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Semi-lattice structure of all extensions of the possibility measure and the consonant belief function on the fuzzy set
Fuzzy Sets and Systems 43 (1991) 183-188 North-Holland
183
Semi-lattice structure of all extensions of the possibility measure and the consonant belief function on the fuzzy set Zhang Guang-Quan Department of Mathematics, Hebei University, Baoding, Hebei, 071002, People's Republic of China Received April 1988 Revised September 1988
Abstract: The concepts of possibility measure and consonant belief function on the fuzzy set are introduced and results analogous to those of Wang Zhenyuan (1985) are obtained.
Keywords: Fuzzy set; possibility measure; consonant belief function.
1. Introduction
In Section 2 of this paper, we introduce the concepts possibility measure and consonant belief function on the class of all fuzzy subsets of X, P-consistency and B-consistency of a fuzzy set function, and give two extension theorems of the possibility measure and consonant belief function. In Section 3, we discuss the semi-lattice structures of all extensions of the possibility measure and the consonant belief function. Throughout this paper, let X be a nonempty set, F(X) be the class of all fuzzy subsets of X, and C be an arbitrary nonempty subset of F(X), Iz be a mapping from C into the unit interval [0, 1], and we make the following convention: U0 {'} = 0, A0 {'} = X, sup0 {/~(')} = 0, inf0 {/~(-)} = 1.
2. Extensions of the possibility measure and consonant belief function Definition 2.1. A possibility measure on F(X) is a non-negative real valued fuzzy set function ~ : F(X)--~ [0, 1] with the property
\t6T
/
t~T
whenever {At; t ~ T} c F ( X ) , where T is an arbitrary index set. 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)
184
Zhang G u a n g - Q u a n
Definition 2.2. A consonant belief function on F ( X ) is a non-negative real valued fuzzy set function fl:F(X)---~ [0, 1] with the property fl(,Or`4') =inffl(`4')',~r whenever {`4,; t • T} ~ F(X),. where T is an arbitrary index set.
Definition 2.3. /~:C---~[0, 1] is called P-consistent (B-consistent), if for every {`4t; t • T} c C, .4 • C, with `4 c U,~r`4, (`4 ~ ['-'l,~r`4,), we have
Theorem 2.1. g can induce a possibility measure ~r on F ( X ) , and er(`4) <~#(.4),
for every 7t • C.
(1)
Proof. If we define :r:F(X)---> [0, 1] by /~ ~ sup xeX
inf sup g(/~s). (Us~,~E_s)(x)~t~(x) s s S x
(2)
EseC
where S x is arbitrary index set, then :r is a possibility measure on F ( X ) , and (1) holds. To conclude the assertions, we first prove that :r is a possibility measure. In fact, the monotonicity of :r is obvious. By the monotonicity of :r, we have, for every {At; t • T} c F ( X ) ,
~(A,) <~~( UTA,), and hence
4u. ,l,
teT
\teT
/
where T is an arbitrary index set. On the other hand, (a) if for every x • X, and for every t • T, there exists {/~s; s • St} c C such that
and for any e > 0, inf
sup g(/~s) i> sup/~(/~s) - e.
(Us~s~ ~s)(x)~.4,(x) s e S x Es~C
Since
s~Sf
Possibility m e a s u r e a n d c o n s o n a n t b e l i e f f u n c t i o n
185
it follows that sup
inf
teT
(Us~.~Es)(x)>-,4,(x) P~eC
=
sup
sup/~(/~s)/> sup s u p / z ( / ~ ) - e s~S~[
t~T
/u(/~) - e ~>
s~U,~rs~
s~S x
inf
sup/~(/~s) - e.
(U,~.~ E~)(x)>~(U,~rA,)(x) s~S ~
This shows that sup sup x~X
inf
t~T
sup/z(/~)
(Us~sxffTs)(X)~.4,(x) E~eC
/> sup
s~S~
inf
x~x
sup/~(/~s) - e.
(3)
(U,~sx k~)(x)>-(U,~TA,)(x) ~ s ~ E,~C
(b) If there exists xo ~ X, to ~ T, for every {/~s; s ~ S,Xl'} = C such that
by using the c o n v e n t i o n inf~ {.} = 1, (3) is true. This yields that
t~T
\t~T
/
\t~T
/
Consequently,
t6T
which m e a n s :t is a possibility m e a s u r e . Next, we p r o v e that (1) holds. In fact, for e v e r y fi~ c C, we have : r ( A ) = sup x~X
Theorem
inf
sup #(/~,) ~< sup #(,4) = #(,4).
