Products of elements in vague semigroups and their implementations in vague arithmetic

Fuzzy Sets and Systems156 (2005) 93–123www.elsevier.com/locate/fss

Products of elements in vague semigroups and theirimplementations in vague arithmetic

Mustafa Demirci∗

Department of Mathematics, Faculty of Sciences and Arts, Akdeniz University, 07058-Antalya, Turkey

Received 16 April 2004; received in revised form 31 March 2005; accepted 1 April 2005Available online 25 April 2005

Abstract

Vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in [Demirci (Internat.J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(1) (2002) 25; Internat. J. General Systems 32(2) (2003)157, 177)] to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers.The main motivating problem of this paper is to introduce the notion of vague product (sum) of a finite number ofreal numbers in vague arithmetic, and to point out their fundamental properties. From a more abstract mathematicalpoint of view, the vague product (sum) of a finite number of real numbers in vague arithmetic and their properties canbe considered as the vague product of a finite number of elements in vague semigroups and their relevant properties.For this reason, a large part of this paper is devoted to the vague product of a finite number of elements in vaguesemigroups and their elementary properties. As a direct implementation of the present results, it is shown that thevague product (sum) of a finite number of real numbers in vague arithmetic can be easily evaluated in terms ofthe underlying many-valued equivalence relations. Furthermore, various non-trivial examples for the vague product(sum) of a finite number of real numbers in vague arithmetic are designed, and a simple technique for the constructionof such non-trivial examples is stated.© 2005 Elsevier B.V. All rights reserved.

Keywords:Semigroup; Fuzzy semigroup; Vague semigroup; Fuzzy arithmetic; Vague arithmetic; Fuzzy equivalence relation

∗ Tel.: +90 242 227 8900; fax: +90 242 227 8911.E-mail address:[email protected].

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.04.001

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94 M. Demirci / Fuzzy Sets and Systems 156 (2005) 93–123

1. Introduction

The concept of indistinguishability is an inevitable part of realistic measurements[13], and is anessential component of fuzzy sets [2,13,18]. Two-valued logic does not provide a suitable logical frame torepresent the notion of indistinguishability, mathematically, and the consideration of indistinguishabilitywithin the context of bivalent logic leads to Poincare-like paradoxes [19,24,25]. Many-valued logicalrepresentation of indistinguishability, known as many-valued equivalence relations (fuzzy equivalencerelations, indistinguishability operators, similarity relations, many-valued equalities, etc.), overcomesthese difficulties, and it has numerous applications in different fields [1–3,6,15,19,23,26,27].

The consideration of many-valued equivalence relations on the setR of all real numbers naturally sug-gests vaguely defined arithmetic operations onR, and the resulting arithmetic operations are called vaguearithmetic operations [7,9–11]. Vague arithmetic and fuzzy arithmetic in the usual sense are completelydifferent from each other in general. In fuzzy arithmetic in the usual sense [17,16,28–31], the numbers arefuzzily defined objects, and the arithmetic operations among them are in the classical sense. In contrast tofuzzy arithmetic, vague arithmetic basically assumes that the numbers are precise objects, but arithmeticoperations among real numbers are vaguely defined.

Whenever the multiplication (addition) operation onR is vaguely defined, the product (sum) of a finitenumber of real numbers is a significant issue for the practical application of vague arithmetic operations.The problem pertaining to the products (sums) of a finite number of real numbers in the context of vaguearithmetic can be abstracted to the formulation of the products of a finite number of elements in vaguesemigroups, which will be the main subject of this paper. In this paper, we will introduce the vagueproduct of a finite number of elements in vague semigroups, and establish its elementary properties.As a direct entailment of the presented results, we will formulate vague sums and products of a finitenumber of real numbers in vague arithmetic, and show how they can be used in practice. In addition tothis, it will be pointed out that non-trivial examples can be built from known examples by using a simpletechnique.

The content of this paper can also briefly be described as follows. We will present the necessarypreliminaries related to vague binary operations, vague arithmetic operations and vague semigroups inSection 2. Section 3 is devoted to the introduction of vague products of a finite number of elements invague semigroups and their subsidiary tools: exact products and successive products. In Section 4, we willestablish the representation properties of vague products. Thereafter, utilizing the representation results,we will formulate the vague sums and products of a finite number of real numbers in vague arithmetic, andshow that they can be easily calculated by means of the underlying many-valued equivalence relations.Various non-trivial examples for the vague sums (products) of a finite number of real numbers and theirpractical realizations will also be the subjects of Section 4. Furthermore, a simple technique will be givento derive non-trivial examples from existing examples.

2. Vague binary operations, vague addition (multiplication) operations and vague semigroups

In this section, we introduce the necessary background materials pertaining to vague binary operations,vague addition (multiplication) operations and vague semigroups presented in [7,9,10], which will beneeded in the sequel. The many-valued logical base of these vague algebraic notions [9,10] is describedby a fixed triple (called an integral, commutative cqm-lattice, in brief an iccqm-lattice)M = (L, �, ∗)

M. Demirci / Fuzzy Sets and Systems 156 (2005) 93–123 95

satisfying the following properties:

(i) (L, �) is a complete lattice with the meet operation∧ onL, the join operation∨ onL, the bottomelement0 of L and the top element1 of L, and0 �= 1.

(ii) (L, ∗) is a commutative monoid with the identity1.(iii) ∗ is isotone w.r.t. both first and second arguments, i.e.(∀�1, �2, �1, �2 ∈ L)[((�1��2) and

(�1��2)) ⇒ (�1 ∗ �1��2 ∗ �2)].In the particular caseL = [0,1], the monoid operation∗ is known as a t-norm. In the whole of this

paper, we always assume thatM = (L, �, ∗) is a fixed iccqm-lattice, unless otherwise mentioned. Forn ∈ N+ and�1, �2, . . . , �n ∈ L, we will use the notation∗ni=1 �i instead of�1 ∗ �2 ∗ · · · ∗ �n for the sakeof simplicity. Throughout this paper, the capital lettersX andY always denote nonempty usual sets. Afunction�:X → L is called anL-fuzzy set ofX, and the set of allL-fuzzy sets ofX is denoted byLX.The kernel (core)ker(�) of anL-fuzzy set� of X is defined as the crisp subset{x ∈ X | �(x) = 1} of X.An L-fuzzy set ofX × Y is called anL-fuzzy relation (or simply, a fuzzy relation) fromX to Y. Givena fuzzy relation◦ ∈ L(X×X)×X, for all x, y, z ∈ X, we denote the element◦((x, y), z) of L by simply◦(x, y, z).

In caseM = (L, �, ∗) is an integral, commutative cl-monoid[20] (i.e. there exists an additional binaryoperation→ on L provided that� ∗ �� ⇔ �� → � for all �, �, � ∈ L), a mapE:X × X → L (anL-fuzzy set ofX×X) is called anM-valued similarity relation [20] onX iff the following three conditionsare satisfied:(E.1) E(x, x) = 1, ∀x ∈ X,(E.2) E(x, y) = E(y, x), ∀x, y ∈ X,(E.3) E(x, y) ∗ E(y, z)�E(x, z), ∀x, y, z ∈ X.

For a general iccqm-latticeM = (L, �, ∗), anM-valued similarity relation is called anM-equivalencerelation[8].

An M-equivalence relationE onX is called anM-equality onX iff E is separated, i.e.(E.1

′) E(x, y) = 1⇒ x = y, ∀x, y ∈ X.

For the particular cases of the iccqm-latticeM = (L, �, ∗), anM-equivalence relation onX is alsocalled a globalM-valued equality [21,22], an equality relation onXw.r.t.∗ [26], anL-fuzzy equivalence onX w.r.t.∗ [27], a fuzzy equivalence relation onX w.r.t.∗ [2], a∗-equivalence [4], a∗-indistinguishabilityoperator onX [32] and a similarity relation [33]. Similarly, anM-equality onX is also known as a globalM-valued equality onX [20], a separated globalM-valued equality onX [21], a ∗-equality [1,5] and a∗-fuzzy equality onX [6–8].

For a givenM-equivalence relationE on X and forx ∈ X, the L-fuzzy set[x]E of X, defined by[x]E(y) = E(x, y), ∀y ∈ X, is called the extensional hull ofx w.r.t. E [27]. The kernel of anM-equivalence relationE on X defines an equivalence relation onX, and it will be denoted by≈E . AnM-equivalence relationE onX is said to be right (left) regular w.r.t. a crisp binary operation◦ onX iff

E(x, y)�E(x ◦ u, y ◦ u) (E(x, y)�E(u ◦ x, u ◦ y)), ∀x, u, y ∈ X.

E is regular w.r.t.◦ iff it is both right and left regular w.r.t.◦ [10].Because of the fact that vague binary operations and vague semigroups are formulated by the notions

of strong and perfect fuzzy functions [7,9], it will be useful to recall these notions studied in [6–8,14] atfirst:


Definition 2.1. Let E andF be twoM-equivalence relations onX andY, respectively.(i) A fuzzy relation� ∈ LX×Y is called a strong fuzzy function fromX toYw.r.t.E andF iff � satisfies

the next conditions:(F.1) For eachx ∈ X, ∃y ∈ Y such that�(x, y) = 1,(F.2) �(x, y) ∗ �(x′, y′) ∗ E(x, x′)�F(y, y′), ∀x, x′ ∈ X, ∀y, y′ ∈ Y .

