# Rough Operations and Uncertainty Measures on MV-Algebras

Authors: Maosen Xie
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We define a lower approximate operation and an upper approximate operation based on a partition on MV-algebras and discuss their properties. We then introduce a belief measure and a plausibility measure on MV-algebras and investigate the relationship between rough operations and uncertainty measures.


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## Body

## 1. Introduction

The rough set theory, introduced by Pawlak, has been conceived as a tool to conceptualize, organize, and analyze various types of data, in particular, to deal with inexact, uncertain, or vague knowledge in applications related to artificial intelligence. Since then, the subject has been investigated in many studies and various rough set models have been used in machine learning, knowledge discovery, decision support systems, and pattern recognition. In Pawlak’s rough set model, a key concept is an equivalence relation, and given an equivalence relation on a universe, we can define a pair of rough approximations which provide a lower bound and an upper bound for each subset of the universe. Rough approximations can also be defined equivalently by a partition of the universe which is corresponding to the equivalence relation [1–6].Dempster-Shafer theory of evidence is a method developed to model and manipulate uncertain, imprecise, incomplete, and even vague information. It was originated by Dempster’s concept of lower and upper probabilities and extended by Shafer as a theory. The basic representational structure in this theory is a belief structure, which consists of a family of subsets, called focal elements, with associated individual positive weights summing to 1. The fundamental numeric measures derived from the belief structure are a dual pair of belief and plausibility functions [7]. Combining the Dempster-Shafer theory and fuzzy set theory has been suggested to be a way to deal with different kinds of uncertain information in intelligent systems in a number of studies [8–10]. In [11–14], by introducing a pair of dual rough operations on Boolean algebras and using them to interpret some uncertainty measures on Boolean algebras, Bayesian theory and Dempster-Shafer theory are extended to be constructed on Boolean algebras. This provides a more general framework to deal with uncertainty reasoning and a better understanding of both rough operations and uncertainty measures on Boolean algebras.The present paper extends the rough operations and Dempster-Shafer theory with respect to Boolean algebras to MV-algebra. The paper is arranged as follows. In Section2, some results of MV-algebra which will be used in this paper are recollected. In Section 3, we introduce a pair of approximate operations based on a partition on MV-algebra, which is the generation of rough operations on Boolean algebras, and discuss their properties. In Section 4, we define a belief measure and a plausibility measure on MV-algebras and investigate the relationship between rough operations and uncertainty measures. In Section 5 concludes.


## 2. MV-Algebra and Its Partitions

In this section, we recall firstly the basic notions on MV-algebras. See [15–18] for further results on MV-algebras. An MV-algebra M=(M;0,¬,⊕) is an algebra where ⊕ is an associative and commutative binary operation on M having 0 as the neutral element, a unary operation ¬ is involutive with a⊕¬0=¬0 for all a∈M, and moreover the identity a⊕¬(a⊕¬b)=b⊕¬(b⊕¬a) is satisfied for all a,b∈M.A partial order is defined onM by a≤b if and only if ¬a⊕b=1. An additional constant 1 and two binary operations ⊗, →, ∨ and ∧ are defined as follows:
(1)1=¬0,a⊗b=¬(¬a⊕¬b),a⟶b=¬a⊕b,a∨b=(a⊗¬b)⊕b,a∧b=(a⊕¬b)⊗b.IfM is also totally ordered then M is called a totally ordered MV-algebra. An MV-algebra M is called σ-complete if every nonempty countable subset of M has a supremum in M.An MV-subalgebra ofM is a subset M1 of M containing the neutral element 0 of M, closed under the operations of M and endowed with the restriction of these operations to M1.The (finite) Cartesian productM1×M2×⋯×Mk of MV-algebras Mi, endowed with the partial order and the MV-algebra operations defined pointwise, really is also an MV-algebra and will be called product, for short.An elementa∈M satisfying a⊕a=a is called an idempotent element (Boolean element); the set B(M)={a∈M∣a⊕a=a} of all idempotent elements of M endowed with the natural restriction of operations inherited from M is a Boolean algebra.Example 1.
The real unit interval[0,1] with Łukasiewicz operation a→b=RL(a,b)=(1-a+b)∧1, and a⊗b=(a+b-1)∨0, where a,b∈[0,1], is a σ-complete MV-algebra called the MV-unit interval.Example 2.
