Rough Operations and Uncertainty Measures on MV-Algebras

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.

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 Section 2, 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 on 𝑀 by 𝑎≤𝑏 if and only if ¬𝑎⊕𝑏=1. An additional constant 1 and two binary operations ⊗, →, ∨ and ∧ are defined as follows: ()()()1=¬0,𝑎⊗𝑏=¬¬𝑎⊕¬𝑏,𝑎⟶𝑏=¬𝑎⊕𝑏,𝑎∨𝑏=𝑎⊗¬𝑏⊕𝑏,𝑎∧𝑏=𝑎⊕¬𝑏⊗𝑏. (1)

If 𝑀 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 of 𝑀 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 product 𝑀₁×𝑀₂×⋯×𝑀_𝑘 of MV-algebras 𝑀_𝑖, 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 element 𝑎∈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=𝑅_𝐿(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. Let 𝑋≠0/, 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(𝑥)(𝑥),(𝑥)(𝑥)(𝑥),(𝑥)(𝑥),𝐴≤𝐵iff𝑥∈𝑋,𝐴≤𝐵𝐴⊕𝐵)(=𝐴⊗𝐵¬𝐴)(=1−𝐴𝑥∈𝑋. (2) Then 𝑀 is an MV-algebra called the MV-cube.

Example 3. Let 𝑀={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 

Table 2 

Figure 1 

Proposition 4 (see [15–18]). Let 𝑀 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∈𝑁𝑥_𝑖)⊗𝑎=⋁_𝑖∈𝑁(𝑥_𝑖⊗𝑎), if the supremum exists in equality;(P9)𝑎⊕(⋀_𝑖∈𝑁𝑦_𝑖)=⋀_𝑖∈𝑁(𝑎⊕𝑦_𝑖), if the infimum exists in equality.

Two elements 𝑎 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,…,𝑎_𝑛} of elements of an MV-algebra 𝑀 is said to be ⊕-orthogonal if ∀𝐼⊂{1,2,…,𝑛}, ⨁_𝑖∈𝐼𝑎_𝑖⟂𝑎_𝑗, 𝑗∈{1,2,…,𝑛}−𝐼.

Definition 5. A finite collection 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} 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)⋁^𝑛_𝑖=1𝑎_𝑖=1.

Lemma 6. Every element in partition 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} is idempotent element; that is, ∀𝑎_𝑖∈𝜉, 𝑎_𝑖⊗𝑎_𝑖=𝑎_𝑖, 𝑎_𝑖⊕𝑎_𝑖=𝑎_𝑖.

Proof. ∀𝑎_𝑖∈𝜉, we have 𝑎_𝑖=𝑎_𝑖⊗1=𝑎_𝑖⊗⋁^𝑛_𝑗=1𝑎_𝑗=⋁^𝑛_𝑗=1(𝑎_𝑖⊗𝑎_𝑗)=𝑎_𝑖⊗𝑎_𝑖, and it also means that 𝑎_𝑖→¬𝑎_𝑖=¬𝑎_𝑖. It follows that 𝑎_𝑖∧¬𝑎_𝑖=𝑎_𝑖⊗(𝑎_𝑖→¬𝑎_𝑖)=𝑎_𝑖⊗¬𝑎_𝑖=0. By (𝑎_𝑖⊕𝑎_𝑖)⊗¬𝑎_𝑖=¬𝑎_𝑖⊗(¬𝑎_𝑖→𝑎_𝑖)=¬𝑎_𝑖∧𝑎_𝑖=0 we know 𝑎_𝑖⊕𝑎_𝑖≤¬𝑎_𝑖→0=𝑎_𝑖. Obviously, 𝑎_𝑖≤𝑎_𝑖⊕𝑎_𝑖. Hence, 𝑎_𝑖⊕𝑎_𝑖=𝑎_𝑖.