(ii) A fuzzy relation� ∈ LX×Y is called a perfect fuzzy function fromX toYw.r.t.E andF iff � fulfillsthe condition (F.1) and the following conditions:(EX.1) �(x, y) ∗ E(x, x′)��(x′, y),(EX.2) �(x, y) ∗ F(y, y′)��(x, y′),

(PF)�(x, y) ∗ �(x, y′)�F(y, y′) for all x, x′ ∈ X and for ally, y′ ∈ Y .

An ordinary functionf :X → Y is said to be extensional w.r.t.M-equivalence relationsE onX andFonY iff the inequalityE(x, x′)�F(f (x), f (x′)) is satisfied for allx, x′ ∈ X [3,20,19,24,26]. The notionof extensionality of an ordinary function can be informally expressed as the satisfaction of the intuitiveexpectation “If two points are similar to each other, so are their images”. The extensionality of ordinaryfunctions plays a crucial role for the representations of strong and perfect fuzzy functions:

Theorem 2.2(Demirci [8] ). For a given strong (perfect) fuzzy function� from X toY w.r.t. E and F, thereexists at least one ordinary functionf :X → Y extensional w.r.t. E and F such that for allx, y, z ∈ X,

�(x, f (x)) = 1 and �(x, y)�E(f (x), y) (�(x, y) = E(f (x), y)). (2.1)

Conversely, for a given ordinary functionf :X → Y extensional w.r.t. E and F, an L-fuzzy relation� ∈ LX×Y , satisfying the conditions (the equality) in(2.1), is a strong (perfect) fuzzy function� from Xto Y w.r.t. E and F.

Given a strong fuzzy function� from X toYw.r.t.E andF, an extensional ordinary functionf :X → Y

w.r.t. E andF, fulfilling conditions (2.1) in Theorem 2.2, is called an ordinary description of�, and theset of all ordinary descriptions of� is denoted byORD(�). If the M-equivalence relationF onY is anM-equality, then� has a unique ordinary description, denoted byord(�) (see [8, Remark 4.13(ii)]). Strongand perfect fuzzy functions possess various interesting properties. In this paper, we restrict ourselves tothe present concepts. A thorough information about them can be supplied from [6–8,14].

After these preparatory results, we are now able to introduce the notions of vague binary operationsand vague semigroups:

Definition 2.3 (Demirci [9] ). Let P andE beM-equivalence relations onX ×X andX, respectively.(i) A strong fuzzy function (perfect fuzzy function)◦ from X × X to X w.r.t. P andE is said to be an

M-vague binary operation (perfectM-vague binary operation) onX w.r.t. P andE.(ii) An M-vague binary operation◦ onX w.r.t. P andE is said to be transitive of the first order iff(T.1) ◦(a, b, c) ∗ E(c, d)� ◦(a, b, d), ∀a, b, c, d ∈ X.(iii) An M-vague binary operation◦ onX w.r.t. P andE is said to be transitive of the second order iff(T.2) ◦(a, b, c) ∗ E(b, d)� ◦(a, d, c), ∀a, b, c, d ∈ X.(iv) An M-vague binary operation◦ onX w.r.t. P andE is said to be transitive of the third order iff(T.3) ◦(a, b, c) ∗ E(a, d)� ◦(d, b, c), ∀a, b, c, d ∈ X.


(v) X together with anM-vague binary operation (a perfectM-vague binary operation), denoted by(X, ◦), is called anM-vague semigroup (a perfectM-vague semigroup) w.r.t.P andE iff the followingcondition of vague associativity is satisfied:

(VAS) (∀a, b, c, d,m, q,w ∈ X)((◦(b, c, d) ∗ ◦(a, d,m) ∗ ◦(a, b, q) ∗ ◦(q, c, w))�E(m,w)).

As a direct consequence of Theorem2.2, vague binary operations can be represented by means of crispbinary operations as follows.

Corollary 2.4. For a given M-vague binary operation◦ (a perfect M-vague binary operation) on X w.r.t.P and E, there exists at least one crisp binary operation◦ on X extensional w.r.t. P and E such that forall x, y, z ∈ X,

◦(x, y, x ◦ y) = 1 and ◦(x, y, z)�E(x ◦ y, z) (◦(x, y, z) = E(x ◦ y, z)). (2.2)

Conversely, for a given crisp binary operation◦ on X extensional w.r.t. P and E, an L-fuzzy relation◦ ∈ L(X×X)×X, satisfying the conditions(the equality) in (2.2), is an M-vague binary operation◦ (aperfect M-vague binary operation) on X w.r.t. P and E.

Owing to Corollary 2.4, any perfectM-vague binary operation◦ is obviously anM-vague binaryoperation, and the converse of this statement will be true if◦ is transitive of the first order [9]. Thus thenotions ofM-vague binary operation and perfectM-vague binary operation differ from each other just bythe first-order transitivity.

If X is particularly taken as the setR of all real numbers, then anM-vague binary operation. (+) onR w.r.t. M-equivalence relationsER2 on R2 andER on R is called anM-vague multiplication (addition)operation onR w.r.t.ER2 andER iff the usual multiplication (addition) operation “.” (“ +”) is an ordinarydescription of. (+) [9–11]. An M-vague multiplication (addition) operation. (+) is said to be perfectiff . (+) is a perfectM-vague binary operation. (+). As a direct consequence of Corollary 2.4, it is easyto observe that a perfectM-vague multiplication (addition) operation. (+) on R w.r.t. ER2 andER isexplicitly given by

.(x, y, z) = ER(x.y, z)(+(x, y, z) = ER(x + y, z)), ∀x, y, z ∈ R.

We refer to the papers[7,9–11] for examples ofM-vague multiplication (addition) operations and for thedetails.

In order to avoid the notation complexity, we always assume in the remaining part of this paper thatP andE are, respectively, two arbitrarily fixedM-equivalence relations onX × X andX, and(X, ◦) isanM-vague semigroup w.r.t.P andE, unless otherwise stated. It is shown in [9] that if the underlyingM-equivalence relationE onX is separated, then the (unique) ordinary description◦ of ◦ is associative.However, ordinary descriptions of◦ cannot be associative ifE is not anM-equality, but they satisfy thefollowing associativity-like property:

Proposition 2.5. If ◦i ∈ ORD(◦) for i = 1, . . . ,4, then

a ◦1 (b ◦2 c) ≈E (a ◦3 b) ◦4 c,∀a, b, c ∈ X.


Proof. Considering Corollary2.4, and by the condition (VAS) of vague associativity, we may write

1= ◦(b, c, b ◦2 c) ∗ ◦(a, b ◦2 c, a ◦1 (b ◦2 c))

∗◦(a, b, a ◦3 b) ∗ ◦(a ◦3 b, c, (a ◦3 b) ◦4 c)

� E(a ◦1 (b ◦2 c), (a ◦3 b) ◦4 c),

so the equivalencea ◦1 (b ◦2 c) ≈E (a ◦3 b) ◦4 c follows. �

3. Vague products of elements in vague semigroups

In this section, we introduce the notion of vague product of elements in vague semigroups, whichprovides a gradual approach to the notion of product of a finite number of elements. For this purpose, itis first necessary to consider the following essential component of this notion:

Definition 3.1. Forn�2, let{◦i | i = 1, . . . , n− 1} ⊆ ORD(◦) anda1, a2, . . . , an ∈ X.(i) The element(· · · ((a1 ◦1 a2) ◦2 a3) · · ·) ◦n−1 an (a1 ◦n−1 (a2 ◦n−2 (a3 ◦n−3 (· · · (an−2 ◦2 (an−1 ◦1

an)) · · ·)))) of X, denoted by[n∏

i=1

ai

]R{◦i |i=1,...,n−1}[ n∏

i=1

ai

]L{◦i |i=1,...,n−1}

,

is called the exact right (left) product ofa1, a2, . . . , an w.r.t. {◦i | i = 1, . . . , n− 1}.(ii) If[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

=[

n∏i=1

ai

]L{◦i |i=1,...,n−1}

,

then [n∏

i=1

ai

]R{◦i |i=1,...,n−1}

is called the exact product ofa1, a2, . . . , an w.r.t. {◦i | i = 1, . . . , n− 1}, and is denoted by[n∏

i=1

ai

]{◦i |i=1,...,n−1}

.


(iii) If ◦1 = ◦2 = · · · = ◦n−1 = �, then[n∏

i=1

ai

]R{◦i |i=1,...,n−1}

,

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

and

[n∏

i=1

ai

]{◦i |i=1,...,n−1}

(if exists) are, respectively, denoted by[n∏

i=1

ai

]R�

,

[n∏

i=1

ai

]L�

and

[n∏

i=1

ai

]�

.

For the sake of completeness of the notations in Definition3.1, we will assume that forn = 1,[n∏

i=1

ai

]R{◦i |i=1,...,n−1}

=[

n∏i=1

ai

]L{◦i |i=1,...,n−1}

=[

n∏i=1

ai

]{◦i |i=1,...,n−1}

= a1.