LetX≠∅, M=[0,1]X, where [0,1] is the MV-unit interval. The order and the operation ⊕,¬ on M are defined in pointwise: for A,B∈M(2)A≤Biffx∈X,A(x)≤B(x),(A⊕B)(x)=A(x)⊗B(x),(¬A)(x)=1-A(x),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkx∈X.
Then M is an MV-algebra called the MV-cube.Example 3.
LetM={0,a,b,c,d,1}. The order and the operations ∨,∧ on M are defined as Figure 1, and the operations ⊗,→ on M are defined as Tables 1 and 2, respectively. Then M is an MV-algebra.Table 1
0
a
b
c
d
1
¬
1
c
d
a
b
0Table 2
⊕
0
a
b
c
d
1
0
0
a
b
c
d
1
a
a
a
1
1
a
1
b
b
1
1
b
1
1
c
c
1
b
c
b
1
d
d
a
1
b
a
1
1
1
1
1
1
1
1Figure 1Proposition 4 (see [15–18]).
LetM be an MV-algebra. For any x,y,z∈M, (P1)
x≤yifandonlyifx→y=1;(P2)
1→x=x,1⊗x=x;(P3)
x⊗y→z=x→(y→z);(P4)
y→y⊗x=x∨¬y;(P5)
x∧y→z=(x→z)∨(y→z);(P6)
(x→y)→y=(y→x)→x=x∨y;(P7)
(M,⊕,0)is a commutative semigroup with unit element 0;(P8)
(⋁i∈Nxi)⊗a=⋁i∈N(xi⊗a),
if the supremum exists in equality;(P9)
a⊕(⋀i∈Nyi)=⋀i∈N(a⊕yi),
if the infimum exists in equality.Two elementsa and b in an MV-algebra M are called orthogonal (denoted as a⊥b) if a≤¬b. Obviously, a⊥b if and only if a⊗b=0. A finite subset ξ={a1,a2,…,an} of elements of an MV-algebra M is said to be ⊕-orthogonal if ∀I⊂{1,2,…,n}, ⨁i∈Iai⊥aj, j∈{1,2,…,n}-I.Definition 5.
A finite collectionξ={a1,a2,…,an} of nonneutral elements of an MV-algebra M is said to be a partition of M if and only if(i)
ξ is ⊕-orthogonal;(ii)
⋁i=1nai=1.Lemma 6.
Every element in partitionξ={a1,a2,…,an} is idempotent element; that is, ∀ai∈ξ, ai⊗ai=ai, ai⊕ai=ai.Proof.
∀
a
i
∈
ξ, we have ai=ai⊗1=ai⊗⋁j=1naj=⋁j=1n(ai⊗aj)=ai⊗ai, and it also means that ai→¬ai=¬ai. It follows that ai∧¬ai=ai⊗(ai→¬ai)=ai⊗¬ai=0. By (ai⊕ai)⊗¬ai=¬ai⊗(¬ai→ai)=¬ai∧ai=0 we know ai⊕ai≤¬ai→0=ai. Obviously, ai≤ai⊕ai. Hence, ai⊕ai=ai.Theorem 7.
Letξ={a1,a2,…,an} be a partition of M. Suppose that
(3)M0={suchthatx=⨁k∈Ixakx∣thereexistfiniteelements{ai1,…,ail}⊆ξsuchthatx=⨁k∈Ixak}∪{0},
where Ix={i1,…,il}⊆{1,2,…,n}. Then M0 is a Boolean algebra.Proof.
At first, we prove that¬x=⨁k∈{1,2,…,n}-Ixak for x=⨁k∈Ixak∈M0, where Ix={i1,…,il}⊆{1,2,…,n}. From x⊕⨁k∈{1,2,…,n}-Ixak=⨁k∈{1,2,…,n}ak=1 we know ¬x≤⨁k∈{1,2,…,n}-Ixak. On other hand, from ai⊗⨁k∈{1,2,…,n}-Ixak=0 we know ⨁k∈{1,2,…,n}-Ixak≤¬ai for any i∈Ix. Hence, ⨁k∈{1,2,…,n}-Ixak≤⋀i∈Ix¬ai=¬(⋁i∈Ixai)≤¬(⨁i∈Ixai)=¬x. This proves that ¬x=⨁k∈{1,2,…,n}-Ixak. Hence, ¬x∈M0 for every x∈M0.
