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Structures of fuzzy order homomorphisms
Fuzzy Sets and Systems21 (1987) 43-51 North-Holland
43
STRUCTURES OF FUZZY ORDER HOMOMORPHISMS* LIU Ying-Ming Institute of Mathematics, Sichuan Universi~, Chengdu, China
Received May 1985 Revised September 1985 First the concept of fuzzy order homomorphism is introduced which is a variety of the generalized order homomorphism given in [7]. Then we show that the fuzzy order homomorphism possesses a nicely local structure. Furthermore, we establish a necessary and sufficient condition under which the Fuzz function is a Zadeh type function. The present condition is simpler than that given in [6]. Finally. for each fuzzy order homomorphism.f, we yield some characterizations about (f-l)-i =.fwhich sharpen the correspondingtheorem in [6]. Keywords: Fuzzy order homomorphism, Zadeh's type function, Fuzz function.
Introduction Zadeh's type function was introduced in the early stage of fuzzy set theory. Both it and its inverse are union-preserving. Using these properties, some authors define the Fuzz function [1] on Fuzzes (i.e., completely distributive lattice with order-reversing involution) and the order homomorphism [5] on L-fuzzy sets respectively. Furthermore, Wang [6] abandons the assumption about existence of involution and gives the notion of generalized order homomorphism on a completely distributive lattice. But the study of mapping on L-fuzzy sets has special importance in itself. In this paper, we shall investigate the mappings (called the fuzzy order homomorphism) on L-fuzzy sets where L may be varied. The fuzzy order homomorphism is essentially a variety of generalized order homomorphisms, but the current terminology may reflect more features on domains and ranges of mappings. Our investigation shows that the fuzzy order homomorphisms possess nicely local structures. Precisely, each fuzzy order homomorphism can be naturally expressed in terms of its restrictions at each point and these restrictions inherit many nice properties from the fuzzy order homomorphism. Based on these results, for a Fuzz function being a Zadeh type function, we get a necessary and sufficient condition which is simpler than the one given in [6]. Finally, we yield some characterizations about f = ( f - 1 ) - l where f is a fuzzy order homomorphism. This aspect of our discussion seems to have independent interest from the view point of algebra. Throughout the paper, L, L~, /--2_always denote the complete lattices; 0 and 1 are its smallest element and ~ e a t e s t element respectively. The identity map on Lj * This research is supported by the Science Fund of the Chinese Academy of ,Science. 0165-0114/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
44
Liu Ying-Ming
(j = 1, 2) is denoted by ij. X and Y will be non-empty crisp sets. For the L-fuzzy set A on X and x • X, the membership degree A(x) at x is also denoted as MemxA. The support of A is { x • X : A ( x ) > O } , denoted by suppA. When suppA is a singleton, A is called L-fuzzy point and denoted by xz where x = suppA and Z =A(x). We assume that for the empty family 0, V 0 = 0 and AO=I.
1. Fuzzy order homomorphisms Definition 1. Suppose f : L~-~ L 2 is a mapping, f is called order-preserving, if for a, b • L, a <- b ~ f(a) <-f(b ). f is called union-preserving, if for each non-empty index set T and Vt • T, at • L1, f ( V at) = V f(at). Similarly, the meet-preserving mapping can be defined. Definition 2. Suppose that f : LI"~ L 2 is a mapping. The inverse off, denoted by f-1 : L2---~L1, is defined as follows: f-l( b)=v{a•L,:f(a)~
b•L2.
Lemma 1. Suppose that f : L1---~ L2 is a mapping. Then (1)f-1 is order-preserving and f - 1 ( 1 ) = 1. (2) If f is order-preserving and L1 is a completely distributive lattice, then f-1 is meet-preserving. (3) If f is order-preserving and La is an infinitely distributive lattice, then f-~ is finitely meet-preserving. Proof. (1) Proof is clear. (2) Take b, • L2 (t • T). For each t, denote the family {a :f(a) ~
Af-'(b,) = t~Tr~Rt A V at,~= r.p~~/{{RI} t~T A a,,~(o t <~V {a • L,:f(a) <~A bt}
=f-'(A bt).
The other containment is clear. (3) Proof is similar to that of (2).
Lemma 2. Suppose that f :L1---~ L2. Then ( 1 ) f - l f ~> il and if f is union-preserving and in]ective, then f - i f = il. (2) If f is union-preserving, then if-1 <~i2; moreover, if f is also sur]ective, then if-1 = i2. Proof. (1) By the definition of f - l , for each a • L1, f - i f ( a ) = V {c • L1 :f(c) ~< f(a)} ~>a. If f is union-preserving, then f(f-lf(a)) = V {f(c):f(c) ~
Structures of fuzzy order homomorphisms
45
Definition 3. A mapping f : L 1 ~ L2 is called generalized order homomorphism if f and its inverse f-1 are both union-preserving, f ( 0 ) = 0 , and f - l ( 0 ) = 0 . f:LXl--~L g is called fuzzy order homomorphism if f is generalized order homomorphism. Remark. (1) The concept of generalized order homomorphism is introduced in [7] for completely distributive lattices L1 and L2. (2) Because any lattice L can be seen as L x, where X is a singleton, each generalized order homomorphism is also a fuzzy order homomorphism. Adopting the new terminology is to emphasize the distinct structures on domains and ranges of the mappings. Definition 4. Suppose that L 1 and L 2 a r e Fuzzes. f : LX---~L2r is called Fuzz function, if (1) f ( 0 ) = 0, (2) f is union-preserving, and (3) for each B • L Y, f-~(B') = (f-~(B))', where the symbol ' denotes the involution, that is f-x is complement-preserving. Remark. We do not assume that L1 = L2. However, in [1], the concept of Fuzz function is given for the case L1 = L2. Notice that if X and Y are singletons, then the mapping f in Definition 4 is reduced to an order homomorphism [5]. Proposition 1. Each Fuzz function f is a fuzzy order homomorphism. Proof. Since f - l ( 1 ) = 1, it follows that f - l ( 0 ) = f - l ( l ' ) = 0 . In view of the complete distributiveness of L 1 and Lemma 1, f-1 is meet-preserving. Thus, because f-~ is complement-preserving and the De Morgan law holds for Fuzzes (see [2]), f-1 is union-preserving. So f is a fuzzy order homomorphism. The following lemma is adapted from Lemma 2.1 of [5]. Lemma 3. Suppose that f :L---~L is a Fuzz function. If Va • L , f(a)>~a, then f and f-1 are identity mappings. ProoL Consider the mapping f-1 first. For b e L, we have f - l ( b ) = ~ / ( a :f(a) ~I b, that is f - l ( b ) = b. Furthermore by Lemma 2, for a • L, ff-l(a) ~ ~ is
Liu Ying-Ming
46
a fuzzy order homomorphism. If there exist fuzzy points xx and x, such that xx <<-f-l(B1) and x, <~f-l(B2), where B1, B2 e L r, then B1 ^ B2d~ O.