In Definition 3.1, if theM-equivalence relationE is strenghtened to be anM-equality, then theM-vaguebinary operation◦ has a unique ordinary description◦, i.e.ORD(◦) = {◦}, and(X, ◦) forms a semigroupin the classical sense (see [9, Theorem 2.10(vi)]). In this case, for all{◦i | i = 1, . . . , n−1} ⊆ ORD(◦),since◦1 = ◦2 = · · · = ◦n−1 = ◦, we obviously have[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

=[

n∏i=1

ai

]L{◦i |i=1,...,n−1}

=[

n∏i=1

ai

]◦.

Furthermore, the exact product[∏n

i=1 ai]◦ of a1, a2, . . . , an ∈ X in theM-vague semigroup(X, ◦) w.r.t.

◦ is the same as the product∏n

i=1 ai of a1, a2, . . . , an ∈ X in the semigroup(X, ◦). These observationsshow that in anM-vague semigroup(X, ◦) w.r.t. P andE, if E is anM-equality, the distinction between[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

and

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

is unnecessary, and they can be simply denoted by∏n

i=1 ai as in the classical group theory. IfE is not anM-equality but just anM-equivalence relation,[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

does not coincide with[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

in general, but they relate to each other by the equivalence relation≈E (see Proposition3.6).After the introduction of exact products, we are now ready to give the definition of vague product of a

finite number of elements in vague semigroups:


Definition 3.2. For n ∈ N+ and a1, a2, . . . , an ∈ X, the map�R(a1, a2, . . . , an):X → L (�L

(a1, a2, . . . , an):X → L), defined by

[�R(a1, a2, . . . , an)](x)=

∨◦[n−1∏

i=1

ai

]R{◦i |i=1,...,n−2}

, an, x

∣∣∣∣∣∣{◦i | i = 1, . . . , n− 2} ⊆ ORD(◦)

[�L(a1, a2, . . . , an)](x) =

∨◦a1,

[n−1∏i=1

ai+1

]L{◦i |i=1,...,n−2}

, x

∣∣∣∣∣∣{◦i | i = 1, . . . , n− 2} ⊆ ORD(◦)

, ∀x ∈ X

for n�2, and

[�R(a1)](x)

([�L

(a1)](x))={1, if x = a1,

0, if x �= a1,∀x ∈ X

for n = 1, is called the right (left) vague product ofa1, . . . , an. If �R(a1, a2, . . . , an) coincides with

�L(a1, a2, . . . , an), then it is called the vague product ofa1, . . . , an, and is denoted by�(a1, a2, . . . , an).

It is easy to note in Definition3.2 that

[�R(a1, a2, . . . , an)]

[ n∏i=1

ai

]R{◦i |i=1,...,n−1}

= [�L

(a1, a2, . . . , an)][ n∏

i=1

ai

]L{◦i |i=1,...,n−1}

= 1.

Therefore,[�R(a1, a2, . . . , an)](x) ([�L

(a1, a2, . . . , an)](x)) can be interpreted as the degree for whichx is an exact right (left) product ofa1, a2, . . . , an ∈ X.

If the vague binary operation◦ is represented by a vague additive notation+, we adopt the word “sum”instead of “product” in Definitions 3.1 and 3.2, and replace the notation� by �. In this paper, we alwaysassume that binary operations in both ordinary sense and vague sense are represented by multiplicativenotations, unless otherwise mentioned.

The non-ideal behavior of measurement devices and their discrete scales are the source of the uncertaintyon the indistinguishability of any two possible values of a measured quantityq, andM-equivalencerelations are natural representations of this kind of uncertainty [8,12,13]. For the sake of simplicity,assume thatR is the set of all possible values ofq. For a suitably chosenM-equivalence relationE onR and for any two possible valuesx andy of q, E(x, y) can be interpreted as the degree for whichx and y are indistinguishable (or equal, identical). In this case, a suitably chosenM-vague additionoperation+ (M-vague multiplication operation.) on R becomes a natural representation of the addition


(multiplication) operation for the possible values ofq in the measurement process. If we particularlytake + (.) as a perfectM-vague addition (multiplication) operation onR, then for a finite number ofpossible valuesa1, a2, . . . , an of q, we later show that the right vague sum (product) ofa1, a2, . . . , ancoincides with the left vague sum (product) ofa1, a2, . . . , an (see Corollary4.2). Since the underlyingaddition (multiplication) operation+ (.) is vaguely defined, the product (sum) ofa1, a2, . . . , an will notbe the usual producta1.a2 . . . an (the usual suma1 + a2 + · · · + an) of a1, a2, . . . , an, but the vagueproduct�(a1, a2, . . . , an) (the vague sum�(a1, a2, . . . , an)) of a1, a2, . . . , an w.r.t. + (.). Moreover, wewill also show in this case (see Theorem 4.7(i)) that�(a1, a2, . . . , an) (�(a1, a2, . . . , an)) is nothing butthe extensional hull of the usual producta1.a2 . . . an (the usual suma1 + a2 + · · · + an) according tothe underlyingM-equivalence relationE on R. Therefore,�(a1, a2, . . . , an) (�(a1, a2, . . . , an)) can bethought of as the fuzzy set of all real numbers approximately equal toa1.a2 . . . an (a1 + a2 + · · · + an).The actual valueak of the quantityq corresponding to its measured valueak can be practically conceivedas the fuzzy set of all real numbers approximately equal toak, and modelled as the extensional hull ofakw.r.t. E for k = 1, . . . , n. It is intuitively reasonable to expect that the product (sum) of the actual valuesa1, a2, . . . , an is the fuzzy set of all real numbers approximately equal toa1.a2 . . . an (a1+a2+· · ·+an).This intuitive expectation completely fits the formation of�(a1, a2, . . . , an) (�(a1, a2, . . . , an)), i.e.

�(a1, a2, . . . , an) = ã1.a2 . . . an (�(a1, a2, . . . , an) = ã1 + a2 + · · · + an).

It should be noted here that if the product (sum) ofa1, a2, . . . , an is computed according to the fuzzymultiplication� (the fuzzy addition⊕) operation in the usual sense[17,16], then the result cannot be theextensional hull ofa1.a2 . . . an (a1 + a2 + · · · + an) w.r.t. E, i.e. the equality

a1 � a2 � · · · � an = ã1.a2 . . . an (a1 ⊕ a2 ⊕ · · · ⊕ an = ã1 + a2 + · · · + an)

does not hold in general. Thus fuzzy arithmetic in the usual sense does not provide a satisfactory tool tomodel the above intuitive expectation.

These considerations form the motivating ideas of vague products and their representation properties,which will be the main subject of the next section. Examples for vague products and their practicalrealizations require the representation properties of vague products. For this reason, they are postponedto the next section.

In order to study the representation properties of vague products, we will need the following notion:

Definition 3.3. Define the maps�R,�L:Xn ×Xn → L by

�R[(a1, a2, . . . , an), (u1, . . . , un)] ={

n−1∗i=1

◦(ui, ai+1, ui+1), if u1 = a1,

0, if u1 �= a1,

�L[(a1, a2, . . . , an), (u1, . . . , un)] ={

n−1∗i=1

◦(ai, ui+1, ui, ), if un = an,

0, if un �= an

for all a1, a2, . . . , an, u1, . . . , un ∈ X and forn�2, and

�R(a, u) = �L(a, u) ={1, if u = a,

0, if u �= a

for all a, u ∈ X and forn = 1.


(i) An n-tuple(u1, . . . , un) ∈ Xn is said to be an exact right (left) successive product ofa1, a2, . . . , an ∈X iff u1 = a1 (un = an) and �R[(a1, a2, . . . , an), (u1, . . . , un)] = 1 (�L[(a1, a2, . . . , an),

(u1, . . . , un)] = 1).(ii) �R[(a1, a2, . . . , an), (u1, . . . , un)] (�L[(a1, a2, . . . , an), (u1, . . . , un)]) is said to be the degree

for which then-tuple(u1, . . . , un) ∈ Xn is a right (left) successive product ofa1, a2, . . . , an ∈ X.

Forn�2, a1, a2, . . . , an ∈ X and for all{◦i | i = 1, . . . , n− 2} ⊆ ORD(◦), noting the fact that

�R

(a1, a2, . . . , an),

a1,

[2∏

i=1

ai

]R{◦1}

,

[3∏

i=1

ai

]R{◦i |i=1,2}

, . . . ,

[n−1∏i=1

ai

]R{◦i |i=1,...,n−2}

, x

= ◦[n−1∏

i=1

ai

]R{◦i |i=1,...,n−2}

, an, x

and

�L

(a1, a2, . . . , an),

x,[n−1∏i=1

ai+1

]L{◦i |i=1,...,n−2}

, . . . ,

[3∏

i=1

an−3+i

]L{◦i |i=1,2}

,

[2∏

i=1

an−2+i

]L{◦1}

, an

= ◦a1,

[n−1∏i=1

ai+1

]L{◦i |i=1,...,n−2}

, x

,

we easily see that the right vague product�R(a1, a2, . . . , an) of a1, . . . , an and the left vague product

�L(a1, a2, . . . , an) of a1, . . . , an can also be written as

[�R(a1, a2, . . . , an)](x) =

∨�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)]

| vk =[

k∏i=1

ai

]R{◦i |i=1,...,k−1}

, k = 2, . . . , n− 1,

{◦i | i = 1, . . . , n− 2} ⊆ ORD(◦) ,

[�L(a1, a2, . . . , an)](x) =

∨�L[(a1, a2, . . . , an), (x, v2, v3, . . . , vn−1, an)]


| vn−k =[k+1∏i=1

an−k−1+i

]L{◦i |i=1,...,k}

, k = 1, . . . , n− 2,

{◦i | i = 1, . . . , n− 2} ⊆ ORD(◦) .