Secondly, we prove thatM0 is closed under ∧ and ∨. Let x=⨁k∈Ixak, y=⨁i∈Iyai∈M0. Note that ai∧aj=(ai⊕¬aj)⊗aj=(ai⊕⨁k∈{1,2,…,n}-{j}ak)⊗aj=(⨁k∈{1,2,…,n}-{j}ak)⊗aj=¬aj⊗aj=0 for i,j∈{1,2,…,n} and i≠j. If Ix∩Iy≠∅ then x∧y=(⨁k∈Ixak)∧(⨁i∈Iyai)=⨁k∈Ix,i∈Iy(ak∧ai)=⨁k∈Ix∩Iyak or else x∧y=0. Hence, x∧y∈M0 for all x,y∈M0. By the duality of ∧ and ∨ we know x∨y∈M0 for all x,y∈M0.
Finally, from the above proof we easily obtain thatx∧¬x=0 and x∨¬x=1. Therefore, M0 is a Boolean algebra.Example 8.
LetM=[0,1]X be the MV-cube in Example 2 and {A1,A2,…,An} be a partition of X in the usual sense. Denote
(4)χAi(x)={1,x∈Ai,0,x∉Ai.
Then ξ={χA1,χA2,…,χAn} is a partition of M.Example 9.
LetM be the MV-algebra in Example 3. Then {a,c} is a partition of M.Example 10.
LetM be the MV-algebra in Example 3 and B={0,1} the classic Boolean algebra. Then M×B is an MV-algebra, and {(a,0),(c,0),(0,1)} is a partition of M×B.


## 3. Approximate Operations on MV-Algebras

In Pawlak’s rough set theory, a rough set is induced by a partition of the universe. In this section, we will extend Pawlak’s rough set theory by defining a pair of approximate operations induced by a partition of the unity of an MV-algebra.Definition 11.
LetM be an MV-algebra and ξ={a1,a2,…,an} a partition of M. Then a pair of operations Hξ:M→M and Lξ:M→M, such that
(5)Hξ(x)=∨{ai∣ai∈ξ,ai⊗x≠0},Lξ(x)=∨{ai∣ai∈ξ,ai⟶x≠1}⟶0,
are called a lower approximation and an upper approximation based on partition ξ, respectively. If Hξ(x)=Lξ(x) then x is called a definable element with respect to (M,ξ) else x is called a rough element. If no confusion arisen then the operations Hξ and Lξ can be abbreviated as H and L, respectively.Remark 12.
LetM be a Boolean algebra of some subsets of a nonempty set U and let  ξ be a partition of U in the usual sense. In this case, the conditions ai⊗x≠0 and ai→x≠1 in Definition 11 identity with that ai∩x≠∅ and ai⊆x do not hold in the usual set meaning, respectively. Hence,
(6)H(x)=∨{ai∣ai∈ξ,ai⊗x≠0}=∪{ai∣ai∈ξ,ai∩x≠∅},L(x)=∨{ai∣ai∈ξ,ai⟶x≠1}⟶0=M-∪{ai∣ai∈ξ,ai⊆xdoesnothold}=∪{ai∣ai∈ξ,ai⊆x}.
This means that the operations H and L introduced for MV-algebras in Definition 11 are extensions of the operations H and L in the typical set notations, respectively.Theorem 13.
If we denote¬ξ={¬a1,¬a2,…,¬an}, then
(7)L(x)=∧{ai∣ai∈¬ξ,x⊕ai≠1}.Proof.
Consider the following:(8)L(x)=∨{ai∣ai∈ξ,ai⟶x≠1}⟶0=∧{ai⟶0∣ai∈ξ,ai⟶x≠1}=∧{¬ai∣ai∈ξ,x⊕¬ai≠1}=∧{¬ai∣¬ai∈¬ξ,x⊕¬ai≠1}=∧{ai∣ai∈¬ξ,x⊕ai≠1}.Theorem 14.