Theorem 7. Let 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} be a partition of 𝑀. Suppose that 𝑀₀=𝑥∣𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑓𝑖𝑛𝑖𝑡𝑒𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠𝑎_𝑖1,…,𝑎_𝑖𝑙⊆𝜉𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑥=_{𝑘∈𝐼𝑥}𝑎_𝑘∪{0}, (3) where 𝐼_𝑥={𝑖₁,…,𝑖_𝑙}⊆{1,2,…,𝑛}. Then 𝑀₀ is a Boolean algebra.

Proof. At first, we prove that ¬𝑥=⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘 for 𝑥=⨁_{𝑘∈𝐼𝑥}𝑎_𝑘∈𝑀₀, where 𝐼_𝑥={𝑖₁,…,𝑖_𝑙}⊆{1,2,…,𝑛}. From 𝑥⊕⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘=⨁_{𝑘∈{1,2,…,𝑛}}𝑎_𝑘=1 we know ¬𝑥≤⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘. On other hand, from 𝑎_𝑖⊗⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘=0 we know ⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘≤¬𝑎_𝑖 for any 𝑖∈𝐼_𝑥. Hence, ⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘≤⋀_{𝑖∈𝐼𝑥}¬𝑎_𝑖=¬(⋁_{𝑖∈𝐼𝑥}𝑎_𝑖)≤¬(⨁_{𝑖∈𝐼𝑥}𝑎_𝑖)=¬𝑥. This proves that ¬𝑥=⨁_{𝑘∈{1,2,…,𝑛}−𝐼𝑥}𝑎_𝑘. Hence, ¬𝑥∈𝑀₀ for every 𝑥∈𝑀₀.
Secondly, we prove that 𝑀₀ is closed under ∧ and ∨. Let 𝑥=⨁_{𝑘∈𝐼𝑥}𝑎_𝑘, 𝑦=⨁_{𝑖∈𝐼𝑦}𝑎_𝑖∈𝑀₀. Note that 𝑎_𝑖∧𝑎_𝑗=(𝑎_𝑖⊕¬𝑎_𝑗)⊗𝑎_𝑗=(𝑎_𝑖⊕⨁_{𝑘∈{1,2,…,𝑛}−{𝑗}}𝑎_𝑘)⊗𝑎_𝑗=(⨁_{𝑘∈{1,2,…,𝑛}−{𝑗}}𝑎_𝑘)⊗𝑎_𝑗=¬𝑎_𝑗⊗𝑎_𝑗=0 for 𝑖,𝑗∈{1,2,…,𝑛} and 𝑖≠𝑗. If 𝐼_𝑥∩𝐼_𝑦≠0/ then 𝑥∧𝑦=(⨁_{𝑘∈𝐼𝑥}𝑎_𝑘)∧(⨁_{𝑖∈𝐼𝑦}𝑎_𝑖)=⨁_{𝑘∈𝐼𝑥,𝑖∈𝐼𝑦}(𝑎_𝑘∧𝑎_𝑖)=⨁_{𝑘∈𝐼𝑥∩𝐼𝑦}𝑎_𝑘 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 that 𝑥∧¬𝑥=0 and 𝑥∨¬𝑥=1. Therefore, 𝑀₀ is a Boolean algebra.

Example 8. Let 𝑀=[0,1]^𝑋 be the MV-cube in Example 2 and {𝐴₁,𝐴₂,…,𝐴_𝑛} be a partition of 𝑋 in the usual sense. Denote 𝜒_𝐴𝑖(𝑥)=1,𝑥∈𝐴_𝑖,0,𝑥∉𝐴_𝑖. (4) Then 𝜉={𝜒_𝐴1,𝜒_𝐴2,…,𝜒_𝐴𝑛} is a partition of 𝑀.

Example 9. Let 𝑀 be the MV-algebra in Example 3. Then {𝑎,𝑐} is a partition of 𝑀.