Thus, right (left) vague products can be expressed in terms of graded right (left) successive products.Furthermore, exact right (left) products and exact right (left) successive products are not independentnotions, and they are associated with each other in the following sense:

Proposition 3.4. For n�2 and a1, a2, . . . , an, u1, . . . , un ∈ X, (u1, . . . , un) is an exact right(left)successive product ofa1, a2, . . . , an iff there exists{◦i | i = 1, . . . , n− 1} ⊆ ORD(◦) such that

uk =[

k∏i=1

ai

]R{◦i |i=1,...,k−1}

uk =[n−k+1∏i=1

ak−1+i

]L{◦i |i=1,...,n−k}

for all k = 1, . . . , n.

Proof. We give the proof only for the case of exact right successive products. The proof for exact leftsuccessive products can be similarly seen. Let(u1, . . . , un) be an exact right successive product ofa1, a2, . . . , an, i.e.u1 = a1 and

�R[(a1, a2, . . . , an), (u1, . . . , un)] = n−1∗i=1

◦(ui, ai+1, ui+1) = 1,

i.e. ◦(a1, a2, u2) = ◦(u2, a3, u3) = · · · = ◦(un−1, an, un) = 1.

Since ◦(a1, a2, u2) = 1, and using[9, Proposition 2.3(ii)],∃◦1 ∈ ORD(◦) such thatu2 = a1 ◦1

a2 =[∏2

i=1 ai

]R{◦1}. Similarly, considering◦(u2, a3, u3) = 1, and using [9, Proposition 2.3(ii)],∃◦2 ∈

ORD(◦) such that

u3 = u2 ◦2 a3 =[

2∏i=1

ai

]R{◦1}

◦2 a3 =[

3∏i=1

ai

]R{◦i |i=1,...,2}

.

Pursuing this process,∃◦k ∈ ORD(◦) such that

uk = uk−1 ◦k−1 ak =[

k∏i=1

ai

]R{◦i |i=1,...,k−1}

for all k = 2, . . . , n,

so the assertion follows. Conversely, ifu1, . . . , un ∈ X are given by

uk =[

k∏i=1

ai

]R{◦i |i=1,...,k−1}


for some{◦i | i = 1, . . . , n − 1} ⊆ ORD(◦) and for allk = 1, . . . , n, then since◦(a1, a2, u2) =◦(u2, a3, u3) = · · · = ◦(un−1, an, un) = 1, it is obvious that(u1, . . . , un) is an exact right successiveproduct ofa1, a2, . . . , an. �

The maps�R and�L defined in Definition3.3 are different from each other in general, but relate toeach other by the following rule:

Theorem 3.5. The maps�R and�L satisfy the relation

�R[(a1, . . . , an), (u1, . . . , un)] ∗ �L[(a1, . . . , an), (v1, . . . , vn)]�E(un, v1)

for n ∈ N+ and for allai, ui, vi ∈ X, i = 1,2, . . . , n.

Proof. For the caseu1 �= a1 orvn �= an orn = 1, the required inequality is trivial. Suppose thatu1 = a1,vn = an andn�2. Let� ∈ ORD(◦). If n = 2, then using Corollary2.4, we have

�R[(a1, a2), (u1, u2)] ∗ �L[(a1, a2), (v1, v2)]= ◦(a1, a2, u2) ∗ ◦(a1, a2, v1)

�E(a1�a2, u2) ∗ E(a1�a2, v1)�E(u2, v1),

so the assertion is straightforward. Now, let us consider the casen�3. We first show that for alli =1, . . . , n− 2,

◦(ui, ai+1, ui+1) ∗ ◦(ai+1, vi+2, vi+1)�E(ui�vi+1, ui+1�vi+2). (3.1)

From Corollary2.4, we have

◦(ui, vi+1, ui�vi+1) = ◦(ui+1, vi+2, ui+1�vi+2) = 1.

Then, using this fact, and by applying the vague associativity (VAS), we may write

◦(ui, ai+1, ui+1) ∗ ◦(ai+1, vi+2, vi+1)

= ◦(ai+1, vi+2, vi+1) ∗ ◦(ui, vi+1, ui�vi+1) ∗ ◦(ui, ai+1, ui+1)

∗◦(ui+1, vi+2, ui+1�vi+2)

�E(ui�vi+1, ui+1�vi+2),

so inequality (3.1) is obtained. Since inequality (3.1) is valid for alli = 1, . . . , n − 2, inequality (3.1)implies the following set of inequalities:

◦(u1, a2, u2) ∗ ◦(a2, v3, v2) � E(u1�v2, u2�v3),

◦(u2, a3, u3) ∗ ◦(a3, v4, v3) � E(u2�v3, u3�v4),...

◦(un−2, an−1, un−1) ∗ ◦(an−1, vn, vn−1) � E(un−2�vn−1, un−1�vn).

Now multiplying all of these inequalities side by side w.r.t.∗, we obtain that

[◦(u1, a2, u2) ∗ ◦(u2, a3, u3) ∗ · · · ∗ ◦(un−2, an−1, un−1)]∗[◦(a2, v3, v2) ∗ ◦(a3, v4, v3) ∗ · · · ∗ ◦(an−1, vn, vn−1)]


�E(u1�v2, u2�v3) ∗ E(u2�v3, u3�v4) ∗ · · · ∗ E(un−2�vn−1, un−1�vn)

�E(u1�v2, un−1�vn) = E(a1�v2, un−1�an).

Thus, if we denote the left-hand side of this inequality byR, the last inequality can be expressed as

R�E(a1�v2, un−1�an). (3.2)

Furthermore, taking into consideration the definitions ofR, �R[(a1, . . . , an), (u1, . . . , un)] and�L[(a1, . . . , an), (v1, . . . , vn)], we easily note that

�R[(a1, . . . , an), (u1, . . . , un)] ∗ �L[(a1, . . . , an), (v1, . . . , vn)]= ◦(un−1, an, un) ∗ ◦(a1, v2, v1) ∗ R. (3.3)

Hence, combining (3.2) and (3.3), using Corollary 2.4, we see that

�R[(a1, . . . , an), (u1, . . . , un)] ∗ �L[(a1, . . . , an), (v1, . . . , vn)]= ◦(un−1, an, un) ∗ ◦(a1, v2, v1) ∗ R� ◦(un−1, an, un) ∗ ◦(a1, v2, v1) ∗ E(a1�v2, un−1�an)

�E(un−1�an, un) ∗ E(a1�v2, v1) ∗ E(a1�v2, un−1�an)

�E(un, v1). �

With the help of Proposition 3.4 and Theorem 3.5, we now show that forn ∈ N+, a1, a2, . . . , an ∈ X

and for all {◦i | i = 1, . . . , n − 1}, {�i | i = 1, . . . , n − 1} ⊆ ORD(◦), [∏ni=1 ai]R{�i |i=1,...,n−1},[∏n

i=1 ai]L{◦i |i=1,...,n−1},

[∏ni=1 ai]R{◦i |i=1,...,n−1} and

[∏ni=1 ai]L{�i |i=1,...,n−1} belong to the same equiv-

alence class according to≈E :

Proposition 3.6. For n ∈ N+, a1, a2, . . . , an ∈ X and for all {◦i | i = 1, . . . , n − 1}, {�i | i =1, . . . , n− 1} ⊆ ORD(◦), the relations[

n∏i=1

ai

]R{�i |i=1,...,n−1}

≈ E

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

≈E

[n∏

i=1

ai

]R{◦i |i=1,...,n−1}

≈ E

[n∏

i=1

ai

]L{�i |i=1,...,n−1}

are satisfied.

Proof. Let us choose the elementsu1, . . . , un, v1, . . . , vn ∈ X in Theorem3.5 such that

uk =[

k∏i=1

ai

]R{�i |i=1,...,k−1}

andvk =[n−k+1∏i=1

ak−1+i

]L{◦i |i=1,...,n−k}

for all k = 1, . . . , n. From Proposition3.4, we have

�R[(a1, . . . , an), (u1, . . . , un)] = �L[(a1, . . . , an), (v1, . . . , vn)] = 1.


Then, by Theorem3.5, we get

�R[(a1, . . . , an), (u1, . . . , un)] ∗ �L[(a1, . . . , an), (v1, . . . , vn)] = 1�E(un, v1).

Therefore,

E(un, v1) = E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

,

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

= 1,

i.e.

[n∏

i=1

ai

]R{�i |i=1,...,n−1}

≈E

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

.