(Us~.~ ~_D(x)>~A(x) s~S • E, e C
x~X
2.2. I f ~ is P-consistent, then :r o f T h e o r e m 2.1 is an extension o f I~.
P r o o f . W e w a n t to p r o v e that ~ of T h e o r e m 2.1 is an extension of/~, that is to p r o v e ~ ( A ) = / ~ ( A ) , for every A e C. W e first p r o v e that ~(A)~> # ( A ) . In fact, for any e > 0 and e v e r y x • X t h e r e exists {/~s; s e S x } c C such that
and inf
sup/u(/~)/> sup/ul,tz,)'¢'" - e.
(U~.~, ~_s)(x)>-A(x) s E s x Es~C
s~S x
186
Zhang Guang-Quan
Hence, by using the P-consistency o f / t , z~(A) = sxuE Xp
(u.~
~s~x)--m(x) sup ~ ( L ) J ~ E, eC
/> sup sup/*(/~) - e = xeX
s~S x
sup
/~(£~) - e = #(A) - e.
seU.~xS •
Therefore
Consequently, from (1), we have =
Now, we give two extension theorems. T h e o r e m 2.3. /~ can be extended to a possibility measure on F(X), if and only if
is P-consistent. Proof. Necessity. Obvious. Sufficiency. Theorems 2.1 and 2.2. Theorem 2.4. /t can be extended to a consonant belief function on F(X), if and
only if # is B-consistent. Proof. The proof is similar to that of Theorems 2.1 and 2.2. We only note that if /~ is B-consistent, then fl is a consonant belief function on F(X), and it is an extension of ~, where fl : F(X) ~ [0, 1] is defined by A ~ inf x~X
sup (ns~
inf/*(L).
(4)
ED(x)<-A(x') s~S ~ Es~C
3. Semi-lattice structure o f all extensions
In the usual case, the extension of a mapping /~ with P-consistency from an arbitrary nonempty class of the fuzzy subsets of X into the unit interval [0, 1] to a possibility measure on F(X) may not be unique. Similarly, the extension of a mapping ~ with B-consistency from an arbitrary nonempty class of the fuzzy subsets of X into the unit interval [0, 1] to a consonant belief function on F ( X ) may not be unique either. All extensions of the possibility measure (consonant belief function) are denoted E~(~) (EB(/t)). By using Theorem 2.3 (Theorem 2.4), we know that E~(/~) (Et~(~)) is nonempty, if /~ is P-consistent (Bconsistent). For two mappings /AI:F(X)---->[0 , 1] and / A 2 : F ( X ) - " > [ 0 , 1], we define the ordering relation ~<: /~l ~
Possibility measure and consonant belief function
187
It is easy to prove that ~< is a partial ordering relation on E~(#) (similarly, on Etff#)). Therefore the least upper bound on #~, #2 e E,,(#) can be defined by (sup{#,, #/})(A) = #,(A) v #2(fit),
V.4 e F ( X ) .
Similarly, the greatest lower bound can be defined by (inf{#l, #z})(A) = #~(,4) ^ #2(A),
V~, ~ F ( X ) .
Theorem 3.1. (E#(#), ~<) is an upper semi-lattice, and the extension ar defined by (2)/s the greatest element of E,~(#). Proof. (a) Obviously, (E,~(#), ~< ) is an upper semi-lattice. (b) The extension Jr defined by (2) is the greatest element of the E~(#). For arbitrary Jr' e E~(#), /~ e F ( X ) , we define
iffy = x , I
Es6C
k0
iffy 4=x,
for every x e X. If/~(x) ~< (U~sx F.s)(X), E~ ~ C, we have
s~S x
Hence sr'(/Tx) ~< supsCs~ sr'(/~s), and therefore sr'(Px) ~<
inf
sup zt'(/~) =
(U~,~ ~_~)(x)>~b(x)~s~ Es~C
inf
sup # ( L ) ,
(U~s~ E_A(x)>-~(x) .~S~ E~eC
for every x e X. It follows, by using
(U.~.~ ~,)(x)~B(x) E, e C
that :r(/~) = sup
inf
sup #(Es)
x~X (U,~,~ E , ) ( x ) ~ ( x ) s~Sx ~eC
Similarly, we can prove: Theorem 3.2. (Eta(#), <~) is a lower semi-lattice, and the extension fl defined by (4) is the least element of Etff#).
188
Zhang Guang-Quan
Acknowledgment The author would like to express his gratitude to Professor Wang Zhenyuan for guidance.
References [1] Wang Zhenyuan, Semi-lattice structure of all extensions of possibility measure and consonant belief function, BUSEFAL 24 (1985) 18-22. [2] Wang Zhenyuan, Semi-lattice isomorphism of the extension of possibility measure and the solutions of fuzzy relation equation, in: R. Trappl, Ed., Cybernetics and Systems '86 (Reidel, Dordrecht, 1986) 581-583.