Letξ={a1,a2,…,an} be a partition of MV-algebra M and L and H the lower and upper approximations induced by ξ, respectively. Then(1)
L(1)=1, H(0)=0;(2)
∀ai∈ξ, L(ai)=ai, H(ai)=ai;(3)
ifx≤y then L(x)≤L(y), H(x)≤H(y);(4)
L(x)=¬(H(¬x));(5)
L(x)≤x≤H(x);(6)
L(x∧y)=L(x)∧L(y), H(x)∨H(y)=H(x∨y);(7)
H(H(x))=H(x), L(L(x))=L(x);(8)
H(L(x))=L(x), L(H(x))=H(x).Proof.
The proof of (1), (2), and (3) is obvious as follows.
(4)¬(H(¬x))=∨{ai∣ai∈ξ,ai⊗¬x≠0}→0=∨{ai∣ai∈ξ,ai→x≠1}→0=L(x).
(5)x=x⊗∨{ai∣ai∈ξ}=∨{ai⊗x∣ai∈ξ}=∨{ai⊗x∣ai∈ξ,ai⊗x≠0}≤∨{ai∣ai∈ξ,ai⊗x≠0}=H(x).
By the duality ofL and H, we have L(x)≤x≤H(x).
(6) By (3) we haveH(x)∨H(y)≤H(x∨y). For any ai∈ξ, if ai⊗(x∨y)≠0 then it follows from ai⊗(x∨y)=(ai⊗x)∨(ai⊗y) that ai⊗x≠0 or ai⊗y≠0. Hence,
(9){ai∣ai∈ξ,ai⊗(x∨y)≠0}⊆{ai∣ai∈ξ,ai⊗x≠0}∪{ai∣ai∈ξ,ai⊗y≠0}.
Thus,
(10)∨{ai∣ai∈ξ,ai⊗(x∨y)≠0}≤(∨{ai∣ai∈ξ,ai⊗x≠0})∨(∨{ai∣ai∈ξ,ai⊗y≠0}).
This shows that H(x∨y)≤H(x)∨H(y). Therefore, H(x)∨H(y)=H(x∨y).
By the duality ofL and H we know that another equation holds.
(7) By (3) and (5) we haveH(x)≤H(H(x)). For any aj∈ξ,
(11)aj⊗H(x)=aj⊗∨{ai∣ai∈ξ,ai⊗x≠0}=∨{aj⊗ai∣ai∈ξ,ai⊗x≠0}≠0,
if and only if there exists ai0∈ξ and ai0⊗x≠0 such that aj⊗ai0≠0. Hence, {aj∈ξ,aj⊗H(x)≠0}⊆{ai∈ξ,ai⊗x≠0}. It follows that
(12)H(H(x))=∨{aj∣aj∈ξ,aj⊗H(x)≠0}≤∨{ai∣ai∈ξ,ai⊗x≠0}=H(x).
This shows that H(H(x))=H(x).
(8) By (3) we haveL(x)≤H(L(x)). If aj0∈{aj∣aj∈ξ,aj⊗L(x)≠0}, then aj0⊗L(x)=aj0⊗∧{¬ai∣ai∈ξ,ai→x≠1}≠0. It follows from aj0⊗¬aj0=0 that ¬aj0∉{¬ai∣ai∈ξ,ai→x≠1}. Since ξ is a partition we have aj0≤¬ai(i≠j0). Hence, aj0≤∧{¬ai∣ai∈ξ,ai→x≠1}. This means that ∨{aj∣aj∈ξ,aj⊗L(x)≠0}≤∧{¬ai∣ai∈ξ,ai→x≠1}. This shows that H(L(x))≤L(x). Therefore, H(L(x))=L(x).Example 15.
Let the MV-algebraM and the partition of M be as defined in Example 9. Then H(b)=1, H(d)=a; L(b)=c, L(d)=0.Example 16.
The MV-algebraM×B and the partition ξ={(a,0),(c,0),(0,1)} of M×B as defined in Example 10. Then H((1,0))=(1,0),H((a,1))=(a,1),H((b,1))=(1,1),H((b,0))=(1,0),H((c,1))=(c,1),H((d,1))=(a,1),H((d,0))=(a,0);   L((1,0))=(1,0),L((a,1))=(a,1),L((b,1))=(c,1),L((b,0))=(c,0),L((c,1))=(c,1),L((d,1))=(0,1),L((d,0))=(0,0).