Proof. By Lemma 1, f-~ is finitely meet-preserving. If B1 ^ BE = 0, we have 0 = f - l ( 0 ) = f - l ( B l ^ B2) = f - l ( B 0 ^f-l(B2)>>-xx^,. In view of the regularity of L~, either Z or tt is zero. This is a contradiction. The following proposition was given in [6] for the case that L~ = Fuzz function.
L2
and f is a
Proposition 2. Suppose that L1 is infinitely distributive and regular, and f : LX---> L~ is a fuzzy order homomorphism. Then (1) f takes fuzzy points on X to fuzzy points on Y. (2) suppf(xx) = suppf(x,) for each x e X and ;t, it e L1. (3) from f, a crisp mapping f : X---> Y and for each x e X, f,. : L1 ~ L2 can be induced as follows: fx(O) = o,
and
(,)
Proof. (1) Assume f(xx)= B. By the definition o f f -1, we have Xx <~f - l ( B ) = f - l (
V
\yesuppB
ynty)) =
V
y~suppB
f-'(YBcy,).
Hence there is at least one point y e suppB such that Mem,.f-~(y~ty))= 3., ~ 0. By Lemma 4, there is no z e s u p p B such that ~ ~ 0 and z ~ y . Thus xx<~ f-l(ya~y)). By Lemma 2, f(xx) ~ Y and for each x e X , f~:L1--~ L2. The mapping fx:Ll-~ L2 is called the restriction of f at x, and f is called associated crisp mapping with f. Lemma $. Under the assumption of Proposition 2 and denoting f(x) by y, we
have
(1) L(z)= Mem,f(x ) ZeL1, Z#0. (2) f~-'(p) = Memxf-'(y,)/~ e L2, # # 0.
Proof. (1) Proof is clear from formula (*). (2) f~-'(#) = V {P e L, :fx(P) <~/~} = V {P :f(xo) ~
Proof. By (1) of Lemma 5 and the union-Preserving property of f, it is easy to show that the restriction ~ is union-preserving. Obviously f~(0) = 0 and f~(~.) 4:0
Structures of fuzzy order homomorphisms
47
for each 3, ~ 0. Hence f~-x(0)= 0. As to the union-preserving of f.7~, it follows easily from (2) of Lemma 2 and the union-preserving property o f f -1. In the following, we discuss the behavior of f and f~ under the case that the fuzzy order homomorphism f is bijective. First, consider an example:
Example 1. Suppose that L 1 = {0, 1}, L2 = {0, a, b, l : a ~ b , b-'~a}, X = {x, z} and Y = { y } . We define f:LX--~L~ by f(xl)=ya, f(zl)=yb, f ( 1 ) = l and f(0) = 0. Obviously, (1) L1 is regular, (2) f is bijective, (3) f is a fuzzy order homomorphism, and (4) L2 is not regular. We see that f ( x ) = y =f(z), hence f is not injective. Moreover, the image of fx (or f~) at most consists of two elements, hence it is not surjective. Lemma 6. Suppose that f : LX---~ L~ is in]ective and union-preserving. Then for ml, A2 ~ L x, 31 <~A2g:~f(A1) <~f(A2). Proof. If AI~
Theorem 2. Suppose that L1 is infinitely distributive and regular, L2 is regular, and f: LX--->L~ is a fuzzy order homomorphism. Then f is bijective iff the induced map f and each restriction fx are bijective. Proof. Sufficiency follows directly from formula (*) of Proposition 2 and the union-preserving property of f. To show the necessary part, we proceed as follows: (1) f is surjective. In fact, for y e Y, since f is surjective there is A e L x such that f ( A ) = Y l . Hence we have x ~suppA, 3'=A(x) such that f(xx)=yp. Thus f(x) = y. (2) f is injective. Assume f(x) = y =f(z). We write f(xx) = yu and f(zl) = Yr. By the regularity of Lz, # ^ v ~ O. Since f is surjective, we have A e L x, A 4=0 such that f(A) =yu^v. Since f(A)<~f(xl), by Lemma 6, A<~xl. Similarly, A ~ MemA.
Proposition 3. Suppose that L is infinitely distributive and regular, and f: LX---~ L Y is a fuzzy order homomorphism. Then f is increasing iff each fx:L--->L is increasing.