Example 10. Let 𝑀 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. Let 𝑀 be an MV-algebra and 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} a partition of 𝑀. Then a pair of operations 𝐻_𝜉∶𝑀→𝑀 and 𝐿_𝜉∶𝑀→𝑀, such that 𝐻_𝜉(𝑥)=∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖,𝐿⊗𝑥≠0_𝜉(𝑥)=∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⟶𝑥≠1⟶0, (5) are called a lower approximation and an upper approximation based on partition 𝜉, respectively. If 𝐻_𝜉(𝑥)=𝐿_𝜉(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 𝐻_𝜉 and 𝐿_𝜉 can be abbreviated as 𝐻 and 𝐿, respectively.

Remark 12. Let 𝑀 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 𝑎_𝑖⊗𝑥≠0 and 𝑎_𝑖→𝑥≠1 in Definition 11 identity with that 𝑎_𝑖∩𝑥≠0/ and 𝑎_𝑖⊆𝑥 do not hold in the usual set meaning, respectively. Hence, 𝐻(𝑥)=∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖𝑎⊗𝑥≠0=∪_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖∩𝑥≠0/,𝐿(𝑥)=∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖𝑎⟶𝑥≠1⟶0=𝑀−∪_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖𝑎⊆𝑥doesnothold=∪_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊆𝑥. (6) 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,…,¬𝑎_𝑛}, then 𝐿(𝑥)=∧𝑎_𝑖∣𝑎_𝑖∈¬𝜉,𝑥⊕𝑎_𝑖≠1. (7)

Proof. Consider the following: 𝐿(𝑥)=∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖𝑎⟶𝑥≠1⟶0=∧_𝑖⟶0∣𝑎_𝑖∈𝜉,𝑎_𝑖⟶𝑥≠1=∧¬𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑥⊕¬𝑎_𝑖≠1=∧¬𝑎_𝑖∣¬𝑎_𝑖∈¬𝜉,𝑥⊕¬𝑎_𝑖𝑎≠1=∧_𝑖∣𝑎_𝑖∈¬𝜉,𝑥⊕𝑎_𝑖≠1. (8)

Theorem 14. Let 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} 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)∀𝑎_𝑖∈𝜉, 𝐿(𝑎_𝑖)=𝑎_𝑖, 𝐻(𝑎_𝑖)=𝑎_𝑖;(3)if 𝑥≤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))=∨{𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗¬𝑥≠0}→0=∨{𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}→0=𝐿(𝑥).
(5) 𝑥=𝑥⊗∨{𝑎_𝑖∣𝑎_𝑖∈𝜉}=∨{𝑎_𝑖⊗𝑥∣𝑎_𝑖∈𝜉}=∨{𝑎_𝑖⊗𝑥∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}≤∨{𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}=𝐻(𝑥).
By the duality of 𝐿 and 𝐻, we have 𝐿(𝑥)≤𝑥≤𝐻(𝑥).
(6) By (3) we have 𝐻(𝑥)∨𝐻(𝑦)≤𝐻(𝑥∨𝑦). For any 𝑎_𝑖∈𝜉, if 𝑎_𝑖⊗(𝑥∨𝑦)≠0 then it follows from 𝑎_𝑖⊗(𝑥∨𝑦)=(𝑎_𝑖⊗𝑥)∨(𝑎_𝑖⊗𝑦) that 𝑎_𝑖⊗𝑥≠0 or 𝑎_𝑖⊗𝑦≠0. Hence, 𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥∨𝑦≠0⊆𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖∪𝑎⊗𝑥≠0_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑦≠0. (9) Thus, ∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥∨𝑦≠0≤∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖∨∨𝑎⊗𝑥≠0_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑦≠0. (10) This shows that 𝐻(𝑥∨𝑦)≤𝐻(𝑥)∨𝐻(𝑦). Therefore, 𝐻(𝑥)∨𝐻(𝑦)=𝐻(𝑥∨𝑦).
By the duality of 𝐿 and 𝐻 we know that another equation holds.
(7) By (3) and (5) we have 𝐻(𝑥)≤𝐻(𝐻(𝑥)). For any 𝑎_𝑗∈𝜉, 𝑎_𝑗⊗𝐻(𝑥)=𝑎_𝑗⊗∨𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖𝑎⊗𝑥≠0=∨_𝑗⊗𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0≠0, (11) if and only if there exists 𝑎_𝑖0∈𝜉 and 𝑎_𝑖0⊗𝑥≠0 such that 𝑎_𝑗⊗𝑎_𝑖0≠0. Hence, {𝑎_𝑗∈𝜉,𝑎_𝑗⊗𝐻(𝑥)≠0}⊆{𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}. It follows that 𝐻(𝐻(𝑥))=∨𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗(𝑥)𝑎⊗𝐻≠0≤∨_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0=𝐻(𝑥). (12) This shows that 𝐻(𝐻(𝑥))=𝐻(𝑥).
(8) By (3) we have 𝐿(𝑥)≤𝐻(𝐿(𝑥)). If 𝑎_𝑗0∈{𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗⊗𝐿(𝑥)≠0}, then 𝑎_𝑗0⊗𝐿(𝑥)=𝑎_𝑗0⊗∧{¬𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}≠0. It follows from 𝑎_𝑗0⊗¬𝑎_𝑗0=0 that ¬𝑎_𝑗0∉{¬𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}. Since 𝜉 is a partition we have 𝑎_𝑗0≤¬𝑎_𝑖(𝑖≠𝑗₀). Hence, 𝑎_𝑗0≤∧{¬𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}. This means that ∨{𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗⊗𝐿(𝑥)≠0}≤∧{¬𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}. This shows that 𝐻(𝐿(𝑥))≤𝐿(𝑥). Therefore, 𝐻(𝐿(𝑥))=𝐿(𝑥).