By Definition11 we know that every element in ξ is definable and the definable elements of M can also be obtained by the following theorem.Theorem 17.
LetM be an MV-algebra and ξ a partition of M. The definable elements of M with respect to (M,ξ) are
(13)L*={x∈M∣∀ai∈ξ,ai⊗x≠0impliesai≤x}={x∈M∣∀ai∈ξ,ai⊗x=0orai≤x},
and L* forms a Boolean algebra.Proof.
At first, we prove that the definable elements set with respect to(M,ξ) are L*. Suppose that x is a definable element. Then x=H(x)=∨{ai∣ai∈ξ,ai⊗x≠0}. ∀ai∈ξ, if ai⊗x≠0 then ai≤H(x)=x. This means that x∈L*. Conversely, let x∈L*. Then ai⊗x≠0 implies ai≤x. It follows that H(x)=∨{ai∣ai∈ξ,ai⊗x≠0}≤x. By Theorem 14(4) we have H(x)=x.
Sincex is a definable element we also know x=L(x)=∧{ai∣ai∈¬ξ,x⊕ai≠1}. Hence, for any ai∈¬ξ, if x⊕ai≠1 then x≤ai. Let ai∈ξ and ai⊗x≠0. Then ¬ai⊕¬x≠1. Hence, ¬x≤¬ai; that is ai≤x. This means that x∈L*. Conversely, let x∈L*. Then ai⊗x≠0 implies ai≤x. If ai∈¬ξ and x⊕ai≠1, then ¬ai⊗¬x≠0. Hence, ¬ai≤x does not hold. By ¬ai∈ξ and the definition of L* we know ¬ai⊗x=0; hence, x≤ai. This shows that x≤∧{ai∣ai∈¬ξ,x⊕ai≠1}=L(x). By Theorem 14(4) we have L(x)=x.
This shows that the definable elements set with respect to(M,ξ) are L*.
Next we prove thatL* forms a Boolean algebra. Let x1,x2∈L*. For any ai∈ξ, if ai⊗(x1∧x2)=(ai⊗x1)∧(ai⊗x2)≠0, then ai⊗x1≠0 and ai⊗x2≠0. By the definition of L* we know ai≤x1 and ai≤x2; that is ai≤x1∧x2. Hence, x1∧x2∈L*. This shows that L* is closed under operation ∧. If ai⊗(x1∨x2)=(ai⊗x1)∨(ai⊗x2)≠0, then ai⊗x1≠0 or ai⊗x2≠0. By the definition of L* we know ai≤x1 or ai≤x2; that is ai≤x1∨x2. Hence, x1∨x2∈L*. This shows that L* is closed under operation ∨. This shows that L* forms a sublattice of M.
By Theorem14 we know 0,1∈L*. Let x∈L*. Then ∀ai∈ξ, ai⊗x=0 or ai≤x. This means that ∀ai∈ξ, ai⊗x=0 or ai⊗¬x=0. It follows from ∀ai∈ξ, ai⊗¬x=0or ai⊗¬¬x=ai⊗x=0 that ¬x∈L*. Since ∀ai∈ξ, ai⊗x=0 or ai⊗¬x=0, we have ∀ai∈ξ, (ai⊗x)∧(ai⊗¬x)=ai⊗(x∧¬x)=0. Hence, 0=⋁i=1n(ai⊗(x∧¬x))=(⋁i=1nai)⊗(x∧¬x)=1⊗(x∧¬x)=x∧¬x. Analogously, 1=x∨¬x.
This shows thatL* is a Boolean algebra.


## 4. Approximate Operations and Uncertainty Measures

In this section, we will discuss the relationship between rough operations and uncertainty measures. Given an MV-algebra, we may only know the measures of some elements when information is absent in an MV-algebra. For those elements that we do not know the measure, what we can do is to define belief measure and plausibility measure on them.Definition 18.
The functionm:M→[0,1] such that(1)
m(0)=0,(2)
m(1)=1,(3)
m is finitely additive, that is, if ξ={a1,a2,…,an} is ⊕-orthogonal then m(⨁k=1nak)=∑k=1n‍m(ak)is called a finitely additive measure on MV-algebra M.For finitely additive measure, we have the following conclusion.Theorem 19 (see [19, 20]).