Liu Ying-Ming
48
Proof. Necessity is clear. For A e L x, we have
f ( A ) = f ( V (Xa(~):x e supp A)) = V {f(x)rx(A(x)):X e supp A}. Hence M e m f ( A ) = V {f,,(A(x)):x e supp A} ~> V ( A ( x ) : x e supp A} = M e m A . Lenuna 7. Under the assumption of Proposition 2, for x e X, f ( x ) = y and B e L r" if B ^ y = 0 , then f - l ( B ) ^ x = 0 . Proof. The contrary means there is a fuzzy point xa =f-X(B) ^ x. Thus f(Xo) f f - l ( B ) <~B. Noting f ( x ) = y, f(Xo) = yp, we have y ^ B >~f(Xo) -~ O. The contradiction completes the proof. Theorem 3. Suppose that L1 is a regular Fuzz and f : LXl --> L~ is a fuzzy order homomorphism. Then f is a Fuzz function iff Vx e X, f~:L1-->Lz is a Fuzz function. Proof. Necessity. By virtue of Theorem 1, it suffices to show that fx is complement-preserving. Write f ( x ) =y. When # = 1, f71(# ') =f~-x(0) = 0 = (f~-l(/z))'. When # = 0, we have similarly f~-l(/~,) = (fxl(/z)),. Now assume/z :~ 0, 1. Noting y ' ~ = y , , v B , where B e L ~ and B ^ y = 0 , we get Memxf-X(y~,) = M e m x f - l ( y , , ) by Lemma 7. Since f - x is complement-preserving, by Lemma 5, we have (fxl(~))
' =
Memx(f-l(y,)) ' = Memx f-X(y~) = Mem~ f - l ( y , , ) = f x X ( / ~ ' ) ,
that is f~ is complement-preserving. Sufficiency. It suffices to show that f is complement-preserving, i.e., to show Mem~(f-l(B)) ' = Memxf-X(B ') Vx e X. (1) When f ( x ) = y e supp B', write B' = y, v H where H ^ y = 0. By Lemma 7, f - l ( H ) ^ x = 0, hence M e m x f - l ( B ') = Memxf-X(y~,) =f21(#). Meanwhile, we can assume B = y , , v K where K ^ y = 0 and when #' = 0, we understand y~,, = 0. Similar to the foregoing argument, we have M e m x f - l ( B ) =f21(/z, ) = (f21(#)),, hence the desired formula follows. (2) When f ( x ) = y ~ s u p p B ' , we have y A B ' = 0 . Similarly we get M e m ~ f - l ( B ' ) = 0 . Meanwhile B = y v K where K ^ y = 0, hence M e m ~ f - l ( B ) =f21(1) = 1 = 0'. This completes the proof.
3. Zadeh's type functions Suppose that f :X---> Y is a mapping. Take L = [0, 1]. From f, we can induce a mappingf:LX---> L r" as follows: for A e L x, y e Y, M e m f f ( A ) = sup{A(x):f(x) = y}.
(**)
The mapping f is called Zadeh's type function induced by f (see [4] and [6]). Now
Structures of fuzzy order homomorphisms
49
we can introduce the corresponding concept on more general setting as follows: Definition 7. Suppose that L is infinitely distributive and regular, and f : X--~ Y is an (ordinary) mapping. Then the mapping f : L X - - * L r given by the above formula (**) is called Zadeh's type function induced by f.
Proposition 4. A Zadeh type function is a f u z z y order homomorphism. Moreover, if L is a Fuzz, then a Zadeh type function is a Fuzz function. Proof. Suppose that f : LX---~ L Y is a Zadeh type function induced by f:X---~ Y. It follows directly that f is union-preserving and f(0) = 0. Now for B • L v, x • X, we have f - l ( B ) ( x ) = Mem~ V {A • L x :f(A) ~
z
Bff(x))} =
B](x),
i.e., f-X(B) = Bf. Thereafter it is easy to see that f-1 is union-preserving and f-l(O) = O. So f is a fuzzy order homomorphism. When L is a Fuzz, we have f - ~ ( B ' ) ( x ) = B ' l ( x ) = ( B l ( x ) ) ' = (f-a(B))'(x),
i.e., f - l ( B ' )
=
Oe-l(B)) ', so f is a Fuzz function.
Proposition 5. Suppose that L is infinitely distributive and regular, and f : LX---~ L r is a f u z z y order homomorphism. Then f is a Zadeh type function iff Vx • X, f~ : L---~ L is identity mapping where f~ is the restriction o f f at x.
Proof. Suppose that f is a Zadeh type function induced by g :X--* Y. For x e X, write g ( x ) = y . By the formula (**), f ( x x ) ( y ) = V { x x ( z ) : g ( z ) = y } = L Therefore, by the definition of fx, fx()O = 4, i.e., f~ is identity mapping. Conversely, suppose that each ~ is identity mapping. In view of the unionpreserving property of f and Proposition 2, for A e L x, we have f ( A ) = V ff(XA(x))
:X •
supp A } = V {f(x)i~(A(x)):X • supp A }
= V {f(x)a(~):x • supp A}, where f is the associated crisp mapping with f. Hence for y • Y, we have Memrf(A ) ~ ~/{A(x) :f(x) = y). This just shows that f is a Zadeh type function induced by f.
Theorem 4. Suppose that L is a regular Fuzz and f : LX--> L Y is a Fuzz function. Then (1) (2) (3)
the following are equivalent: f is a Zadeh type function. Each .ix : L--> L is increasing. f is increasing.
Proof. (1)¢~ (2). By Theorem 3, each f~ is a Fuzz function. We have following equivalence: (2)¢~each f~ is an identity mapping (Lemma 3)¢:> (1) (Proposition 5). As to (2)¢~, (3), this is given in Proposition 3.