Example 15. Let the MV-algebra 𝑀 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-algebra 𝑀×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 Definition 11 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. Let 𝑀 be an MV-algebra and 𝜉 a partition of 𝑀. The definable elements of 𝑀 with respect to (𝑀,𝜉) are 𝐿^∗=𝑥∈𝑀∣∀𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝑎_𝑖=≤𝑥𝑥∈𝑀∣∀𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥=0𝑜𝑟𝑎_𝑖≤𝑥, (13) 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)=∨{𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}. ∀𝑎_𝑖∈𝜉, if 𝑎_𝑖⊗𝑥≠0 then 𝑎_𝑖≤𝐻(𝑥)=𝑥. This means that 𝑥∈𝐿^∗. Conversely, let 𝑥∈𝐿^∗. Then 𝑎_𝑖⊗𝑥≠0 implies 𝑎_𝑖≤𝑥. It follows that 𝐻(𝑥)=∨{𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}≤𝑥. By Theorem 14(4) we have 𝐻(𝑥)=𝑥.
Since 𝑥 is a definable element we also know 𝑥=𝐿(𝑥)=∧{𝑎_𝑖∣𝑎_𝑖∈¬𝜉,𝑥⊕𝑎_𝑖≠1}. Hence, for any 𝑎_𝑖∈¬𝜉, if 𝑥⊕𝑎_𝑖≠1 then 𝑥≤𝑎_𝑖. Let 𝑎_𝑖∈𝜉 and 𝑎_𝑖⊗𝑥≠0. Then ¬𝑎_𝑖⊕¬𝑥≠1. Hence, ¬𝑥≤¬𝑎_𝑖; that is 𝑎_𝑖≤𝑥. This means that 𝑥∈𝐿^∗. Conversely, let 𝑥∈𝐿^∗. Then 𝑎_𝑖⊗𝑥≠0 implies 𝑎_𝑖≤𝑥. If 𝑎_𝑖∈¬𝜉 and 𝑥⊕𝑎_𝑖≠1, then ¬𝑎_𝑖⊗¬𝑥≠0. Hence, ¬𝑎_𝑖≤𝑥 does not hold. By ¬𝑎_𝑖∈𝜉 and the definition of 𝐿^∗ we know ¬𝑎_𝑖⊗𝑥=0; hence, 𝑥≤𝑎_𝑖. This shows that 