Ifm is a finitely additive measure on the MV-algebra M then(1)
ifa≤b then m(b⊖a)=m(b)-m(a), where b⊖a=b⊗¬a;(2)
ifa≤b then m(a)≤m(b);(3)
m(a)+m(b)=m(a⊕b)+m(a⊗b);(4)
m(a)+m(b)=m(a∨b)+m(a∧b).Example 20.
LetM be an MV-algebra and [0,1] the MV-unit interval. If a mapping v:M→[0,1] is a homomorphism of type (¬,⊕), that is, v(¬a)=¬v(a), v(a⊕b)=v(a)⊕v(b), for any a,b∈M, then v is called a Łukasiewicz-valuation (see [21]). Then it is easy to prove that the Łukasiewicz-valuation v is a finitely additive measure.Example 21.
LetM be an MV-algebra, let ξ={a1,a2,…,an} be a partition of M, let M0 be the smallest subalgebra of M containing ξ, and let Ω0 be the set of all homomorphisms of type (⊕,¬) from M0 to the MV-unit interval [0,1]. An element x in M0 can be viewed as a function from Ω0 to [0,1]; that is, for any x∈M0, a function x:Ω0→[0,1], x(v)=v(x) can be defined. Suppose that (Ω0,A,μ) is a probability measure space, which satisfies the fact that x is a A-measurable function on Ω0; that is, ∀α∈[0,1],{v∈Ω0∣x(v)≥α}∈A for every x∈M0. In this case, the element x can also be viewed as a random variable from Ω0 to [0,1] (see [21]). Hence, we can define m(x)=∫Ω0‍xdμ for every x∈M0. It is easy to prove that m:M0→[0,1] is a finitely additive measure.Example 22.
LetM be an MV-algebra. A state m on a σ-complete MV-algebra M is a mapping m:M→[0,1] such that for all a,b,an∈M:(1)
m(1)=1,(2)
ifa⊗b=0 then m(a⊕b)=m(a)+m(b),(3)
ifan↗a then m(an)↗m(a),where an↗a stands for an is a nondecreasing sequence and a=⋁n∈Nan (see [20, 21]). Then a state m is a finitely additive measure by the following Lemma 23.Lemma 23.
For any⊕-orthogonal subset ξ={a1,a2,…,an} and any state m of M, it holds that
(14)m(⨁i=1nai)=∑i=1n‍m(ai).Proof.
Sinceξ={a1,a2,…,an} is ⊕-orthogonal we know (⨁i∈Iai)⊗aj=0. From the definition of a state, we have inductively
(15)m(⨁i=1nai)=∑i=1n‍m(ai).Proposition 24.
Letξ={a1,a2,…,an} be a partition of MV-algebra M and let M0 be the Boolean algebra defined in Theorem 7. If function m:ξ→[0,1] satisfies ∑i=1n‍m(ai)=1 then we can extend m to M0 by defining m(0)=0, m(x)=m(⨁k∈Ixak)=∑k∈Ix‍m(ak), for every x=⨁k∈Ixak∈M0, and m is a finitely additive measure on M0.Proof.
Obviouslym(0)=0 and m(1)=1. For x,y∈M0 then there are Ix,Iy⊆{1,2,…,n} such that x=⨁k∈Ixak and y=⨁k∈Iyak. If x⊗y=0 then we assert Ix∩Iy=∅. In fact, if Ix∩Iy≠∅ then there is i0∈Ix∩Iy such that x⊗y≥ai0⊗ai0≠0. Hence, m(x⊕y)=m(⨁k∈Ixak⊕⨁k∈Iyak)=m(⨁k∈Ix∪Iyak)=∑k∈Ix∪Iy‍m(ak). This shows that m is a finitely additive measure on M0.Definition 25.
(1) A functionBel:M→[0,1] is called a belief measure if Bel(0)=0, Bel(1)=1, and
(16)Bel(x1∨x2∨⋯∨xl)≥∑‍{(-1)|J|+1Bel(⋀j∈Jxj)∣∅≠J⊆{1,2,…,l}},
for every positive integer l and for every l-tuple x1,x2,…,xl, of subsets of M.