50
Liu Y'mg-Ming
Remark. The above theorem shows the relative result in [6]. We especially show the complement-preserving requirement off~ in Theorem 2.1 of [6] is superfluous.
4. Twofold inverse of fuzzy, order homomorphism In the present section, we ~ve characterizations about (f-1)-1 = f and some further results on the structure of fuzzv order homomorphisms.
Proposition 6. Suppose that f:Ll---~L,_ satisf3"ing ( f - i ) - l = f . Then: (1) If f is union-preserving, then f is surjective and .f-~ is injectit,e. (2) /f f-~ is unionpreserving, then f is injective and f -1 is surjective. Proof. Applying (1) of Lemma 2 to the mapping f-I:L,_---,L1, we get ( f - 1 ) - I f - l ~ i2, i . e . . f f - l ~ i 2 . On the other hand. by (2) of Lemma 2 and f b e i n g union-preserving, it folloss"sif-1 ~ i2. To sum up. if-1 = i:, so f is surjective and f-1 is injective. (2) Similarly, using Lemma 2 we get f - ~ f ~>il. and (f--1)(f--1)-i ~< il, that is f-if<< il. Hence f - i f = il and this completes the proof. This assumption about (f-1)-1 = f does not in itself imply that f is either surjective or injective. Example 2. Suppose that LI = L2 = [0, 1]. Putf:L1--->L2 by
f(x)=
1,x
x = 1,
x<1.
Then, we have 1, f--x0')= 2y,
y~>½, y
It is easy to see the following: (1) (f-1)-t = f , b u t / i s not surjective. Notice t h a t f is not union-preserving. (2) Write g=f-l:L2--~L1; we have g - l = f and (g-l)-1 =f--i =g. However, g is not injective. qUaeorem 5. Suppose that f : L1---~ L2 is a fuzzy order homomorphism. Then the following are equivalent: (1) f is bijective.
(2) F -1 is hi#cave. (3) f = (f--i)-1.
Moreover, if either [ or ]--1 is bi]ective, then it is just inverse correspondence of the other one in the set-theoretical sense. Proof. (1) If f is bijective, then by Lemma 2 we have f - - i f = il and i f - l = i2. These show f - i is bijective and is the inverse correspondence of f in the set theoretical sense.
Structures of fuz=y order homomorphisms
51
(2) Suppose that f i s bijective. We have shown t h a t f "q is bijective. Then using the argument above, we see that (f-~)-t is the inverse correspondence of .f-k Because both f and (f-1)-I are the inverse correspondence off, f = ( f - l ) - k (3) Suppose that f = ( f - ~ ) - ~ . Since f and f--~ are union-preserving, by Proposition 6 it follows that both f and f-~ are bijective. (4) Finally. under the assumption t h a t f -~ is bijective, we need to show that f i s also bijective. For a e L l , there exists b t e L 2 such that a=.f-~(bt). Write f(a) = b and f - l ( b ) = al. Since f is union-preser~fing, by Lemma 2, ff-l(b 0 <~bx, i.e., b ~b~. Since f --~ is union-preser~fing, a =f-~(bO ~ f - ~ ( b ) = a . On the other hand, al =f--l(b) = f - q f ( a ) e a, hence a = al, that is f - i f ( a ) = a, f - i f = il. Furthermore, for b e/-e_ we have f - l ( b ) = ( f - l f ) f - l ( b ) =f-q(ff-l(b)). Since f - i is injective, b =ff-~(b), i.e., ff-~=i2- Thus f is bijective and is the inverse correspondence o f f - k Remark. The above implication ( 1 ) ~ (3) has already been ~ven in [6], Theorem 6. Suppose that f :L~--~ L~ is a fuzzy order homomorphism, L1 and 1-,2_ are regular, and Ll is infinitely distributive. Then ( f - l ) - i = f iff the associated crisp mapping f : X---~ Y is bijective and for each restriction f~, (fZ~l) - l = f~.
ProoL By Theorem 5. (f-~)-~ = f implies that f is bijective. By Theorem 2, f is bijective implies that each f~ and f are bijective. Since each fx :L~---~/-.2_ is a fuzzy. order homomorphism (Theorem 1), the claim that f~ is bijective is equivalent with (f~ ~)-~ =f~. This completes the proof. Remark. The mapping f : X - - > Y may be seen as a mapping: 2x-->2 r via union-preserving extension. Thus we can define f'-~, and the part on f being bijective in Theorem 6 can be restated in a more tidy form: f satisfies the formula
=f.
References [1] M.A. Erceg. Functions. equix~ence relations, quotient spaces and subsets in fuzzy set theory. Fuzz)." Sets and Systems 3 (1980) 75--92_. [2] l i u Ying-hfing. Fu~3." Stone-t~ch compactifications, Acta Math. Sinica 26 (1983) 507-512 (in Chinese). Abstract in English: Ke.,me Tongbao _Y7(1982) 799. [3] Liu Ying-hfing and hiring He. Induced mappin~ on completely dism'butive lattices, Proc. of the 15th Intemat. Syrup. on Multiple-Value Logi'c (1985) 346-353. [4] Pu Pao-t~fing and Liu Ying-hfing, Fuzz)."Topolo~' 1I, Product and quotient spaces, J. Math. Anal. Appl. 77 (1980) 20-37. [5] Wang Guoqun, Topolo~cal molecular lattices, J. Shaamd Normal Univ. (1979) 1-15 (in Chinese) and Kex-ue Tongbao 28 (1983) 1089-1091 (in Chinese). [6] Wang Guo-Jtm. Order-homomorphism of Fuzzes. Fuzzy. Sets and Systems 12 (1984) 281-288. [7] Wang Guoqun. Pointwise topoloD" on a completeb" distributive lattice. J. Shaanri Normal Univ. (1984) No. 1 and 2 (in Chinese).