Since the equivalence[n∏

i=1

ai

]R{�i |i=1,...,n−1}

≈E

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

is valid for all {◦i | i = 1, . . . , n − 1}, {�i | i = 1, . . . , n − 1} ⊆ ORD(◦), this equivalence yields theequivalences[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

≈E

[n∏

i=1

ai

]L{◦i |i=1,...,n−1}

and [n∏

i=1

ai

]R{�i |i=1,...,n−1}

≈E

[n∏

i=1

ai

]L{�i |i=1,...,n−1}

.

Thus, the required result follows from the transitivity of≈E . �

4. Representations and practical realizations of vague products

Representations and constructions of vague products by means of the products in the classical senseand the underlyingM-equivalence relations are significant issues for the practical applications of vagueproducts (sums) of a finite number of real numbers in vague arithmetic. For this reason, we concentrate onthese issues in this section, and show how vague products can be represented in terms of exact products(and so the products in the classical sense) and the underlyingM-equivalence relations. Exemplificationsof the presented results and their practical realizations will also be subjects of this section. We start withthe following central result for the representations of vague products.


Theorem 4.1. For n ∈ N+ withn�2,a1, a2, . . . , an ∈ X and for all{�i | i = 1, . . . , n−2} ⊆ ORD(◦),the following properties are valid:

(i) If ◦ is transitive of the third(second) order, then the right(left) vague product ofa1, a2, . . . , an canbe represented as

[�R(a1, a2, . . . , an)](x)=

∨{�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)]

| v2, v3, . . . , vn−1 ∈ X}

= ◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

(4.1)

[�L(a1, a2, . . . , an)](x) =

∨{�L[(a1, a2, . . . , an), (x, v2, v3, . . . , vn−1, an)]

| v2, v3, . . . , vn−1 ∈ X}

= ◦a1,

[n−1∏i=1

ai+1

]L{�i |i=1,...,n−2}

, x

,∀x ∈ X. (4.2)

(ii) If (X, ◦) is a perfect M-vague semigroup, then the right and left vague products ofa1, a2, . . . , anare equal, and the vague product ofa1, a2, . . . , an is given by

[�(a1, a2, . . . , an)](x) = E

[ n∏i=1

ai

]U{�i |i=1,...,n−1}

, x

for all U ∈ {L,R} and for allx ∈ X.

Proof. (i) We prove the assertion for only right vague products. The proof for left vague products canbe easily stated in an analogous manner to right vague products, so it is left to readers as an exercise.Assume that◦ is transitive of the third order. From the definition of�

R(a1, a2, . . . , an), it is easy to see

that

◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

�[�R

(a1, a2, . . . , an)](x)�∨

{�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)] | v2, v3, . . . , vn−1 ∈ X}.Thus, to verify the assertion, it is sufficient to show that∨

{�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)] | v2, v3, . . . , vn−1 ∈ X}

� ◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

. (4.3)


Utilizing the third-order transitivity of◦, and by Corollary2.4, we may write

�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)]= ◦(a1, a2, v2) ∗ ◦(v2, a3, v3) ∗ · · · ∗ ◦(vn−1, an, x)

�E(a1�1a2, v2) ∗ ◦(v2, a3, v3) ∗ · · · ∗ ◦(vn−1, an, x)

� ◦(a1�1a2, a3, v3) ∗ ◦(v3, a4, v4) · · · ∗ ◦(vn−1, an, x)

�E((a1�1a2)�

2a3, v3) ∗ ◦(v3, a4, v4) · · · ∗ ◦(vn−1, an, x)

...

�E((· · · ((a1�1a2)�

2a3) · · ·)�n−2an−1, vn−1) ∗ ◦(vn−1, an, x)

� ◦((· · · ((a1�1a2)�

2a3) · · ·)�n−2an−1, an, x)

= ◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

for all v2, v3, . . . , vn−1 ∈ X, so we get

�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)]� ◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

.

It is obvious that this inequality implies (4.3).(ii) Let (X, ◦) be a perfectM-vague semigroup. We first show that�

R(a1, a2, . . . , an) and

�L(a1, a2, . . . , an) are, respectively, given by (4.1) and (4.2). Forn = 2, equalities (4.1) and (4.2)

are trivial. Supposen�3. In a similar fashion to (i), it is sufficient to see inequality (4.3). Applying thevague associativity condition (VAS), and using Proposition 2.5, we easily observe that

◦[k−1∏

i=1

ai

]R{�i |i=1,...,k−2}

, ak, vk

∗ ◦(vk, ak+1, vk+1)

= ◦(ak, ak+1, ak�kak+1) ∗ ◦

[k−1∏i=1

ai

]R{�i |i=1,...,k−2}

, ak�kak+1,

[k−1∏i=1

ai

]R{�i |i=1,...,k−2}

�k−1(ak�kak+1)

∗◦[k−1∏

i=1

ai

]R{�i |i=1,...,k−2}

, ak, vk

∗ ◦(vk, ak+1, vk+1)


�E

[k−1∏i=1

ai

]R{�i |i=1,...,k−2}

�k−1(ak�kak+1), vk+1

= E

[k−1∏i=1

ai

]R{�i |i=1,...,k−2}

�k−1ak

�kak+1, vk+1

= E

[ k∏i=1

ai

]R{�i |i=1,...,k−1}

�kak+1, vk+1

= ◦[ k∏

i=1

ai

]R{�i |i=1,...,k−1}

, ak+1, vk+1

for all vk, vk+1 ∈ X and for allk = 2, . . . , n− 2. Thus, we possess the inequality

◦[k−1∏

i=1

ai

]R{�i |i=1,...,k−2}

, ak, vk

∗ ◦(vk, ak+1, vk+1)

� ◦[ k∏

i=1

ai

]R{�i |i=1,...,k−1}

, ak+1, vk+1

. (4.4)

Now exploiting inequality (4.4), we obtain the inequalities

�R[(a1, a2, . . . , an), (a1, v2, v3, . . . , vn−1, x)]= ◦(a1, a2, v2) ∗ ◦(v2, a3, v3) ∗ · · · ∗ ◦(vn−1, an, x)

= [◦(a1, a2, v2) ∗ ◦(v2, a3, v3)] ∗ [◦(v3, a4, v4) ∗ · · · ∗ ◦(vn−1, an, x)]� ◦(a1�

1a2, a3, v3) ∗ [◦(v3, a4, v4) ∗ · · · ∗ ◦(vn−1, an, x)]= [◦(a1�

1a2, a3, v3) ∗ ◦(v3, a4, v4)] ∗ ◦[(v4, a5, v5) ∗ · · · ∗ ◦(vn−1, an, x)]� ◦((a1�

1a2)�2a3, a4, v4) ∗ [◦(v4, a5, v5) ∗ · · · ∗ ◦(vn−1, an, x)]

...

� ◦((· · · ((a1�1a2)�

2a3) · · ·)�n−3an−2, an−1, vn−1)) ∗ ◦(vn−1, an, x)

� ◦((· · · ((a1�1a2)�

2a3) · · ·)�n−2an−1, an, x)

= ◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

,


and therefore we reach inequality (4.3) again. Now using equalities (4.1) and (4.2), and by the fact that◦ is a perfectM-vague binary operation, we get from Corollary 2.4 that

[�R(a1, a2, . . . , an)](x)= ◦

[n−1∏i=1

ai

]R{�i |i=1,...,n−2}

, an, x

=E

[n−1∏i=1

ai

]R{�i |i=1,...,n−2}

�n−1an, x

=E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, x

. (4.5)

Similarly,

[�L(a1, a2, . . . , an)](x) = E

[ n∏i=1

ai

]L{�i |i=1,...,n−1}

, x

. (4.6)

On the other hand, we have from Proposition3.6 that

E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

,

[n∏

i=1

ai

]L{�i |i=1,...,n−1}

= 1, i.e.

E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, x

= E

[ n∏i=1

ai

]L{�i |i=1,...,n−1}

, x

.

Thus the assertion follows from (4.5) and (4.6) at once.�

Corollary 4.2. Let (X, ◦) be a perfect M-vague semigroup w.r.t. P and E such that there exists anassociative ordinary description� ∈ ORD(◦) of ◦. Then forn�2, the vague product�(a1, a2, . . . , an)

of a1, a2, . . . , an ∈ X is explicitly given by

[�(a1, a2, . . . , an)](x) = E

([n∏

i=1

ai

]�

, x

), ∀x ∈ X. (4.7)

Proof. If we put�i = � for all i = 1, . . . , n− 1 in Theorem4.1(ii), then we obviously have

[�(a1, a2, . . . , an)](x) = E

[ n∏i=1

ai

]R�

, x

= E

[ n∏i=1

ai

]L�

, x

, ∀x ∈ X. (4.8)


Since(X,�) forms a semigroup,[∏n

i=1 ai]R� coincides with

[∏ni=1 ai]L� , i.e.

