AbstractBy means of the probability measure theory of MV-algebras we introduce the concepts of probability truth degrees of the elements of MV-algebra and probability similarity degrees between elements, and then define therefrom three probability metric spaces on MV-algebra, and get some good results

Zuo, Weibing

Elsevier - Publisher Connector 

 Physics Procedia  25 ( 2012 )  2138 – 2145 
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Garry Lee
doi: 10.1016/j.phpro.2012.03.361 
 
2012 International Conference on Solid State Devices and Materials Science 
Probability Metric Space on MV-Algebras 
Weibing Zuo 
College of Mathematics and Information Science 
North China University of Water Resources and Electric Power, Zhengzhou, P.R.China 
Abstract 
By means of the probability measure theory of MV-algebras we introduce the concepts of probability truth degrees of 
the elements of MV-algebra and probability similarity degrees between elements, and then define therefrom three 
probability metric spaces on MV-algebra, and get some good results. 
 
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] 
 
Keywords-MV-algebra; probability truth degree; probability similarity degree; probability metric space; 
1. Introduction 
MV-algebra [1] were introduced by Chang.C.C in 1958 as the algebraic counterpart of propositional 
àukasiewicz logic. An MV-algebra is a many-valued generalization of a Boolean algebra, MV-algebras 
can be viewed as algebraic generalizations of certain collections of fuzzy sets, so-called àukasiewicz 
tribes, as well. Currently, MV-algebra has many applications and many algebras is closely linked with the 
MV-algebra. States on MV-algebras were investigated by Mundici in [2] as [0,1]-valued additive 
functionals on formulas in àukasiewicz propositional logic with the intention to capture the notion of 
average truth degree of a formula. In [3] a probability theory on MV-algebras is systematically developed. 
In [4,5] the conditional probability on MValgebra and ı MV-algebra is systematically developed also. In 
recent years, a focus of attention is study of grated of mathematical logic in the background of uncertainty 
reasoning. Based on the measure theory Wang [6] introduced the truth degree of elements in MV-algebras, 
for established the probability on MV-algebra provided an effective method. In the present paper, by 
means of the probability measure theory we introduce the concepts of probability truth degrees of the 
elements in MV-algebra and probability similarity degrees between elements, and then define therefrom 
three probability metric spaces on MV-algebra, and get some good results.  
The paper is structured as follows. Basic notions and results on MV-algebra are repeated in section 2, 
the concepts of probability truth degree of elements and probability similarity degrees between elements 
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 Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 2139
 
on MV-algebras are introduced in section 3, three probability metric spaces on MV-algebra are introduced 
and some good results are get.  
2. Basic notions of MV-algebra 
Definition 2.1 (Chang C.C [1]) An MV-algebra is an algebra (M, , , 0), where M is a non-empty set, 
 is binary operation having 0 as the neutral element,  is a unary operation, satisfying the following 
properties: 
(i) (M, , 0) is a commutative semi-group, 
(ii) a0 = 0, 
(iii)  (a) = a, 
(iv)  (a  b)  b =  (b  a)  a, a, b  M. 
Let (M, , , 0) be an MV-algebra, the constant 1 is defined as follows: 1 =0, a partial order d can 
be introduced on M by defining a d b if and only if a  b = 1, then (M, d) is a allocation lattice. Binary 
operations ,  are defined as follows: ab =  (ab)  b, a^b =  (ab).  
Proposition 2.1 (Chang C.C [1]) Let (M, , , 0) be an MV-algebra, binary operations : , o are 
defined as follows: 
  , ,x y x y x y x y     o   :                   (1) 
then 
(i) x: y d z iff x d y o z, 
(ii) 1 o x = x, x o 0 = x, x :  0 = 0, 
(iii) x o y = 1 iff x d y, 
(iv) x o y = y o x, 
(v) x o (y o z) = y o (x o z), 
(vi) x: (y  z) = (x: y)  (x: z), 
(vii) x: (y z) = (x: y)(x: z), 
(viii) (x o y): (y o z) d x o z. 
Example 2.1 Any Boolean algebra B is a MV-algebra, where 0 =; 1 = X,  is the union . 
Example 2.2 The real unit interval [0,1] equipped with the àukasiewicz operation a  b = min(a + b, 1) 
and the standard complement a = 1- a, where a, b  [0, 1], is a ı-complete MV-algebra called the 
standard MV-algebra. 
Definition 2.2(Wang and Zhou [6]) Let M be MV-algebra, [0, 1]MV standard MV-algebra, the 
homomorphism ȣ: M o [0, 1] is called a assignment of M, i.e., ȣ(xy) =ȣ(x)ȣ(y)= (ȣ(x)+ȣ(y))1, 
ȣ(x)=1-ȣ(x). the set of all assignments of M is denote : 
ȣ:, ȣ satisfying all operations of M, i.e., 
ȣ(x: y) = ȣ(x): ȣ(y) = (ȣ(x) + ȣ(y) - 1) 0, 
ȣ(xoy) = ȣ(x)oȣ(y) = (1-ȣ(x) + ȣ(y) ) 1, 
ȣ(xy)= max{ȣ(x) , ȣ(y)}, 
ȣ(xy) = min{ȣ(x) , ȣ(y)}. 
Proposition 2.2 (Wang and Zhou [6]) Let M be MV-algebra, defining :x : o [0, 1] is following 
    ,x v v x                                      (2) 
Let ^ `M x x M  , and defining 
 ,x y x y x x      , 
2140   Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 
 