43
STRUCTURES OF FUZZY ORDER HOMOMORPHISMS* LIU Ying-Ming Institute of Mathematics, Sichuan Universi~, Chengdu, China
Received May 1985 Revised September 1985 First the concept of fuzzy order homomorphism is introduced which is a variety of the generalized order homomorphism given in [7]. Then we show that the fuzzy order homomorphism possesses a nicely local structure. Furthermore, we establish a necessary and sufficient condition under which the Fuzz function is a Zadeh type function. The present condition is simpler than that given in [6]. Finally. for each fuzzy order homomorphism.f, we yield some characterizations about (f-l)-i =.fwhich sharpen the correspondingtheorem in [6]. Keywords: Fuzzy order homomorphism, Zadeh's type function, Fuzz function.
Introduction Zadeh's type function was introduced in the early stage of fuzzy set theory. Both it and its inverse are union-preserving. Using these properties, some authors define the Fuzz function [1] on Fuzzes (i.e., completely distributive lattice with order-reversing involution) and the order homomorphism [5] on L-fuzzy sets respectively. Furthermore, Wang [6] abandons the assumption about existence of involution and gives the notion of generalized order homomorphism on a completely distributive lattice. But the study of mapping on L-fuzzy sets has special importance in itself. In this paper, we shall investigate the mappings (called the fuzzy order homomorphism) on L-fuzzy sets where L may be varied. The fuzzy order homomorphism is essentially a variety of generalized order homomorphisms, but the current terminology may reflect more features on domains and ranges of mappings. Our investigation shows that the fuzzy order homomorphisms possess nicely local structures. Precisely, each fuzzy order homomorphism can be naturally expressed in terms of its restrictions at each point and these restrictions inherit many nice properties from the fuzzy order homomorphism. Based on these results, for a Fuzz function being a Zadeh type function, we get a necessary and sufficient condition which is simpler than the one given in [6]. Finally, we yield some characterizations about f = ( f - 1 ) - l where f is a fuzzy order homomorphism. This aspect of our discussion seems to have independent interest from the view point of algebra. Throughout the paper, L, L~, /--2_always denote the complete lattices; 0 and 1 are its smallest element and ~ e a t e s t element respectively. The identity map on Lj * This research is supported by the Science Fund of the Chinese Academy of ,Science. 0165-0114/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)
44
Liu Ying-Ming
(j = 1, 2) is denoted by ij. X and Y will be non-empty crisp sets. For the L-fuzzy set A on X and x • X, the membership degree A(x) at x is also denoted as MemxA. The support of A is { x • X : A ( x ) > O } , denoted by suppA. When suppA is a singleton, A is called L-fuzzy point and denoted by xz where x = suppA and Z =A(x). We assume that for the empty family 0, V 0 = 0 and AO=I.
1. Fuzzy order homomorphisms Definition 1. Suppose f : L~-~ L 2 is a mapping, f is called order-preserving, if for a, b • L, a <- b ~ f(a) <-f(b ). f is called union-preserving, if for each non-empty index set T and Vt • T, at • L1, f ( V at) = V f(at). Similarly, the meet-preserving mapping can be defined. Definition 2. Suppose that f : LI"~ L 2 is a mapping. The inverse off, denoted by f-1 : L2---~L1, is defined as follows: f-l( b)=v{a•L,:f(a)~
b•L2.
Lemma 1. Suppose that f : L1---~ L2 is a mapping. Then (1)f-1 is order-preserving and f - 1 ( 1 ) = 1. (2) If f is order-preserving and L1 is a completely distributive lattice, then f-1 is meet-preserving. (3) If f is order-preserving and La is an infinitely distributive lattice, then f-~ is finitely meet-preserving. Proof. (1) Proof is clear. (2) Take b, • L2 (t • T). For each t, denote the family {a :f(a) ~
Af-'(b,) = t~Tr~Rt A V at,~= r.p~~/{{RI} t~T A a,,~(o t <~V {a • L,:f(a) <~A bt}
=f-'(A bt).
The other containment is clear. (3) Proof is similar to that of (2).