(2) A functionPl:M→[0,1] is called a plausibility measure if Pl(0)=0, Pl(1)=1, and
(17)Pl(x1∧x2∧⋯∧xl)≤∑‍{(-1)|J|+1Pl(⋁j∈Jxj)∣∅≠J⊆{1,2,…,l}},
for every positive integer l and for every l-tuple x1,x2,…,xl, of subsets of M.Theorem 26.
Suppose thatξ={a1,a2,…,an} is a partition of MV-algebra M and function m:ξ→[0,1] satisfies ∑i=1n‍m(ai)=1. By using function m we define a function Pl:M→[0,1] on M as follows:
(18)Pl(x)=∑‍{m(ai)∣ai∈ξ,ai⊗x≠0},x∈M.
Then Pl is a plausibility measure on M.Proof.
It is easy to checkPl(0)=0, Pl(1)=1, and 0≤Pl(x)≤1. For any x1,x2,…,xl∈M, it is easy to check
(19){ai∣ai⊗(⋀j=1lxj)≠0}={ai∣⋀j=1l(ai⊗xj)≠0}⊆⋂j=1l‍{ai∣ai⊗xj≠0}.
By the well-known inclusion-exclusion formulas in probability theory we have
(20)Pl(⋀j=1lxj)=∑‍{m(ai)∣ai⊗(⋀j=1lxj)≠0}≤∑‍{m(ai)∣ai∈⋂j=1l‍{ai∣ai⊗xj≠0}}=∑‍{(∑‍{m(ai)∣∃j∈J,ai⊗xj≠0})(-1)|J|+1×(∑‍{m(ai)∣∃j∈J,ai⊗xj≠0})∣∅≠J⊆{1,2,…,l}(∑‍{m(ai)∣∃j∈J,ai⊗xj≠0})}=∑‍{(-1)|J|+1×(∑‍{m(ai)∣⋁j∈J(ai⊗xj)≠0})∣∅≠J⊆{1,2,…,l}(-1)|J|+1}=∑‍{(-1)|J|+1Pl(⋁j∈Jxj)∣∅≠J⊆{1,2,…,l}}.
This shows that Pl is a plausibility measure on M.Theorem 27.
Suppose thatξ={a1,a2,…,an} is a partition of MV-algebra M and function m:ξ→[0,1] satisfies ∑i=1n‍m(ai)=1. By using function m we define a function Pl:M→[0,1] on M as follows:
(21)Bel(x)=1-∑‍{m(ai)∣ai∈ξ,ai⟶x≠1}.
Then Bel is a belief measure on M.Proof.
It is easy to checkBel(0)=0, Bel(1)=1, and 0≤Bel(1)≤1. For any x1,x2,…,xl∈M, it is easy to check
(22){ai∣ai⟶⋁j=1lxj≠1}={ai∣⋁j=1l(ai⟶xj)≠1}⊆⋂j=1l‍{ai∣ai⟶xj≠1}.
By the well-known inclusion-exclusion formulas in probability theory we have
(23)Bel(⋁j=1lxi)=1-∑‍{m(ai)∣ai⟶⋁j=1lxi≠1}≥1-∑‍{m(ai)∣ai∈⋂j=1l‍{ai∣ai⟶xj≠1}}=1-∑‍{(∑‍{m(ai)∣∃j∈J,ai⟶xj≠1})(-1)|J|+1×(∑‍{m(ai)∣∃j∈J,ai⟶xj≠1})∣∅≠J⊆{1,2,…,l}(∑‍{m(ai)∣∃j∈J,ai⟶xj≠1})}=1-∑‍{(-1)|J|+1×(∑‍{m(ai)∣ai⟶⋀j∈Jxj≠1})∣∅≠J⊆{1,2,…,l}(-1)|J|+1}=1-∑‍{(1-Bel(⋀j∈Jxj))∣∅≠J⊆{1,2,…,l}(-1)|J|+1×(1-Bel(⋀j∈Jxj))∣∅≠J⊆{1,2,…,l}}=1-∑‍{(-1)|J|+1∣∅≠J⊆{1,…,l}}+∑‍{(-1)|J|+1Bel(⋀j∈Jxj)∣∅≠J⊆{1,…,l}}=∑‍{(-1)|J|+1Bel(⋀j∈Jxj)∣∅≠J⊆{1,2…,l}}.