183
Semi-lattice structure of all extensions of the possibility measure and the consonant belief function on the fuzzy set Zhang Guang-Quan Department of Mathematics, Hebei University, Baoding, Hebei, 071002, People's Republic of China Received April 1988 Revised September 1988
Abstract: The concepts of possibility measure and consonant belief function on the fuzzy set are introduced and results analogous to those of Wang Zhenyuan (1985) are obtained.
Keywords: Fuzzy set; possibility measure; consonant belief function.
1. Introduction
In Section 2 of this paper, we introduce the concepts possibility measure and consonant belief function on the class of all fuzzy subsets of X, P-consistency and B-consistency of a fuzzy set function, and give two extension theorems of the possibility measure and consonant belief function. In Section 3, we discuss the semi-lattice structures of all extensions of the possibility measure and the consonant belief function. Throughout this paper, let X be a nonempty set, F(X) be the class of all fuzzy subsets of X, and C be an arbitrary nonempty subset of F(X), Iz be a mapping from C into the unit interval [0, 1], and we make the following convention: U0 {'} = 0, A0 {'} = X, sup0 {/~(')} = 0, inf0 {/~(-)} = 1.
2. Extensions of the possibility measure and consonant belief function Definition 2.1. A possibility measure on F(X) is a non-negative real valued fuzzy set function ~ : F(X)--~ [0, 1] with the property
\t6T
/
t~T
whenever {At; t ~ T} c F ( X ) , where T is an arbitrary index set. 0165-0114/91/$03.50 © 1991--Elsevier Science Publishers B.V. (North-Holland)
184
Zhang G u a n g - Q u a n
Definition 2.2. A consonant belief function on F ( X ) is a non-negative real valued fuzzy set function fl:F(X)---~ [0, 1] with the property fl(,Or`4') =inffl(`4')',~r whenever {`4,; t • T} ~ F(X),. where T is an arbitrary index set.
Definition 2.3. /~:C---~[0, 1] is called P-consistent (B-consistent), if for every {`4t; t • T} c C, .4 • C, with `4 c U,~r`4, (`4 ~ ['-'l,~r`4,), we have
Theorem 2.1. g can induce a possibility measure ~r on F ( X ) , and er(`4) <~#(.4),
for every 7t • C.
(1)
Proof. If we define :r:F(X)---> [0, 1] by /~ ~ sup xeX
inf sup g(/~s). (Us~,~E_s)(x)~t~(x) s s S x
(2)
EseC
where S x is arbitrary index set, then :r is a possibility measure on F ( X ) , and (1) holds. To conclude the assertions, we first prove that :r is a possibility measure. In fact, the monotonicity of :r is obvious. By the monotonicity of :r, we have, for every {At; t • T} c F ( X ) ,
~(A,) <~~( UTA,), and hence
4u. ,l,
teT
\teT
/
where T is an arbitrary index set. On the other hand, (a) if for every x • X, and for every t • T, there exists {/~s; s • St} c C such that
and for any e > 0, inf
sup g(/~s) i> sup/~(/~s) - e.
(Us~s~ ~s)(x)~.4,(x) s e S x Es~C
Since
s~Sf
Possibility m e a s u r e a n d c o n s o n a n t b e l i e f f u n c t i o n
185
it follows that sup
inf
teT
(Us~.~Es)(x)>-,4,(x) P~eC
=
sup
sup/~(/~s)/> sup s u p / z ( / ~ ) - e s~S~[
t~T
/u(/~) - e ~>
s~U,~rs~
s~S x
inf
sup/~(/~s) - e.
(U,~.~ E~)(x)>~(U,~rA,)(x) s~S ~
This shows that sup sup x~X
inf
t~T
sup/z(/~)
(Us~sxffTs)(X)~.4,(x) E~eC
/> sup
s~S~
inf
x~x
sup/~(/~s) - e.
(3)
(U,~sx k~)(x)>-(U,~TA,)(x) ~ s ~ E,~C
(b) If there exists xo ~ X, to ~ T, for every {/~s; s ~ S,Xl'} = C such that
by using the c o n v e n t i o n inf~ {.} = 1, (3) is true. This yields that
t~T
\t~T
/
\t~T
/
Consequently,
t6T
which m e a n s :t is a possibility m e a s u r e . Next, we p r o v e that (1) holds. In fact, for e v e r y fi~ c C, we have : r ( A ) = sup x~X
Theorem
inf
sup #(/~,) ~< sup #(,4) = #(,4).