Lemma 2. Suppose that f :L1---~ L2. Then ( 1 ) f - l f ~> il and if f is union-preserving and in]ective, then f - i f = il. (2) If f is union-preserving, then if-1 <~i2; moreover, if f is also sur]ective, then if-1 = i2. Proof. (1) By the definition of f - l , for each a • L1, f - i f ( a ) = V {c • L1 :f(c) ~< f(a)} ~>a. If f is union-preserving, then f(f-lf(a)) = V {f(c):f(c) ~
Structures of fuzzy order homomorphisms
45
Definition 3. A mapping f : L 1 ~ L2 is called generalized order homomorphism if f and its inverse f-1 are both union-preserving, f ( 0 ) = 0 , and f - l ( 0 ) = 0 . f:LXl--~L g is called fuzzy order homomorphism if f is generalized order homomorphism. Remark. (1) The concept of generalized order homomorphism is introduced in [7] for completely distributive lattices L1 and L2. (2) Because any lattice L can be seen as L x, where X is a singleton, each generalized order homomorphism is also a fuzzy order homomorphism. Adopting the new terminology is to emphasize the distinct structures on domains and ranges of the mappings. Definition 4. Suppose that L 1 and L 2 a r e Fuzzes. f : LX---~L2r is called Fuzz function, if (1) f ( 0 ) = 0, (2) f is union-preserving, and (3) for each B • L Y, f-~(B') = (f-~(B))', where the symbol ' denotes the involution, that is f-x is complement-preserving. Remark. We do not assume that L1 = L2. However, in [1], the concept of Fuzz function is given for the case L1 = L2. Notice that if X and Y are singletons, then the mapping f in Definition 4 is reduced to an order homomorphism [5]. Proposition 1. Each Fuzz function f is a fuzzy order homomorphism. Proof. Since f - l ( 1 ) = 1, it follows that f - l ( 0 ) = f - l ( l ' ) = 0 . In view of the complete distributiveness of L 1 and Lemma 1, f-1 is meet-preserving. Thus, because f-~ is complement-preserving and the De Morgan law holds for Fuzzes (see [2]), f-1 is union-preserving. So f is a fuzzy order homomorphism. The following lemma is adapted from Lemma 2.1 of [5]. Lemma 3. Suppose that f :L---~L is a Fuzz function. If Va • L , f(a)>~a, then f and f-1 are identity mappings. ProoL Consider the mapping f-1 first. For b e L, we have f - l ( b ) = ~ / ( a :f(a) ~
Liu Ying-Ming
46
a fuzzy order homomorphism. If there exist fuzzy points xx and x, such that xx <<-f-l(B1) and x, <~f-l(B2), where B1, B2 e L r, then B1 ^ B2d~ O.
Proof. By Lemma 1, f-~ is finitely meet-preserving. If B1 ^ BE = 0, we have 0 = f - l ( 0 ) = f - l ( B l ^ B2) = f - l ( B 0 ^f-l(B2)>>-xx^,. In view of the regularity of L~, either Z or tt is zero. This is a contradiction. The following proposition was given in [6] for the case that L~ = Fuzz function.
L2
and f is a
Proposition 2. Suppose that L1 is infinitely distributive and regular, and f : LX---> L~ is a fuzzy order homomorphism. Then (1) f takes fuzzy points on X to fuzzy points on Y. (2) suppf(xx) = suppf(x,) for each x e X and ;t, it e L1. (3) from f, a crisp mapping f : X---> Y and for each x e X, f,. : L1 ~ L2 can be induced as follows: fx(O) = o,
and
(,)
Proof. (1) Assume f(xx)= B. By the definition o f f -1, we have Xx <~f - l ( B ) = f - l (
V
\yesuppB
ynty)) =
V
y~suppB
f-'(YBcy,).
Hence there is at least one point y e suppB such that Mem,.f-~(y~ty))= 3., ~ 0. By Lemma 4, there is no z e s u p p B such that ~ ~ 0 and z ~ y . Thus xx<~ f-l(ya~y)). By Lemma 2, f(xx) ~
have
(1) L(z)= Mem,f(x ) ZeL1, Z#0. (2) f~-'(p) = Memxf-'(y,)/~ e L2, # # 0.
Proof. (1) Proof is clear from formula (*). (2) f~-'(#) = V {P e L, :fx(P) <~/~} = V {P :f(xo) ~
Proof. By (1) of Lemma 5 and the union-Preserving property of f, it is easy to show that the restriction ~ is union-preserving. Obviously f~(0) = 0 and f~(~.) 4:0
Structures of fuzzy order homomorphisms
47
for each 3, ~ 0. Hence f~-x(0)= 0. As to the union-preserving of f.7~, it follows easily from (2) of Lemma 2 and the union-preserving property o f f -1. In the following, we discuss the behavior of f and f~ under the case that the fuzzy order homomorphism f is bijective. First, consider an example:
Example 1. Suppose that L 1 = {0, 1}, L2 = {0, a, b, l : a ~ b , b-'~a}, X = {x, z} and Y = { y } . We define f:LX--~L~ by f(xl)=ya, f(zl)=yb, f ( 1 ) = l and f(0) = 0. Obviously, (1) L1 is regular, (2) f is bijective, (3) f is a fuzzy order homomorphism, and (4) L2 is not regular. We see that f ( x ) = y =f(z), hence f is not injective. Moreover, the image of fx (or f~) at most consists of two elements, hence it is not surjective. Lemma 6. Suppose that f : LX---~ L~ is in]ective and union-preserving. Then for ml, A2 ~ L x, 31 <~A2g:~f(A1) <~f(A2). Proof. If AI~
Theorem 2. Suppose that L1 is infinitely distributive and regular, L2 is regular, and f: LX--->L~ is a fuzzy order homomorphism. Then f is bijective iff the induced map f and each restriction fx are bijective. Proof. Sufficiency follows directly from formula (*) of Proposition 2 and the union-preserving property of f. To show the necessary part, we proceed as follows: (1) f is surjective. In fact, for y e Y, since f is surjective there is A e L x such that f ( A ) = Y l . Hence we have x ~suppA, 3'=A(x) such that f(xx)=yp. Thus f(x) = y. (2) f is injective. Assume f(x) = y =f(z). We write f(xx) = yu and f(zl) = Yr. By the regularity of Lz, # ^ v ~ O. Since f is surjective, we have A e L x, A 4=0 such that f(A) =yu^v. Since f(A)<~f(xl), by Lemma 6, A<~xl. Similarly, A ~
Proposition 3. Suppose that L is infinitely distributive and regular, and f: LX---~ L Y is a fuzzy order homomorphism. Then f is increasing iff each fx:L--->L is increasing.