Then ( , , ,0M   ) is an MV-algebra. 
3. Probability truth degree of elements on MV-algebra 
Let M be MV-algebra, : be the set of all assignments of M. Suppose A be ı-algebra on :, ȝ be 
probability measure on :, then (:, A, ȝ) is a probability space[7]. xM, let     ,x v v x ȣ:, then 
x  is ȝ-integrable function[7]. 
Definition 3.1 (Wang and Zhou [6]) Defining T: M o [0, 1] is following: 
    , ,T x x v d x MP
:
 ³                           (3) 
 T x  is called ȝ-truth degree of element x, in short, truth degree of x. 
Proposition 3.1 (Wang and Zhou [6]) (i) 0dT(x) d 1, x  M, 
(ii) T (0) = 0, T (1) = 1, 
(iii) T (x) = 1-T (x), 
(iv) If x d y, then T (x) dT (y), x, y  M. 
Proposition 3.2 (Wang and Zhou [6]) (i) T (xy) = T(x)+ T(y)- T (x  y), 
(ii) T (x  y) = T (x) + T (y) - T (x: y), 
(iii) If E =    ^ `v x v y v : d  is ȝ-measurable, then T (x) + T (x o y) = T (y) + T (y o x). 
Proposition 3.3 (Wang and Zhou [6]) Let x, y  M, D,E [0, 1], then 
(i) if T (x) t D, T (x o y) tE, then T (y) tD + E - 1, 
(ii) if T (x o y)tD, T (y o z) tE, then T (x o z) tD+ E - 1. 
Corollary 3.1 Let x, y, z  M, then 
(i) T (x o y) d T (x) o T (y), 
(ii) T (x o y): T (yoz) d T (xoz), 
(iii) T (x o y)d T (yoz) o T (xoz). 
Definition 3.2 Let x, y  M, defining K : MuM o [0, 1] is following: 
(i) K1(x, y) = T((x o y) (y o x)) is called the first ȝ-similarity degree, 
(ii) K2(x, y) = T(x o y) T(y o x) is called the second ȝ-similarity degree, 
(iii) K3(x, y) = (T(x) o T (y)) ( T (y)o T (x)) is called the third ȝ-similarity degree, 
K1, K2, K3 uniformly expressed as K, so called similarity degree. 
If Ki(x, y) = 1, (i = 1, 2, 3) then x and y is called i-similarity, denote x ~i y. 
Lemma 3.1 Let a, b, c, d  [0; 1]MV, then 
(a  c): (b  d) d (a: b)  (c: d). 
Theorem 3.1 Let x, y, z  M, then 
(i) K(x, x) = 1, 
(ii) K(x, y) = K(y, x), 
(iii) K(x, y) = K(x, y), 
(iv) K(x, z)t K(x, y) +K(y, z) - 1. 
Proof (i) and (ii) is clearly established. 
(iii) x, yM, by proposition 1(iv) we obtain x y y xo   o  , so 
 Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 2141
 
 
 
    
(iv) For K1 see [6] proposition 7(iv). 
For K2, from corollary 3.1(ii) we can obtain T(x o z) t T(x o y): T(y o z) and T(z o x)t T(z o y) 
: T(y o x) = T(y o x): T(z o y). From lemma 3.1, 
 
So K2(x, z)t K2(x, y) +K2(y, z) - 1. 
For K3, from proposition 2.1(viii) we obtain T (x) o T (z) t (T(x) o T(y)): (T (y) o T (z)) and T (z) 
o T (x) t (T(z) o T(y)): (T (y) o T (x)), similar to K2, we can proved. 
Theorem 3.2 Let x, y  M, then 
(i)       1 , 1 ,x y x v y v dK P:  ³  
     3 , 1 ,x y T x T yK     
(ii) K1(x, y)d K2(x, y)d K3(x, y). 
Proof (i) Because 1-a+b and 1-b+a not both greater than 1 for a, b  [0, 1]MV , thus 
 