[n∏

i=1

ai

]R�

=[

n∏i=1

ai

]L�

=[

n∏i=1

ai

]�

. (4.9)

Thus, equality (4.7) follows from equalities (4.8) and (4.9), at once.�

Remark 4.3. (i) Let (X, ◦) be a perfectM-vague semigroup w.r.t.P andE. If E is anM-equality onX, then◦ has unique ordinary description� = ord(◦), and it follows from[9, Theorem 2.10(vi)] that(X,�) forms a semigroup, and so the hypothesis of Corollary 4.2 is satisfied. Thus, the vague product�(a1, a2, . . . , an) of a1, a2, . . . , an ∈ X is given by the equality (4.7).

(ii) Let ER be a regularM-equivalence relation onR w.r.t. the usual addition (multiplication) operation“+” (“ .”). For anM-equivalence relationER2 on R2, let “+” (“ .”) be extensional w.r.t.ER2 andER. If

we define the fuzzy relation+ER∈ LR3

( .ER∈ LR3

) by

+ER(x, y, z) = ER(x + y, z) ( .ER

(x, y, z) = ER(x.y, z))

for all x, y, z ∈ R, then due to Corollary2.4,+ER(.ER

) is obviously a perfectM-vague addition (multipli-cation) operation onR w.r.t.ER2 andER. Furthermore, it is easy to deduce from [10, Corollary 2.15] that(R, +ER

) ((R, .ER)) forms a perfectM-vague semigroup. Thus, since+ ∈ ORD(+ER

) (. ∈ ORD(.ER))

is an associative ordinary description of+ER(.ER

), the assumptions in Corollary 4.2 are clearly ful-filled. Therefore, by virtue of Corollary 4.2, we easily evaluate the vague sum�(a1, a2, . . . , an) (product�(a1, a2, . . . , an)) of given real numbersa1, a2, . . . , an ∈ R w.r.t. +ER

(.ER) by the equality

[�(a1, a2, . . . , an)](x) = ER

(n∑

i=1

ai, x

)([�(a1, a2, . . . , an)](x) = ER

(n∏

i=1

ai, x

)).

Example 4.4. (i) Let us consider the particular iccqm-latticeM = ([0,1], �, Lck), whereLck(x, y) =max{x+y−1,0} is Lukasiewicz’st-norm. Define the mapsE(+)

R : R×R → [0,1],E(+)

R2 : R2×R2 → [0,1]and the fuzzy relation+

E(+)R

∈ [0,1]R3by

E(+)R (x, y)= 1−min{|x − y|,1},

E(+)

R2 ((x, y), (x′, y′))=E(+)R (x + y, x′ + y′)

and+E

(+)R

(x, y, z)=E(+)R (x + y, z), ∀x, y, z, x′, y′ ∈ R.

It is shown in[10, Example 4.5(i)] thatE(+)R andE

(+)

R2 are, respectively,M-equivalence relations (or

simplyLck-indistinguishability operators) onR andR2. Furthermore, “+” is extensional w.r.t.E(+)

R2 and

E(+)R , andE(+)

R is regular w.r.t. “+”. Thus, +E

(+)R

is clearly a perfectM-vague addition operation onR

w.r.t.E(+)

R2 andE(+)R , and then by Remark 4.3(ii), the vague sum�(a1, a2, . . . , an) of given real numbers


a1, a2, . . . , an ∈ R w.r.t. +E

(+)R

is simply calculated as

[�(a1, a2, . . . , an)](x) = E(+)R

(n∑

i=1

ai, x

)= 1−min

{∣∣∣∣∣n∑

i=1

ai − x

∣∣∣∣∣ ,1

}, ∀x ∈ R.

(ii) For the productt-normPrdefined byPr(x, y) = x.y, let the underlying iccqm-lattice be particularlychosen asM = ([0,1],≤, P r). The mapsER: R × R → [0,1] andER2: R2 × R2 → [0,1], defined by

ER(x, y) =

min{xy,yx

}, if x.y > 0

0, if x.y < 01, if x = y = 0

and

ER2((x, y), (x′, y′)) = ER(x.y, x

′.y′), ∀x, y, x′, y′ ∈ R,

areM-equivalence relations (or simplyPr-indistinguishability operators) onR andR2. Furthermore “.”is extensional w.r.t.ER2 andER, andER is regular w.r.t. “.”. Therefore, the fuzzy relation.ER

∈ [0,1]R3,

given by

.ER(x, y, z) = ER(x.y, z), ∀x, y, z ∈ R,

is a perfectM-vague multiplication operation onR w.r.t. ER2 andER. Thereafter, considering Remark4.3(ii), we easily formulate the vague product�(a1, a2, . . . , an) of given real numbersa1, a2, . . . , an ∈ R

w.r.t. .ERby

[�(a1, a2, . . . , an)](x) = ER

(n∏

i=1

ai, x

), ∀x ∈ R.

(iii) Now let us deal with the particular iccqm-latticeM = ([0,1], �,∧). For arbitrarily fixed�, �, � ∈[0,1] with ��, it is easy to see that the mapsS(+)

R , S(.)R : R × R → [0,1], defined by

S(+)R (x, y) =

1, if x = y

�, if y − x ∈ Q − {0}�, if y − x /∈ Q

and

S(.)

R2(x, y) =

1, if x = y

�, if yx∈ Q − {0,1}

�, if yx/∈ Q

�, if x.y = 0 andx �= y

, ∀x, y ∈ R,

are M-equivalence relations (known also as similarity relations[33]) on R. TherewithS(+)R and S

(.)R

are, respectively, regular w.r.t. “+” and “.”. In a similar fashion to (i) and (ii), if we consider the sim-ilarity relationsS(+)

R2 , S(.)

R2: R2 × R2 → [0,1] on R2 and the fuzzy relations+S(+)R

, .S(.)R

∈ [0,1]R3,


defined by

S(+)

R2 ((x, y), (x′, y′))= S(+)R (x + y, x′ + y′),

S(.)

R2((x, y), (x′, y′))= S

(.)R (x.y, x′.y′),

+S(+)R

(x, y, z)= S(+)R (x + y, z) and .

S(.)R

(x, y, z) = S(.)R (x.y, z)

for all x, x′, y, y′, z ∈ R, then by Remark4.3(ii), the vague sum�(a1, a2, . . . , an) (the vague product�(a1, a2, . . . , an)) of a1, a2, . . . , an ∈ R w.r.t. the perfectM-vague addition operation+

S(+)R

(the perfect

M-vague multiplication operation.S(.)R

) is given as

[�(a1, a2, . . . , an)](x) = S(+)R

(n∑

i=1

ai, x

)([�(a1, a2, . . . , an)](x) = S

(.)R

(n∏

i=1

ai, x

)).

As a practical realization of Example4.4(i), we may be interested in the measurement of the perimeterof a quadrilateralQ. In order to measure the length of its sides, let us take a rod having cm readings as ourmeasurement instrument. Leta1, a2, a3 anda4 be the measured values of the lengths of sides ofQ. Thediscrete readings of this rod cause the uncertainty on the indistinguishability of any two readings. Thisuncertainty can be simply modelled by a suitably chosenM-equivalence relationE on R. For any tworeadingsxandyon this rod,E(x, y) can be interpreted as the degree for whichxandyare indistinguishable(or equal, identical). The uncertainty resulting from the indistinguishability of any two readings naturallynecessitates the uncertainty on the addition (multiplication) operation on the readings of this rod, whichcan be mathematically represented as a suitably chosenM-vague addition+ (M-vague multiplication.) operation onR. In the measurement of the perimeter ofQ, if we particularly chooseE and+ as theLck-indistinguishability operatorE(+)

R and the perfectM-vague addition operation+E

(+)R

in Example

4.4(i), then the perimeter ofQ in the unit of cm will be the vague sum�(a1, a2, a3, a4) of a1, a2, a3 anda4 w.r.t. +

E(+)R

, given by

[�(a1, a2, a3, a4)](x) = E(+)R

(4∑

i=1

ai, x

)= 1−min

{∣∣∣∣∣4∑

i=1

ai − x

∣∣∣∣∣ ,1

}, ∀x ∈ R.

Here�(a1, a2, a3, a4) is simply a triangular fuzzy number centered ata1+ a2+ a3+ a4, and is sketchedin Fig. 1.

The measurement of the volume of a rectangular prismP can be considered as an example for thepractical implementation of Example4.4(ii). Let us use the same rod in the previous example, and nowoperate with thePr-indistinguishability operatorER and the perfectM-vague multiplication operation.ER

in Example 4.4(ii). If we measure the lengths of sides of the rectangular base ofP and its heightasa1, a2 anda3, then the volume ofP will now be the vague product�(a1, a2, a3) of a1, a2 anda3


x

1

0

(x )

4

i =1 ai − 1Σ

4

i =1 ai + 1Σ

4 i =1

aiΣ

[Σ(a1, ...,a1)]~

Fig. 1. The perimeter of the quadrilateralQ with sides of lengthsa1, . . . , a4.

xa1.a2.a30

1

[∏ (a1,a2,a3)] (x)~

Fig. 2. The volume of the rectangular prismP with sides of lenghtsa1, a2 anda3.

w.r.t. .ER. Here�(a1, a2, a3) can be simply computed as

[�(a1, a2, a3)](x)=ER(a1.a2.a3, x)

={min{a1.a2.a3

x, xa1.a2.a3

}, if x > 0,

0, if x�0

for all x ∈ R (see Fig. 2).If the real numbers�, �, � ∈ [0,1] in Example4.4(iii) are particularly taken as� = � = �, then the

mapsS(+)R andS(.)