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48
Proof. Necessity is clear. For A e L x, we have
f ( A ) = f ( V (Xa(~):x e supp A)) = V {f(x)rx(A(x)):X e supp A}. Hence M e m f ( A ) = V {f,,(A(x)):x e supp A} ~> V ( A ( x ) : x e supp A} = M e m A . Lenuna 7. Under the assumption of Proposition 2, for x e X, f ( x ) = y and B e L r" if B ^ y = 0 , then f - l ( B ) ^ x = 0 . Proof. The contrary means there is a fuzzy point xa =f-X(B) ^ x. Thus f(Xo) f f - l ( B ) <~B. Noting f ( x ) = y, f(Xo) = yp, we have y ^ B >~f(Xo) -~ O. The contradiction completes the proof. Theorem 3. Suppose that L1 is a regular Fuzz and f : LXl --> L~ is a fuzzy order homomorphism. Then f is a Fuzz function iff Vx e X, f~:L1-->Lz is a Fuzz function. Proof. Necessity. By virtue of Theorem 1, it suffices to show that fx is complement-preserving. Write f ( x ) =y. When # = 1, f71(# ') =f~-x(0) = 0 = (f~-l(/z))'. When # = 0, we have similarly f~-l(/~,) = (fxl(/z)),. Now assume/z :~ 0, 1. Noting y ' ~ = y , , v B , where B e L ~ and B ^ y = 0 , we get Memxf-X(y~,) = M e m x f - l ( y , , ) by Lemma 7. Since f - x is complement-preserving, by Lemma 5, we have (fxl(~))
' =
Memx(f-l(y,)) ' = Memx f-X(y~) = Mem~ f - l ( y , , ) = f x X ( / ~ ' ) ,
that is f~ is complement-preserving. Sufficiency. It suffices to show that f is complement-preserving, i.e., to show Mem~(f-l(B)) ' = Memxf-X(B ') Vx e X. (1) When f ( x ) = y e supp B', write B' = y, v H where H ^ y = 0. By Lemma 7, f - l ( H ) ^ x = 0, hence M e m x f - l ( B ') = Memxf-X(y~,) =f21(#). Meanwhile, we can assume B = y , , v K where K ^ y = 0 and when #' = 0, we understand y~,, = 0. Similar to the foregoing argument, we have M e m x f - l ( B ) =f21(/z, ) = (f21(#)),, hence the desired formula follows. (2) When f ( x ) = y ~ s u p p B ' , we have y A B ' = 0 . Similarly we get M e m ~ f - l ( B ' ) = 0 . Meanwhile B = y v K where K ^ y = 0, hence M e m ~ f - l ( B ) =f21(1) = 1 = 0'. This completes the proof.
3. Zadeh's type functions Suppose that f :X---> Y is a mapping. Take L = [0, 1]. From f, we can induce a mappingf:LX---> L r" as follows: for A e L x, y e Y, M e m f f ( A ) = sup{A(x):f(x) = y}.
(**)
The mapping f is called Zadeh's type function induced by f (see [4] and [6]). Now
Structures of fuzzy order homomorphisms
49
we can introduce the corresponding concept on more general setting as follows: Definition 7. Suppose that L is infinitely distributive and regular, and f : X--~ Y is an (ordinary) mapping. Then the mapping f : L X - - * L r given by the above formula (**) is called Zadeh's type function induced by f.
Proposition 4. A Zadeh type function is a f u z z y order homomorphism. Moreover, if L is a Fuzz, then a Zadeh type function is a Fuzz function. Proof. Suppose that f : LX---~ L Y is a Zadeh type function induced by f:X---~ Y. It follows directly that f is union-preserving and f(0) = 0. Now for B • L v, x • X, we have f - l ( B ) ( x ) = Mem~ V {A • L x :f(A) ~
z
Bff(x))} =
B](x),
i.e., f-X(B) = Bf. Thereafter it is easy to see that f-1 is union-preserving and f-l(O) = O. So f is a fuzzy order homomorphism. When L is a Fuzz, we have f - ~ ( B ' ) ( x ) = B ' l ( x ) = ( B l ( x ) ) ' = (f-a(B))'(x),
i.e., f - l ( B ' )
=
Oe-l(B)) ', so f is a Fuzz function.
Proposition 5. Suppose that L is infinitely distributive and regular, and f : LX---~ L r is a f u z z y order homomorphism. Then f is a Zadeh type function iff Vx • X, f~ : L---~ L is identity mapping where f~ is the restriction o f f at x.
Proof. Suppose that f is a Zadeh type function induced by g :X--* Y. For x e X, write g ( x ) = y . By the formula (**), f ( x x ) ( y ) = V { x x ( z ) : g ( z ) = y } = L Therefore, by the definition of fx, fx()O = 4, i.e., f~ is identity mapping. Conversely, suppose that each ~ is identity mapping. In view of the unionpreserving property of f and Proposition 2, for A e L x, we have f ( A ) = V ff(XA(x))
:X •
supp A } = V {f(x)i~(A(x)):X • supp A }
= V {f(x)a(~):x • supp A}, where f is the associated crisp mapping with f. Hence for y • Y, we have Memrf(A ) ~ ~/{A(x) :f(x) = y). This just shows that f is a Zadeh type function induced by f.
Theorem 4. Suppose that L is a regular Fuzz and f : LX--> L Y is a Fuzz function. Then (1) (2) (3)
the following are equivalent: f is a Zadeh type function. Each .ix : L--> L is increasing. f is increasing.
Proof. (1)¢~ (2). By Theorem 3, each f~ is a Fuzz function. We have following equivalence: (2)¢~each f~ is an identity mapping (Lemma 3)¢:> (1) (Proposition 5). As to (2)¢~, (3), this is given in Proposition 3.
50
Liu Y'mg-Ming
Remark. The above theorem shows the relative result in [6]. We especially show the complement-preserving requirement off~ in Theorem 2.1 of [6] is superfluous.