𝑥≤∧{𝑎_𝑖∣𝑎_𝑖∈¬𝜉,𝑥⊕𝑎_𝑖≠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 that 𝐿^∗ forms a Boolean algebra. Let 𝑥₁,𝑥₂∈𝐿^∗. For any 𝑎_𝑖∈𝜉, if 𝑎_𝑖⊗(𝑥₁∧𝑥₂)=(𝑎_𝑖⊗𝑥₁)∧(𝑎_𝑖⊗𝑥₂)≠0, then 𝑎_𝑖⊗𝑥₁≠0 and 𝑎_𝑖⊗𝑥₂≠0. By the definition of 𝐿^∗ we know 𝑎_𝑖≤𝑥₁ and 𝑎_𝑖≤𝑥₂; that is 𝑎_𝑖≤𝑥₁∧𝑥₂. Hence, 𝑥₁∧𝑥₂∈𝐿^∗. This shows that 𝐿^∗ is closed under operation ∧. If 𝑎_𝑖⊗(𝑥₁∨𝑥₂)=(𝑎_𝑖⊗𝑥₁)∨(𝑎_𝑖⊗𝑥₂)≠0, then 𝑎_𝑖⊗𝑥₁≠0 or 𝑎_𝑖⊗𝑥₂≠0. By the definition of 𝐿^∗ we know 𝑎_𝑖≤𝑥₁ or 𝑎_𝑖≤𝑥₂; that is 𝑎_𝑖≤𝑥₁∨𝑥₂. Hence, 𝑥₁∨𝑥₂∈𝐿^∗. This shows that 𝐿^∗ is closed under operation ∨. This shows that 𝐿^∗ forms a sublattice of 𝑀.
By Theorem 14 we know 0,1∈𝐿^∗. Let 𝑥∈𝐿^∗. Then ∀𝑎_𝑖∈𝜉, 𝑎_𝑖⊗𝑥=0 or 𝑎_𝑖≤𝑥. This means that ∀𝑎_𝑖∈𝜉, 𝑎_𝑖⊗𝑥=0 or 𝑎_𝑖⊗¬𝑥=0. It follows from ∀𝑎_𝑖∈𝜉, 𝑎_𝑖⊗¬𝑥=0or 𝑎_𝑖⊗¬¬𝑥=𝑎_𝑖⊗𝑥=0 that ¬𝑥∈𝐿^∗. Since ∀𝑎_𝑖∈𝜉, 𝑎_𝑖⊗𝑥=0 or 𝑎_𝑖⊗¬𝑥=0, we have ∀𝑎_𝑖∈𝜉, (𝑎_𝑖⊗𝑥)∧(𝑎_𝑖⊗¬𝑥)=𝑎_𝑖⊗(𝑥∧¬𝑥)=0. Hence, 0=⋁^𝑛_𝑖=1(𝑎_𝑖⊗(𝑥∧¬𝑥))=(⋁^𝑛_𝑖=1𝑎_𝑖)⊗(𝑥∧¬𝑥)=1⊗(𝑥∧¬𝑥)=𝑥∧¬𝑥. Analogously, 1=𝑥∨¬𝑥.
This shows that 𝐿^∗ 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 function 𝑚∶𝑀→[0,1] such that(1)𝑚(0)=0,(2)𝑚(1)=1,(3)𝑚 is finitely additive, that is, if 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} is ⊕-orthogonal then 𝑚(⨁^𝑛_𝑘=1𝑎_𝑘)=∑^𝑛_𝑘=1𝑚(𝑎_𝑘)is called a finitely additive measure on MV-algebra 𝑀.