R : R×R → [0,1] considered in Example 4.4(iii) simply turn into a constant similarityrelation:

S(+)R (x, y) = S

(.)

R2(x, y) ={

1, if x = y,

� otherwise.


As also observed in Example4.4(iii) and the previous case� = � = �, because of the fact that the regularityrequirement on similarity relations usually compels them to be either constant or discontinuous, similarityrelations do not provide a satisfactorily good modelling of indistinguishability from the viewpoint of thecalculations of vague sum or product of real numbers. For this reason, we pay attention to onlyLck-indistinguishability operators andPr-indistinguishability operators in this paper.

The regularity requirement on the underlyingM-equivalence relation has a central role for the calcu-lation of vague sums and products of real numbers. The following theorem enables the conversion of anM-equivalence relation regular w.r.t. a crisp binary operation� to anotherM-equivalence relation regularw.r.t. another crisp binary operation◦.

Theorem 4.5. Let ◦ and � be two crisp binary operations on X and Y, respectively. Given an M-equivalence relation E on Y and a groupoid homomorphism from X to Y, i.e. (x ◦ y) = (x)�(y),∀x, y ∈ X, the mapE:X × X → L, given byE(x, y) = E((x),(y)), ∀x, y ∈ X, defines anM-equivalence relation on X. Furthermore, if E is regular w.r.t. �, thenE is regular w.r.t. ◦.

Proof. The first part of theorem is given in[8, Theorem 4.4(i)]. In order to prove the remaining assertion,assume thatE is regular w.r.t.�. Then by making use of the regularity ofE w.r.t. � and the fact that isa groupoid homomorphism betweenX andY, we may write

E(x, y)=E((x),(y))�E((x)�(u),(y)�(u))

=E((x ◦ u),(y ◦ u)) = E(x ◦ u, y ◦ u), ∀x, u, y ∈ X.

Similarly, we also get

E(x, y)�E(u ◦ x, u ◦ y), ∀x, u, y ∈ X.

Thus the regularity ofE w.r.t. ◦ follows. �

As an application of Theorem4.5, we may establish an infinitely many number of examples for perfectM-vague addition and multiplication operations and vague sums and products of real numbers accordingto these operations in an analogous manner to Example 4.4:

Example 4.6. (i) Let us consider the extended real lineR∞ = R ∪ {−∞,+∞} in the usual sense.The Lck-indistinguishability operatorE(+)

R considered in Example4.4(i) can be extended to theLck-

indistinguishability operatorE(+)

R∞ on R∞:

E(+)

R∞ (x, y) =E

(+)R (x, y), if x, y ∈ R,

1, if x = y ∈ {−∞,+∞},0 otherwise.

∀x, y ∈ R∞.

For an arbitrarily fixed� ∈ R − {0}, it is easy to perceive that the map�: (R, .) → (R∞,+), defined by

�(x) ={In(|x|�), if x �= 0,−∞, if x = 0,

∀x ∈ R,


1

[∏(a1, ..., an)]

x

(x)~

i =1ai

n−e.∏

i =1ai

n− ∏

i =1ai

n∏

i =1ai

n∏

i =1ai

n

e

∏

ei =1

ai

n∏e.

Fig. 3. The graph of�(a1, a2, . . . , an) for � = 1 andn∏

i=1ai > 0.

is a groupoid homomorphism from(R, .) to (R∞,+). SinceE(+)

R∞ is regular w.r.t. the usual addition

operation “+” on R∞, we obtain from Theorem4.5 that the mapE(.)�

: R × R → [0,1], defined by

E(.)�(x, y)=E

(+)

R∞ (�(x),�(y))

=

1−min{|�|.

∣∣∣In(∣∣∣xy ∣∣∣)∣∣∣ ,1}, if x, y ∈ R − {0},0, if x.y = 0 andx �= y,

1, if x = y = 0

is anLck-indistinguishability operator onR, and is also regular w.r.t. “.”. Thus, if we consider theLck-indistinguishability operatorP (.)

�on R2 and the fuzzy relation.

E(.)�

∈ [0,1]R3defined by

P(.)�((x, y), (x′, y′)) = E

(.)�(x.y, x′.y′) and .

E(.)�

(x, y, z) = E(.)�(x.y, z)

for all x, y, z, x′, y′ ∈ R, then in a similar way to Example4.4(ii), for M = ([0,1], �, Lck), .E

(.)�

is a

perfectM-vague multiplication operation onR w.r.t.P (.)�

andE(.)�

, and the vague product�(a1, a2, . . . , an)

of a1, a2, . . . , an ∈ R w.r.t. .E

(.)�

is computed as

[�(a1, a2, . . . , an)](x) = E(.)�

(n∏

i=1

ai, x

), ∀x ∈ R.

For � = 1 and∏n

i=1 ai > 0, the vague product�(a1, a2, . . . , an) is represented in Fig. 3.(ii) For an arbitrarily fixed� ∈ R, the map�: (R,+) → (R, .), x �→ e�.x , obviously defines a groupoid

homomorphism from(R,+) to (R, .). Therefore, if we reconsider thePr-indistinguishability operatorER given in Example4.4(ii), and define thePr-indistinguishability operatorE(+)

�by

E(+)�

(x, y) = ER(�(x),�(y)) = e−|�.(x−y)|, ∀x ∈ R,


1

x

[ Σ(a1, ..., an)] (x)~

i =1ai

nΣ

Fig. 4. The graph of�(a1, a2, . . . , an) for � = 1 andn∑

i=1ai > 0.

then Theorem4.5 entails thatE(+)�

is regular w.r.t. “+”. Thus, taking into consideration the

Pr-indistinguishability operatorP (+)�

on R2 and the fuzzy relation+E

(+)�

∈ [0,1]R3defined by

P (+)�

((x, y), (x′, y′)) = E(+)�

(x + y, x′ + y′) and+E

(+)�

(x, y, z) = E(+)�

(x + y, z)

for all x, y, z, x′, y′ ∈ R, we observe that forM = ([0,1],≤, P r), +E

(+)�

is a perfectM-vague addition

operation onR w.r.t.P (+)�

andE(+)�

. Thereupon, the vague sum�(a1, a2, . . . , an) of a1, a2, . . . , an ∈ R

w.r.t. +E

(+)�

is now computed as

[�(a1, a2, . . . , an)](x) = E(+)�

(n∑

i=1

ai, x

)= e−|�.((

∑ni=1 ai)−x)|, ∀x ∈ R.

For � = 1 and∑n

i=1 ai > 0, �(a1, a2, . . . , an) is drawn in Fig. 4.

As can be seen from Fig. 3, since the graph of the vague product�(a1, a2, . . . , an) in Example4.6(i)is symmetric according to they-axis,

∏ni=1 ai is indistinguishable from any real numberx in the interval

n∏i=1

ai

e, e.

(n∏

i=1

ai

)insofar as−x, i.e.E(.)

�(∏n

i=1 ai, x) = E(.)�(∏n

i=1 ai,−x), though any pointx between

−

n∏

i=1ai

e


and 0 is distinguishable from∏n

i=1 ai , i.e.E(x,∏n

i=1 ai) = 0. This is intuitively an undesirale situation.

In order to remove this inconvenience, it is enough to modifyE(.)�

slightly. Namely, if we replaceE(.)�

by

theLck-indistinguishability operatorE∗� : R2 → [0,1] given as

E∗� (x, y) =

1−min

{|�|.∣∣∣In(xy )∣∣∣ ,1

}, if x.y > 0,

0, if x.y < 0,1, if x = y = 0,

then the extensional hull of∏n

i=1 ai w.r.t. E∗� (and so the vague product�(a1, a2, . . . , an) w.r.t. .E∗

�)

will not be symmetric according to they-axis. In a similar fashion to Fig. 3, if we choose� = 1 and∏ni=1 ai > 0, then the vague product�(a1, a2, . . . , an) w.r.t. .E∗

�has exactly the same shape as the graph

in Fig. 3 on the set of all non-negative real numbers, while it vanishes on the set of all negative realnumbers.

For n ∈ N+, let CR(a1, . . . , an) (CL(a1, . . . , an)) denote the set of all exact right (left) products ofgiven elementsa1, . . . , an of (X, ◦) w.r.t. all {◦i | i = 1, . . . , n− 1} ⊆ ORD(◦)}, i.e.

CR(a1, . . . , an) =[

n∏i=1

ai

]R{◦i |i=1,...,n−1}

| {◦i | i = 1, . . . , n− 1} ⊆ ORD(◦)CL(a1, . . . , an) =

[

n∏i=1

ai

]L{◦i |i=1,...,n−1}

| {◦i | i = 1, . . . , n− 1} ⊆ ORD(◦) .

