensions.

CHAPTER TWO

LITERATURE REVIEW

The queerness of the concept of vagueness has long been drawing the attention of philosophers,linguists, logicians, and mathematicians. As noted in (Magnus, 1997), Nietsche was the first to recognize the notion of vagueness. In course of time, various other closely related notions such as loose concepts, haziness, borderline cases, flu- ent boundaries, case by grades, etc., appeared. As mentioned earlier, Gottlob Frege (1848–1925) was the first to provide a mathematical definition of vagueness in terms of having an unsharp boundary. A seminal contribution towards investigating the concept of vagueness was made by (Black, 1937). The epicentre of Black’s explica- tion can be seen as a unifying thread between Bertrand Russell’s and C. S. Peirce’s approach.

Menger (1979) argued that the notion of probability could adequately deal with loose concepts. He also introduced the notion of hazy set. However, it was not explicit until the formulation of the theory of fuzzy sets (Zadeh, 1965) that the notion of probability could not deal with vagueness and other loose concepts if the meaning of these concepts is the absence of sharp boundaries. The concept of fuzziness essen- tially refers to the semantic feature of the vagueness of a phenomenon rather than its stochastic explication which is devoid of it.

A distinctive feature of the concept of fuzziness can be seen summarized in the following: In contrast to the stochastic uncertainty-type vagueness, the vagueness concerning the description of the semantic meaning of events, phenomena or state- ments is called fuzziness [(Moreno-Armella and Waldegg, 1993). Kaushal et al. (2010) and Seising (2005) provided a good deal of illustrations to describe the rele- vancy of fuzzy concept in mathematics.]

In view of the pervasive role played by set-theoretic foundation, it was seemingly natural to look for a set theory-like framework to model the class of problems in which the source of vagueness is not the presence of random variables rather the absence of precisely defined criteria of class membership. Fortunately, it was found

forthcoming by way of relaxing the restriction of definiteness imposed on objects

to form a Cantorian set. L. A. Zadeh was the first who formulated a set-theoretic model in (Zadeh, 1965) and titled it fuzzy set theory in contrast to crisp set theory. Fuzzy set theory is a mathematical theory to model vagueness and other loose con- cepts. It deals with fuzzy variables and fuzzy relations.

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CHAPTER THREE

FUNDAMENTALS OF FUZZY SETS, MULTISETS AND FUZZY MULTISETS

In this chapter, we explicate basic notions of fuzzy sets, multisets and fuzzy mul- tisets. In particular, we identify some fundamental notions of the aforesaid non- classical set theories which could be further clarified and extended.

Definition and Representations of Fuzzy Sets

  Definition of a fuzzy set

Definition 3.1.1 A fuzzy set(class) A˜ in X is characterized by a membership func- tion µ_A which associates with each point x in X, a real number µA˜  x   in the interval [0, 1]. The value of µA˜(x) represents the grade of membership of x in A˜.

Intuitively,  it  can  be  said  that,the  closer  the  value  of  µA˜(x) to  unity,  the  higher valued)  characteristic  function  of  an  ordinary  set,   muA˜  can  be  called  generalized

characteristic function first introduced in (Whitney, 1933). In general, a fuzzy set in a universal set can be obtained by applying a fuzzy restriction to x. If the mem bership function of A˜ is limited to take values in the set    0, 1 , it becomes a crisp set and, in this sense, the concept of fuzzy sets is a generalization of ordinary sets.

Remark 3.1.1 It is well-known that ordinary sets exist only as subsets of a given universal set X by applying a crisp restriction to X.

CHAPTER FOUR

α-CUTS, INVERSE α-CUTS AND RELATED RESULTS

The notions of α-cuts and strong α-cuts, apart from their multitudinal applications, can be viewed as a bridge both between fuzzy sets and crisp sets, and fuzzy multisets and crisp multisets. In particular, α-cuts will be used in this direction to generalize some fundamental results of fuzzy set theory to fuzzy multiset theory.

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATIONS

  Summary

Besides chapter one, a comprehensive and critical Literature survey was presented in chapter two. In Chapter three, fundamentals of fuzzy sets, multisets, fuzzy mul- tisets were presented from a specific perspective to provide means to model a large class of real-life problems which intrinsically involve vagueness and uncertainty. In chapter four, alpha-cuts, inverse alpha-cuts in fuzzy sets and fuzzy multisets were presented, and some of their properties were extended from fuzzy sets to fuzzy multi- sets, in particular the two decomposition theorems were extended to fuzzy multiset. Moreover, inverse α-cuts were introduced in fuzzy multisets and their properties studied. In chapter five, algebra of multisets and fuzzy sets were studied viz., fuzzy groups, monoids of fuzzy subsets, multigroups and fuzzy multigroups. Finally, cer- tain monoids of multisets partitions, fuzzy partitions were constructed and abelian fuzzy multigroup introduced.

Conclusion

The following results are the main contributions of this dissertation

Extension of properties of α-cuts for fuzzy sets to fuzzy multisets

Formulation of decomposition theorems for fuzzymultisets

Let A ∈ FM (X) and † be the standard fuzzy multiset union.

First Decomposition Theorem states that

A

α∈[0,1]

_α[A].

Second Decomposition Theorem states that

[Proofs on page 82 ]

A

α∈[0,1]

_α]A[.