This shows that Bel is a belief measure on M.LetM be an MV-algebra and ξ={a1,a2,…,an} a partition of M. Suppose that function m:ξ→[0,1] satisfies ∑i=1n‍m(ai)=1. Then we can extend m to M by defining
(24)m*(x)=Bel(L(x)),m*(x)=Pl(H(x)),
where x∈M, L(x), and H(x) are lower and upper approximations of x, respectively. The values m*(x) and m*(x) of an element x of M can be viewed as our best estimate of the true measures of x, given our lack of knowledge. Moreover, we can get the following theorem.Theorem 28.
m
* and m* defined above are belief measure and plausibility measure on M, respectively.Proof.
(1) We provePl(H(x))=Pl(x). Note that Pl(H(x))=∑‍{m(ai)∣ai∈ξ,ai⊗H(x)≠0}. In the following we prove that ai⊗H(x)≠0 if and only if ai⊗x≠0. In fact, ai⊗H(x)=ai⊗∨{aj∣aj∈ξ,aj⊗x≠0}=∨{ai⊗aj∣aj∈ξ,aj⊗x≠0}={ai⊗ai∣ai⊗x≠0}. This shows that ai⊗H(x)≠0 if and only if ai⊗x≠0. Hence, m*(x)=Pl(H(x))=∑‍{m(ai)∣ai∈ξ,ai⊗H(x)≠0}=∑‍{m(ai)∣ai∈ξ,ai⊗x≠0}=Pl(x). It follows from Theorem 26 that m* is a plausibility measure on M.
(2) We proveBel(L(x))=Bel(x). Note that Bel(L(x))=1-∑‍{m(ai)∣ai∈ξ,ai→L(x)≠1}. In the following we prove that ai→L(x)≠1 if and only if ai→x≠1. In fact, ai→L(x)=ai→(∨{aj∣aj∈ξ,aj→x≠1}→0)=ai⊗∨{aj∣aj∈ξ,aj→x≠1}→0=∨{ai⊗aj∣aj∈ξ,aj→x≠1}→0={ai⊗ai∣ai→x≠1}→0. This shows that ai→L(x)≠1 if and only if ai→x≠1. Hence, m*(x)=Bel(H(x))=1-∑‍{m(ai)∣ai∈ξ,ai→L(x)≠1}=1-∑‍{m(ai)∣ai∈ξ,ai→x≠1}=Bel(x). It follows from Theorem 27 that m* is a belief measure on M.Example 29.
The MV-algebraM×B and the partition ξ={(a,0),(c,0),(0,1)} of M×B as defined in Example 10. Suppose that function m:ξ→[0,1], m((a,0))=1/4, m((c,0))=1/4, m((0,1))=1/2. The uncertainty measures Pl,Bel,m*,m* on M with respect to ξ are shown in Table 3.Table 3
x
(
1,1
)
(
1,0
)
(
a
,
1
)
(
a
,
0
)
(
b
,
1
)
(
b
,
0
)
(
c
,
1
)
(
c
,
0
)
(
d
,
1
)
(
d
,
0
)
(
0,1
)
(
0,0
)
Pl
(
x
)
1
1
/
2
3
/
4
1
/
4
1
1
/
2
3
/
4
1
/
4
3
/
4
1
/
4
1
/
2
0
Bel
(
x
)
1
1
/
2
3
/
4
1
/
4
3
/
4
1
/
4
3
/
4
1
/
4
1
/
2
0
1
/
2
0
m
*
(
x
)
1
1
/
2
3
/
4
1
/
4
1
1
/
2
3
/
4
1
/
4
3
/
4
1
/
4
1
/
2
0
m
*
(
x
)
1
1
/
2
3
/
4
1
/
4
3
/
4
1
/
4
3
/
4
1
/
4
1
/
2
0
1
/
2
0


## 5. Conclusion

In this paper, a pair of dual rough operations on MV-algebras is introduced. The properties of rough operations and the relationship between rough operations and uncertainty measures are discussed. If information is absent in an MV-algebra and we may only know the measures of some elements, then what we can do is to define belief measure and plausibility measure on MV-algebra, which is used to interpret some uncertainty measures on MV-algebras.



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*Source: 597650-2014-06-11.xml*