2142   Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 
 
 
4. Probability metrics of MV-algebra 
Definition 4.1 Let x, y  M, defining 
   , 1 , ,x y x yU K                               (4) 
   , 1 , , 1, 2,3.i ix y x y iU K                           (5) 
Theorem 4.1 ȡi : MuM o [0, 1] is metric of MV-algebra M(i=1,2,3). 
Proof Let x, y, z  M, from theorem 3.1 we see that    , 1 , 0x x x xU K   , and 
       , 1 , 1 , , .x y x y y x y xU K K U     
                 , 1 , 1 , , 1 1 , 1 , , ,x z x z x y y z x y y z x y y zU K K K K K U U  d           
Definition 4.2 (M, ȡi) is called probability metric space of MV-algebra (i=1,2,3). 
Theorem 4.2 Let x, y  M, then 
 
Theorem 4.3 Let x, y  M, then 
(i) ȡ1(x, y)t ȡ2(x, y)t ȡ3(x, y), 
(2) ȡ2(x, y)= 
ଵ
ଶ
 (ȡ1(x, y)+ ȡ3(x, y)). 
Proof (i) From theorem 3.2(ii), we can proved. 
(ii) 
 
 Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 2143
 
 
 
 
Theorem 4.4 (Wang and Zhou [6]) On probability metric space of MV-algebra (M, ȡ1), operations , 
, ,:  ,  and o are continuous. 
Lemma 3.2 Let x, y, u, w  M, D, E[0, 1], if K2(x, u)tD, K2(y, w)tE, then K2(xoy, u o w)tD+ E-1. 
Proof Let [0, 1]MV be standard MV-algebra, a, b, c  [0, 1], then a o b d (b o c) o (a o c) and b o a 
d (a o c) o (b o c). Thus v:, we see that 
 
And 
 
From proposition 3.1, 
 
 
Theorem 4.5 Suppose (M, ȡ2) be probability metric space of MV-algebra, then operations , , ,:  , 
 and o on (M, ȡ2) are continuous. 
2144   Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 
 
Proof First, from theorem 3.1 we see that ȡ2(x, y) =1 - K2(x, y) = 1- K2(x, y) = ȡ2(x, y), therefore 
 is continuous. 
Second, let ȡ2(x, u) dH, ȡ2(y, w) dH, then K2(x, u) t 1-H, K2(y, w) t 1-H, so from lemma 3.2 we have 
 
i.e., ȡ2(x o y, u o w) d 2H, so implication operation o is continuous. 
Because x  y = x o y, x: y = (x oy), x  y = (x o y) o y, xy = (xy), therefore , 
,:  ,  are all continuous. 
Theorem 4.6 (Wang and Zhou [6]) If (M, ȡ1) be complete metric space, then M is a countably complete 
lattice. 
Theorem 4.7 If (M, ȡ2) be complete metric space, then M is also a countably complete lattice. 
Proof Suppose (M, ȡ2) be complete metric space, ǻ={x1,x2,Ă}M. Let 1
n
n iiy V x  , then y1, y2, Ăis 
an increasing sequence. From proposition 3.1 we see that 
T (y1), T (y2),Ăis an increasing sequence in [0,1], so is a Cauchy sequence. And 
 
Therefore, if n d m then 
 
From T (y1), T (y2),Ă is a Cauchy sequence, we see that y1, y2, Ăis a Cauchy sequence in complete 
metric space (M, ȡ2), so it converges to y, where y is a point of M. 
We fix an n, then yn = yn  yn+k, from the continuity of operation see that 
 
So yn d y, i.e., y is a upper bound of ¦ = { y1, y2, Ă}. Taking any a upper bound of ¦, then yn = yn  z, 
so 
 
Thus y d z, i.e., y = sup¦  M. It is clearly supǻ =sup¦M. 
Similarly, we can prove inf ǻ  M. Therefore M is countable complete. 
Theorem 4.8 Suppose (M, ȡ3) be probability metric space of MV-algebra, then operations  is 
continuous, but operation , ,:  ,  and o are not continuous all. 
5. Conclusion 
In this paper, we introduced three probability metric space on MV-algebras, studied the propositions of 
probability metric spaces on MV-algebras, get some good results. We can see that the method of one 
paper can be extended to other algebras, so the method of one paper is meaningful. 
 Weibing Zuo /  Physics Procedia  25 ( 2012 )  2138 – 2145 2145
 
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[5] A.Dvurecenskij and S.Pulmannova, Conditional probability on V-MV-algebras, Fuzzy Sets and Systems, USA, 2005, 
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[8] Weibing Zuo, Approximate reasoning in Godel 4-valued nonlinear ordered set logic system , 2010 International Conference 
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Probability Metric Space on MV-Algebras 

Weibing Zuo

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