For finitely additive measure, we have the following conclusion.

Theorem 19 (see [19, 20]). If 𝑚 is a finitely additive measure on the MV-algebra 𝑀 then(1)if 𝑎≤𝑏 then 𝑚(𝑏⊖𝑎)=𝑚(𝑏)−𝑚(𝑎), where 𝑏⊖𝑎=𝑏⊗¬𝑎;(2)if 𝑎≤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. Let 𝑀 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. Let 𝑀 be an MV-algebra, let 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} 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. Let 𝑀 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,𝑎_𝑛∈𝑀:(1)𝑚(1)=1,(2)if 𝑎⊗𝑏=0 then 𝑚(𝑎⊕𝑏)=𝑚(𝑎)+𝑚(𝑏),(3)if 𝑎_𝑛↗𝑎 then 𝑚(𝑎_𝑛)↗𝑚(𝑎),where 𝑎_𝑛↗𝑎 stands for 𝑎_𝑛 is a nondecreasing sequence and 𝑎=⋁_𝑛∈𝑁𝑎_𝑛 (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,…,𝑎_𝑛} and any state 𝑚 of 𝑀, it holds that 𝑚^𝑛_𝑖=1𝑎_𝑖=^𝑛_𝑖=1𝑚𝑎_𝑖. (14)

Proof. Since 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} is ⊕-orthogonal we know (⨁_𝑖∈𝐼𝑎_𝑖)⊗𝑎_𝑗=0. From the definition of a state, we have inductively 𝑚^𝑛_𝑖=1𝑎_𝑖=^𝑛_𝑖=1𝑚𝑎_𝑖. (15)

Proposition 24. Let 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} 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 ∑^𝑛_𝑖=1𝑚(𝑎_𝑖)=1 then we can extend 𝑚 to 𝑀₀ by defining 𝑚(0)=0, 𝑚(𝑥)=𝑚(⨁_{𝑘∈𝐼𝑥}𝑎_𝑘)=∑_{𝑘∈𝐼𝑥}𝑚(𝑎_𝑘), for every 𝑥=⨁_{𝑘∈𝐼𝑥}𝑎_𝑘∈𝑀₀, and 𝑚 is a finitely additive measure on 𝑀₀.

Proof. Obviously 𝑚(0)=0 and 𝑚(1)=1. For 𝑥,𝑦∈𝑀₀ then there are 𝐼_𝑥,𝐼_𝑦⊆{1,2,…,𝑛} such that 𝑥=⨁_{𝑘∈𝐼𝑥}𝑎_𝑘 and 𝑦=⨁_{𝑘∈𝐼𝑦}𝑎_𝑘. If 𝑥⊗𝑦=0 then we assert 𝐼_𝑥∩𝐼_𝑦=0/. In fact, if 𝐼_𝑥∩𝐼_𝑦≠0/ then there is 𝑖₀∈𝐼_𝑥∩𝐼_𝑦 such that 𝑥⊗𝑦≥𝑎_𝑖0⊗𝑎_𝑖0≠0. Hence, 𝑚(𝑥⊕𝑦)=𝑚(⨁_{𝑘∈𝐼𝑥}𝑎_𝑘⊕⨁_{𝑘∈𝐼𝑦}𝑎_𝑘)=𝑚(⨁_{𝑘∈𝐼𝑥∪𝐼𝑦}𝑎_𝑘)=∑_{𝑘∈𝐼𝑥∪𝐼𝑦}𝑚(𝑎_𝑘). This shows that 𝑚 is a finitely additive measure on 𝑀₀.

Definition 25. (1) A function Bel∶𝑀→[0,1] is called a belief measure if Bel(0)=0, Bel(1)=1, and Bel𝑥₁∨𝑥₂∨⋯∨𝑥_𝑙≥(−1)^|𝐽|+1Bel_𝑗∈𝐽𝑥_𝑗∣0/≠𝐽⊆{1,2,…,𝑙}, (16) for every positive integer 𝑙 and for every 𝑙-tuple 𝑥₁,𝑥₂,…,𝑥_𝑙, of subsets of 𝑀.
(2) A function Pl∶𝑀→[0,1] is called a plausibility measure if Pl(0)=0, Pl(1)=1, and Pl𝑥₁∧𝑥₂∧⋯∧𝑥_𝑙≤(−1)^|𝐽|+1Pl_𝑗∈𝐽𝑥_𝑗∣0/≠𝐽⊆{1,2,…,𝑙}, (17) for every positive integer 𝑙 and for every 𝑙-tuple 𝑥₁,𝑥₂,…,𝑥_𝑙, of subsets of 𝑀.