It is easy to observe that all exact right (left) products ofa1, . . . , an ∈ X w.r.t. all{◦i | i = 1, . . . , n−1} ⊆ORD(◦)belong to the kernel of�R

(a1, a2, . . . , an) (�L(a1, a2, . . . , an)), or formallyCR(a1, . . . , an) ⊆

ker(�R(a1, a2, . . . , an)) (CL(a1, . . . , an) ⊆ ker(�

L(a1, a2, . . . , an))). This means that theL-fuzzy sub-

set�R(a1, a2, . . . , an) (�L

(a1, a2, . . . , an)) of X is a fuzzification ofCR(a1, . . . , an) (CL(a1, . . . , an)).Furthermore, we show in the following theorem that if the underlyingM-vague binary operation◦ istaken as a perfect one, thenCR(a1, . . . , an) coincides withCL(a1, . . . , an), and it is nothing but theequivalence class of any one of its elements according to≈E . Moreover, if we denoteCR(a1, . . . , an)

byC(a1, . . . , an) in this case, then the vague product�(a1, a2, . . . , an) of a1, a2, . . . , an will be just theextensional hull of any elementx of C(a1, . . . , an) w.r.t. E.

Theorem 4.7. The following properties are valid forn ∈ N+ and fora1, a2, . . . , an ∈ X:(i) For all u ∈ CR(a1, . . . , an) (u ∈ CL(a1, . . . , an)) andx ∈ X,

[�R(a1, a2, . . . , an)](x)�E(u, x) ([�L

(a1, a2, . . . , an)](x)�E(u, x)),

and the equality is satisfied if◦ is a perfect M-vague binary operation w.r.t. P and E.(ii) For all u ∈ CR(a1, . . . , an) ∪ CL(a1, . . . , an),

CR(a1, . . . , an) ∪ CL(a1, . . . , an)

⊆ ker(�R(a1, a2, . . . , an)) ∪ ker(�

L(a1, a2, . . . , an)) ⊆ [u]≈E

.


(iii) If ◦ is a perfect M-vague binary operation w.r.t. P and E, then

CR(a1, . . . , an) = CL(a1, . . . , an) = ker(�(a1, a2, . . . , an)) = [u]≈E

for all u ∈ CR(a1, . . . , an) (u ∈ CL(a1, . . . , an)).

Proof. (i) We only prove the inequality[�R(a1, a2, . . . , an)](x)�E(u, x) for u ∈ CR(a1, . . . , an). The

other inequality can be similarly verified, so it is omitted here. Let us takeu ∈ CR(a1, . . . , an) andx ∈ X. Then, by the definition ofCR(a1, . . . , an), ∃{�i | i = 1, . . . , n− 1} ⊆ ORD(◦) such that

u =[

n∏i=1

ai

]R{�i |i=1,...,n−1}

.

By virtue of Corollary2.4, for all{�i | i = 1, . . . , n− 1} ⊆ ORD(◦), we clearly have the inequality

◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

� E

[n−1∏i=1

ai

]R{�i |i=1,...,n−2}

�n−1an, x

= E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, x

.

On the other hand, since(X, ◦) is anM-vague semigroup w.r.t.P andE, we possess from Proposition 3.6that

E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

,

[n∏

i=1

ai

]R{�i |i=1,...,n−1}

= 1, i.e.

E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, x

=E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, x

=E(u, x).

Therefore, we reach the inequality

◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, x

�E(u, x).

Hence, taking supremum over{�i | i = 1, . . . , n− 2} on both sides of this inequality, we get

[�R(a1, a2, . . . , an)](x)�E(u, x).

If ◦ is a perfectM-vague binary operation w.r.t.P andE, then the required equality is straightforwardfrom Theorem4.1(ii).

(ii) SinceCR(a1, . . . , an) ⊆ ker(�R(a1, a2, . . . , an)) andCL(a1, . . . , an) ⊆ ker(�

L(a1, a2, . . . ,

an)), it is enough to show the inclusion

ker(�R(a1, a2, . . . , an)) ∪ ker(�

L(a1, a2, . . . , an)) ⊆ [u]≈E


for all u ∈ CR(a1, . . . , an) ∪ CL(a1, . . . , an). Let z ∈ ker(�R(a1, a2, . . . , an)) ∪

ker(�L(a1, a2, . . . , an)), i.e.

[�R(a1, a2, . . . , an)](z) = 1 or [�L(a1, a2, . . . , an)](z) = 1.

Consider the case[�R(a1, a2, . . . , an)](z) = 1. From (i), we haveE(v, z) = 1 for allv ∈ CR(a1, . . . , an).

It is also clear from Proposition3.6 thatE(v, u) = 1, i.e.E(u, z) = 1, i.e.z ∈ [u]≈E. If [�L

(a1, a2, . . . ,

an)](z) = 1, then by using (i) and Proposition 3.6, we similarly getz ∈ [u]≈E. Thus the required inclusion

follows.(iii) Let ◦ be a perfectM-vague binary operation w.r.t.P and E, andu ∈ CR(a1, . . . , an) (u ∈

CL(a1, . . . , an)). Then considering Theorem 4.1(ii), and by the property (ii), we have

CR(a1, . . . , an) ⊆ ker(�(a1, a2, . . . , an)) ⊆ [u]≈E

and

CL(a1, . . . , an) ⊆ ker(�(a1, a2, . . . , an)) ⊆ [u]≈E.

Thus, to verify the required equalities, it is adequate to show that

[u]≈E⊆ CR(a1, . . . , an) and[u]≈E

⊆ CL(a1, . . . , an).

Let us takez ∈ [u]≈E, i.e.E(u, z) = 1. SinceCR(a1, . . . , an) �= ∅, there exists at least one element

v of CR(a1, . . . , an). From Proposition3.6, it is clear thatE(u, v) = 1. Similar to (i), ∃{�i | i =1, . . . , n− 1} ⊆ ORD(◦) such that

v =[

n∏i=1

ai

]R{�i |i=1,...,n−1}

.

We easily obtain fromE(u, v) = 1andE(u, z) = 1 thatE(v, z) = 1. Owing to the fact that◦ is a perfectM-vague binary operation w.r.t.P andE, we may write

◦[n−1∏

i=1

ai

]R{�i |i=1,...,n−2}

, an, z

=E

[n−1∏i=1

ai

]R{�i |i=1,...,n−2}

�n−1an, z

=E

[ n∏i=1

ai

]R{�i |i=1,...,n−1}

, z

= E(v, z) = 1.


Then, since

◦([n−1∏i=1

ai]R{�i |i=1,...,n−2}, an, z)= 1,

we get from[9, Proposition 2.3(ii)] that∃� ∈ ORD(◦) such that

z =[n−1∏i=1

ai

]R{�i |i=1,...,n−2}

�an ∈ CR(a1, . . . , an).

This gives the inclusion[u]≈E⊆ CR(a1, . . . , an). The inclusion[u]≈E

⊆ CL(a1, . . . , an) can also beshown in a similar fashion to[u]≈E

⊆ CR(a1, . . . , an), and hence the proof is now complete.�

If the underlyingM-vague binary operation◦ is not perfect, the right vague product�R(a1, a2, . . . , an)

does not necessarily coincide with the left vague product�R(a1, a2, . . . , an), but they satisfy the following

relation in general:

Proposition 4.8. For n ∈ N+ anda1, a2, . . . , an ∈ X, the relation

[�R(a1, a2, . . . , an)](x) ∗ [�L

(a1, a2, . . . , an)](y)�E(x, y)

holds for allx, y ∈ X.

Proof. The required inequality is evident forn = 1. Consider the casen�2. For arbitrarily fixedu ∈CR(a1, . . . , an) andv ∈ CL(a1, . . . , an), we have from Theorem4.7(i) that

[�R(a1, a2, . . . , an)](x)�E(u, x) and[�L

(a1, a2, . . . , an)](y)�E(v, y).

Furthermore, by Theorem4.7(ii), we getv ∈ [u]≈E, i.e.E(u, v) = 1, i.e.E(u, x) = E(v, x). Thus

[�R(a1, a2, . . . , an)](x) ∗ [�L

(a1, a2, . . . , an)](y)�E(v, x) ∗ E(v, y)�E(x, y). �

5. Conclusions and future studies

In this paper, starting with vague semigroups, vague products of a finite number of elements in vaguesemigroups and their elementary properties have been introduced. A considerable part of this paperhas been devoted to the representation properties of them and their constructions by means of productsand semigroups in the classical sense and the underlying many-valued equivalence relations. As a naturalentailment of the representation results, vague sums (products) of a finite number of real numbers in vaguearithmetic have been formulated in terms of sums (products) of these real numbers in the usual arithmeticand the underlying many-valued equivalence relations. Furthermore, special attention has been paid tonon-trivial examples for vague sums (products) of a finite number of real numbers in vague arithmeticand their practical realizations.


In an analogous manner to standard arithmetic and classical algebra, vague integral powers (multiples)of real numbers in vague arithmetic can be abstracted to vague integral powers of elements in vaguesemigroups, and they can be defined as a special case of vague products of a finite number of elementsin vague semigroups. The formulations of the properties of vague integral powers of elements in vaguesemigroups need a vague counterpart to the generalized associative law (called vague associative law) invague semigroups. For this reason, accepting the present paper as an outset, a comprehensive study onvague associative law and its applications to vague integral powers of elements in vague semigroups isplanned in a forthcoming paper.

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Products of elements in vague semigroups and their implementations in vague arithmetic