Extension of Properties of inverse α-cuts for fuzzy sets to fuzzy multisets

Proposition 4.2.1

Let A˜, B˜ ∈ F (X) and α, β ∈ [0, 1].  The following properties hold: i.  αA˜−1 ⊆  α⁻ A˜−1

1. α≤ β  implies αA˜−1 ⊆  β A˜−1  and α− A˜−1 ⊆ β− A˜−1

• α A˜ B˜  −1 αA˜−1 αB˜−1,  α   A˜ B˜  −1 αA˜−1 αB˜−1 , and

αA˜−1 ∩α B˜−1 ⊆ α(A˜ ∩ B˜)−1

1. iv.α⁻   A˜   B˜  −1 α⁻ A˜−1 α⁻ B˜−1, α⁻   A˜   B˜  −1 α⁻ A˜−1 α⁻ B˜−1 , and α⁻ A˜−1   α⁻

B˜−1 ⊆ α⁻ (A˜ ∩ B˜)−1

1. v.α((A˜−1)′) =  (1−α)− (A˜−1)′
2. vi.1A˜−1 = X

A˜ ⊆ B˜ iff αB˜−1 ⊆  αA˜−1; A˜ ⊆ B˜ iff α⁻ B˜−1 ⊆  α⁻ A˜−1

A˜ = B˜ iff αB˜−1 = αA˜−1; A˜ = B˜ iff α⁻ B˜−1 =  α⁻ A˜−1

[Proofs on page 84]

Proposition 4.2.2 Let A, B, C ∈ FM (X). The following properties hold for

α, β ∈ (0, 1]:

1. α[A]−1 ∪ α[B]−1 = α[B]−1 ∪ α[A]−1 ; ^α[A]−1 ∩ [ B]−1 = α[B]−1 ∩ α[A]−1
2. α ]A[−1 ∪ α ]B[−1 = α ]B[−1 ∪ α ]A[−1 ; ^α]A[−1 ∩ α ]B[−1 = α ]B[−1 ∩ α ]A[−1

iii. α[A]−1 ∪ (α[B]−1 ∪ α[ C]−1) = (α[A]−1 ∪ α[ B]−1) ∪ α[C]−1;

α[A]−1 ∩ (α[B]−1 ∩ α[ C]−1) = (α[A]−1 ∩ α[ B]−1) ∩ α[C]−1

1. α ]A[−1 ∪ (α ]B[−1 ∪ α ]C[−1) = (α ]A[−1 ∪ α ]B[−1) ∪ α ]C[−1 ;

α ]A[−1 ∩ (α ]B[−1 ∩ α ]C[−1) = (α ]A[−1 ∩ α ]B[−1) ∩ α ]C[−1

1. α [A]−1 ⊆ α ]A[−1
2. α ≤ β implies α [A]−1 ⊆ β [A]−1‘ and α ]A[−1 ⊆ β ]A[−1

[proofs on page 88]

Formulation of Monoids of partitions of a multiset and that of S-H fuzzy partitions of a set.

Monoids of partitions ofmultisets

Let A be a cardinality-bounded nonempty mset and A be the collection of all partitions of A . Then operations ∗ and ⊕ were introduced such tha

• (∏(A), ∗) or (∏(A, ∗, {A}) is a commutative idempotent monoid,and
• A, is a commutative idempotent monoid. [Proofs on page 110]

Monoids of S-H fuzzy Partitions of a set

Let T˜

be  a  S-H  fuzzy  partition  of  a  set  X  and,  let  A˜i   (i = 1, n) denote  the blocks  of  the  partition  T˜.   Let X denote  the  collection  of  all  S-H  fuzzy partitions of X. Then operations ∗, ○, and ⊕ were introduced such that

• (∏(X), ∗) is a commutative idempotentmonoid,
• (∏(X),○) is a commutative idempotent monoid, and
• X , is a commutative idempotent monoid. [Proofs on page 105]

Properties of Abelian Fuzzy Multigroups introduced

Proposition 5.3.26 Let A FM  X . Then x, y X the following assertions are equivalent:

• CM_A(xy) =CM_A(yx),
• CM_A(xyx⁻¹) =CM_A(y),
• CM_A(xyx⁻¹)  ≥CM_A(y),
• CM_A(xyx⁻¹)  ≤CM_A(y),

Proposition 5.3.27Let A FM X .

Then the following assertions are equivalent:

• CM_A(xy)= CM_A(yx), ∀x, y ∈ X,
• A○ B = B ○ A, ∀B ∈ FM (X).

Proposition 5.3.28

Let A AFMG  X  . Then A^∗, and A  α, n , n N are normal subgroups of X.

[Proofs on page 116]

Recommendations

The construction provided in subsection 4.2.2 may be found useful in addressing adequately some real life problems which require symmetry of abstraction. The concept of α-cuts and inverse α-cuts can be very useful in many areas like infor- mation retrieval on the web, data encription, data mining, coding theory, decision making, etc. In view of the multitudinal application of α-cuts and inverse α-cuts, it needs to be investigated whether or not, similar to the result that every fuzzy multiset can be decomposed into its α-cuts obtained in (Singh et al., 2014), holds for inverse α-cuts as well.

In view of the significance of fuzzy multiset theory it is recommended that it should be inculcated into the curricula of studies, both at undergraduate and postgraduate levels.

List of Journal publications

• Singh, D., Alkali, A., Ibrahim, A. M., (2013). An Outline of the Development of the Concept of Fuzzy Multisets, International Journal of Innovation, Management and Technology, Vol. 4, no.2, 309-315.
• Singh, D., Alkali, , Isah, A. I., (2014). Some Applications of α-Cuts in Fuzzy Multiset Theory, Journal of Emerging Trends in Computing and Information Sci- ences, Vol. 5, no. 4,328-335.
• Singh, D., Alkali, , Singh, J. N., (2014). Monoids of Partitions of aMultiset, Journal of Mathematical Sciences & Mathematics Education Vol. 9 No. 1, 9-16.
• Singh, D., Alkali, J., (2015). Monoids of S-H fuzzy Partitions of a set, In- ternational Journal of Pure and Applied Mathematics Vol. 98 No. 1,123-128.

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