Theorem 26. Suppose that 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} is a partition of MV-algebra 𝑀 and function 𝑚∶𝜉→[0,1] satisfies ∑^𝑛_𝑖=1𝑚(𝑎_𝑖)=1. By using function 𝑚 we define a function 𝑃𝑙∶𝑀→[0,1] on 𝑀 as follows: 𝑃𝑙(𝑥)=𝑚𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0,𝑥∈𝑀. (18) Then 𝑃𝑙 is a plausibility measure on 𝑀.

Proof. It is easy to check Pl(0)=0, Pl(1)=1, and 0≤Pl(𝑥)≤1. For any 𝑥₁,𝑥₂,…,𝑥_𝑙∈𝑀, it is easy to check 𝑎_𝑖∣𝑎_𝑖⊗^𝑙_𝑗=1𝑥_𝑗≠0=𝑎_𝑖∣^𝑙_𝑗=1𝑎_𝑖⊗𝑥_𝑗⊆≠0_𝑙𝑗=1𝑎_𝑖∣𝑎_𝑖⊗𝑥_𝑗≠0. (19) By the well-known inclusion-exclusion formulas in probability theory we have Pl^𝑙_𝑗=1𝑥_𝑗=𝑚𝑎_𝑖∣𝑎_𝑖⊗^𝑙_𝑗=1𝑥_𝑗𝑚𝑎≠0≤_𝑖∣𝑎_𝑖∈^𝑙_𝑗=1𝑎_𝑖∣𝑎_𝑖⊗𝑥_𝑗)≠0=(−1^|𝐽|+1×𝑚𝑎_𝑖∣∃𝑗∈𝐽,𝑎_𝑖⊗𝑥_𝑗∣≠0/{)0≠𝐽⊆1,2,…,𝑙}=(−1^|𝐽|+1×𝑚𝑎_𝑖∣_𝑗∈𝐽𝑎_𝑖⊗𝑥_𝑗∣≠0/{()0≠𝐽⊆1,2,…,𝑙}=−1^|𝐽|+1Pl_𝑗∈𝐽𝑥_𝑗∣0/≠𝐽⊆{1,2,…,𝑙}. (20) This shows that Pl is a plausibility measure on 𝑀.

Theorem 27. Suppose that 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} is a partition of MV-algebra 𝑀 and function 𝑚∶𝜉→[0,1] satisfies ∑^𝑛_𝑖=1𝑚(𝑎_𝑖)=1. By using function 𝑚 we define a function 𝑃𝑙∶𝑀→[0,1] on 𝑀 as follows: 𝐵𝑒𝑙(𝑥)=1−𝑚𝑎_𝑖∣𝑎_𝑖∈𝜉,𝑎_𝑖⟶𝑥≠1. (21) Then Bel is a belief measure on 𝑀.