4. Twofold inverse of fuzzy, order homomorphism In the present section, we ~ve characterizations about (f-1)-1 = f and some further results on the structure of fuzzv order homomorphisms.
Proposition 6. Suppose that f:Ll---~L,_ satisf3"ing ( f - i ) - l = f . Then: (1) If f is union-preserving, then f is surjective and .f-~ is injectit,e. (2) /f f-~ is unionpreserving, then f is injective and f -1 is surjective. Proof. Applying (1) of Lemma 2 to the mapping f-I:L,_---,L1, we get ( f - 1 ) - I f - l ~ i2, i . e . . f f - l ~ i 2 . On the other hand. by (2) of Lemma 2 and f b e i n g union-preserving, it folloss"sif-1 ~ i2. To sum up. if-1 = i:, so f is surjective and f-1 is injective. (2) Similarly, using Lemma 2 we get f - ~ f ~>il. and (f--1)(f--1)-i ~< il, that is f-if<< il. Hence f - i f = il and this completes the proof. This assumption about (f-1)-1 = f does not in itself imply that f is either surjective or injective. Example 2. Suppose that LI = L2 = [0, 1]. Putf:L1--->L2 by
f(x)=
1,x
x = 1,
x<1.
Then, we have 1, f--x0')= 2y,
y~>½, y
It is easy to see the following: (1) (f-1)-t = f , b u t / i s not surjective. Notice t h a t f is not union-preserving. (2) Write g=f-l:L2--~L1; we have g - l = f and (g-l)-1 =f--i =g. However, g is not injective. qUaeorem 5. Suppose that f : L1---~ L2 is a fuzzy order homomorphism. Then the following are equivalent: (1) f is bijective.
(2) F -1 is hi#cave. (3) f = (f--i)-1.
Moreover, if either [ or ]--1 is bi]ective, then it is just inverse correspondence of the other one in the set-theoretical sense. Proof. (1) If f is bijective, then by Lemma 2 we have f - - i f = il and i f - l = i2. These show f - i is bijective and is the inverse correspondence of f in the set theoretical sense.
Structures of fuz=y order homomorphisms
51
(2) Suppose that f i s bijective. We have shown t h a t f "q is bijective. Then using the argument above, we see that (f-~)-t is the inverse correspondence of .f-k Because both f and (f-1)-I are the inverse correspondence off, f = ( f - l ) - k (3) Suppose that f = ( f - ~ ) - ~ . Since f and f--~ are union-preserving, by Proposition 6 it follows that both f and f-~ are bijective. (4) Finally. under the assumption t h a t f -~ is bijective, we need to show that f i s also bijective. For a e L l , there exists b t e L 2 such that a=.f-~(bt). Write f(a) = b and f - l ( b ) = al. Since f is union-preser~fing, by Lemma 2, ff-l(b 0 <~bx, i.e., b ~b~. Since f --~ is union-preser~fing, a =f-~(bO ~ f - ~ ( b ) = a . On the other hand, al =f--l(b) = f - q f ( a ) e a, hence a = al, that is f - i f ( a ) = a, f - i f = il. Furthermore, for b e/-e_ we have f - l ( b ) = ( f - l f ) f - l ( b ) =f-q(ff-l(b)). Since f - i is injective, b =ff-~(b), i.e., ff-~=i2- Thus f is bijective and is the inverse correspondence o f f - k Remark. The above implication ( 1 ) ~ (3) has already been ~ven in [6], Theorem 6. Suppose that f :L~--~ L~ is a fuzzy order homomorphism, L1 and 1-,2_ are regular, and Ll is infinitely distributive. Then ( f - l ) - i = f iff the associated crisp mapping f : X---~ Y is bijective and for each restriction f~, (fZ~l) - l = f~.
ProoL By Theorem 5. (f-~)-~ = f implies that f is bijective. By Theorem 2, f is bijective implies that each f~ and f are bijective. Since each fx :L~---~/-.2_ is a fuzzy. order homomorphism (Theorem 1), the claim that f~ is bijective is equivalent with (f~ ~)-~ =f~. This completes the proof. Remark. The mapping f : X - - > Y may be seen as a mapping: 2x-->2 r via union-preserving extension. Thus we can define f'-~, and the part on f being bijective in Theorem 6 can be restated in a more tidy form: f satisfies the formula
=f.
References [1] M.A. Erceg. Functions. equix~ence relations, quotient spaces and subsets in fuzzy set theory. Fuzz)." Sets and Systems 3 (1980) 75--92_. [2] l i u Ying-hfing. Fu~3." Stone-t~ch compactifications, Acta Math. Sinica 26 (1983) 507-512 (in Chinese). Abstract in English: Ke.,me Tongbao _Y7(1982) 799. [3] Liu Ying-hfing and hiring He. Induced mappin~ on completely dism'butive lattices, Proc. of the 15th Intemat. Syrup. on Multiple-Value Logi'c (1985) 346-353. [4] Pu Pao-t~fing and Liu Ying-hfing, Fuzz)."Topolo~' 1I, Product and quotient spaces, J. Math. Anal. Appl. 77 (1980) 20-37. [5] Wang Guoqun, Topolo~cal molecular lattices, J. Shaamd Normal Univ. (1979) 1-15 (in Chinese) and Kex-ue Tongbao 28 (1983) 1089-1091 (in Chinese). [6] Wang Guo-Jtm. Order-homomorphism of Fuzzes. Fuzzy. Sets and Systems 12 (1984) 281-288. [7] Wang Guoqun. Pointwise topoloD" on a completeb" distributive lattice. J. Shaanri Normal Univ. (1984) No. 1 and 2 (in Chinese).