(Us~.~ ~_D(x)>~A(x) s~S • E, e C
x~X
2.2. I f ~ is P-consistent, then :r o f T h e o r e m 2.1 is an extension o f I~.
P r o o f . W e w a n t to p r o v e that ~ of T h e o r e m 2.1 is an extension of/~, that is to p r o v e ~ ( A ) = / ~ ( A ) , for every A e C. W e first p r o v e that ~(A)~> # ( A ) . In fact, for any e > 0 and e v e r y x • X t h e r e exists {/~s; s e S x } c C such that
and inf
sup/u(/~)/> sup/ul,tz,)'¢'" - e.
(U~.~, ~_s)(x)>-A(x) s E s x Es~C
s~S x
186
Zhang Guang-Quan
Hence, by using the P-consistency o f / t , z~(A) = sxuE Xp
(u.~
~s~x)--m(x) sup ~ ( L ) J ~ E, eC
/> sup sup/*(/~) - e = xeX
s~S x
sup
/~(£~) - e = #(A) - e.
seU.~xS •
Therefore
Consequently, from (1), we have =
Now, we give two extension theorems. T h e o r e m 2.3. /~ can be extended to a possibility measure on F(X), if and only if
is P-consistent. Proof. Necessity. Obvious. Sufficiency. Theorems 2.1 and 2.2. Theorem 2.4. /t can be extended to a consonant belief function on F(X), if and
only if # is B-consistent. Proof. The proof is similar to that of Theorems 2.1 and 2.2. We only note that if /~ is B-consistent, then fl is a consonant belief function on F(X), and it is an extension of ~, where fl : F(X) ~ [0, 1] is defined by A ~ inf x~X
sup (ns~
inf/*(L).
(4)
ED(x)<-A(x') s~S ~ Es~C
3. Semi-lattice structure o f all extensions
In the usual case, the extension of a mapping /~ with P-consistency from an arbitrary nonempty class of the fuzzy subsets of X into the unit interval [0, 1] to a possibility measure on F(X) may not be unique. Similarly, the extension of a mapping ~ with B-consistency from an arbitrary nonempty class of the fuzzy subsets of X into the unit interval [0, 1] to a consonant belief function on F ( X ) may not be unique either. All extensions of the possibility measure (consonant belief function) are denoted E~(~) (EB(/t)). By using Theorem 2.3 (Theorem 2.4), we know that E~(/~) (Et~(~)) is nonempty, if /~ is P-consistent (Bconsistent). For two mappings /AI:F(X)---->[0 , 1] and / A 2 : F ( X ) - " > [ 0 , 1], we define the ordering relation ~<: /~l ~
Possibility measure and consonant belief function
187
It is easy to prove that ~< is a partial ordering relation on E~(#) (similarly, on Etff#)). Therefore the least upper bound on #~, #2 e E,,(#) can be defined by (sup{#,, #/})(A) = #,(A) v #2(fit),
V.4 e F ( X ) .
Similarly, the greatest lower bound can be defined by (inf{#l, #z})(A) = #~(,4) ^ #2(A),
V~, ~ F ( X ) .
Theorem 3.1. (E#(#), ~<) is an upper semi-lattice, and the extension ar defined by (2)/s the greatest element of E,~(#). Proof. (a) Obviously, (E,~(#), ~< ) is an upper semi-lattice. (b) The extension Jr defined by (2) is the greatest element of the E~(#). For arbitrary Jr' e E~(#), /~ e F ( X ) , we define
iffy = x , I
Es6C
k0
iffy 4=x,
for every x e X. If/~(x) ~< (U~sx F.s)(X), E~ ~ C, we have
s~S x
Hence sr'(/Tx) ~< supsCs~ sr'(/~s), and therefore sr'(Px) ~<
inf
sup zt'(/~) =
(U~,~ ~_~)(x)>~b(x)~s~ Es~C
inf
sup # ( L ) ,
(U~s~ E_A(x)>-~(x) .~S~ E~eC
for every x e X. It follows, by using
(U.~.~ ~,)(x)~B(x) E, e C
that :r(/~) = sup
inf
sup #(Es)
x~X (U,~,~ E , ) ( x ) ~ ( x ) s~Sx ~eC
Similarly, we can prove: Theorem 3.2. (Eta(#), <~) is a lower semi-lattice, and the extension fl defined by (4) is the least element of Etff#).
188
Zhang Guang-Quan
Acknowledgment The author would like to express his gratitude to Professor Wang Zhenyuan for guidance.
References [1] Wang Zhenyuan, Semi-lattice structure of all extensions of possibility measure and consonant belief function, BUSEFAL 24 (1985) 18-22. [2] Wang Zhenyuan, Semi-lattice isomorphism of the extension of possibility measure and the solutions of fuzzy relation equation, in: R. Trappl, Ed., Cybernetics and Systems '86 (Reidel, Dordrecht, 1986) 581-583.