Proof. It is easy to check Bel(0)=0, Bel(1)=1, and 0≤Bel(1)≤1. For any 𝑥₁,𝑥₂,…,𝑥_𝑙∈𝑀, it is easy to check 𝑎_𝑖∣𝑎_𝑖⟶^𝑙_𝑗=1𝑥_𝑗≠1=𝑎_𝑖∣^𝑙_𝑗=1𝑎_𝑖⟶𝑥_𝑗⊆≠1_𝑙𝑗=1𝑎_𝑖∣𝑎_𝑖⟶𝑥_𝑗≠1. (22) By the well-known inclusion-exclusion formulas in probability theory we have Bel^𝑙_𝑗=1𝑥_𝑖=1−𝑚𝑎_𝑖∣𝑎_𝑖⟶^𝑙_𝑗=1𝑥_𝑖𝑚𝑎≠1≥1−_𝑖∣𝑎_𝑖∈^𝑙_𝑗=1𝑎_𝑖∣𝑎_𝑖⟶𝑥_𝑗)≠1=1−(−1^|𝐽|+1×𝑚𝑎_𝑖∣∃𝑗∈𝐽,𝑎_𝑖⟶𝑥_𝑗∣≠1/{)0≠𝐽⊆1,2,…,𝑙}=1−(−1^|𝐽|+1×𝑚𝑎_𝑖∣𝑎_𝑖⟶_𝑗∈𝐽𝑥_𝑗∣≠1/{()0≠𝐽⊆1,2,…,𝑙}=1−−1^|𝐽|+1×1−Bel_𝑗∈𝐽𝑥_𝑗∣/{})0≠𝐽⊆1,2,…,𝑙=1−(−1^|𝐽|+1∣/{()0≠𝐽⊆1,…,𝑙}+−1^|𝐽|+1Bel_𝑗∈𝐽𝑥_𝑗∣/{}()0≠𝐽⊆1,…,𝑙=−1^|𝐽|+1Bel_𝑗∈𝐽𝑥_𝑗∣0/≠𝐽⊆{1,2…,𝑙}. (23) This shows that Bel is a belief measure on 𝑀.

Let 𝑀 be an MV-algebra and 𝜉={𝑎₁,𝑎₂,…,𝑎_𝑛} a partition of 𝑀. Suppose that function 𝑚∶𝜉→[0,1] satisfies ∑^𝑛_𝑖=1𝑚(𝑎_𝑖)=1. Then we can extend 𝑚 to 𝑀 by defining 𝑚_∗(𝑥)=Bel(𝐿(𝑥)),𝑚^∗(𝑥)=Pl(𝐻(𝑥)), (24) 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 prove Pl(𝐻(𝑥))=Pl(𝑥). Note that Pl(𝐻(𝑥))=∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝐻(𝑥)≠0}. In the following we prove that 𝑎_𝑖⊗𝐻(𝑥)≠0 if and only if 𝑎_𝑖⊗𝑥≠0. In fact, 𝑎_𝑖⊗𝐻(𝑥)=𝑎_𝑖⊗∨{𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗⊗𝑥≠0}=∨{𝑎_𝑖⊗𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗⊗𝑥≠0}={𝑎_𝑖⊗𝑎_𝑖∣𝑎_𝑖⊗𝑥≠0}. This shows that 𝑎_𝑖⊗𝐻(𝑥)≠0 if and only if 𝑎_𝑖⊗𝑥≠0. Hence, 𝑚^∗(𝑥)=Pl(𝐻(𝑥))=∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝐻(𝑥)≠0}=∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖⊗𝑥≠0}=Pl(𝑥). It follows from Theorem 26 that 𝑚^∗ is a plausibility measure on 𝑀.
(2) We prove Bel(𝐿(𝑥))=Bel(𝑥). Note that Bel(𝐿(𝑥))=1−∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝐿(𝑥)≠1}. In the following we prove that 𝑎_𝑖→𝐿(𝑥)≠1 if and only if 𝑎_𝑖→𝑥≠1. In fact, 𝑎_𝑖→𝐿(𝑥)=𝑎_𝑖→(∨{𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗→𝑥≠1}→0)=𝑎_𝑖⊗∨{𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗→𝑥≠1}→0=∨{𝑎_𝑖⊗𝑎_𝑗∣𝑎_𝑗∈𝜉,𝑎_𝑗→𝑥≠1}→0={𝑎_𝑖⊗𝑎_𝑖∣𝑎_𝑖→𝑥≠1}→0. This shows that 𝑎_𝑖→𝐿(𝑥)≠1 if and only if 𝑎_𝑖→𝑥≠1. Hence, 𝑚_∗(𝑥)=Bel(𝐻(𝑥))=1−∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝐿(𝑥)≠1}=1−∑{𝑚(𝑎_𝑖)∣𝑎_𝑖∈𝜉,𝑎_𝑖→𝑥≠1}=Bel(𝑥). It follows from Theorem 27 that 𝑚_∗ is a belief measure on 𝑀.

Example 29. The MV-algebra 𝑀×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 

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.