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The Topology of AFS Structure and AFS Algebras
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
217, 479]489 Ž1998.
AY975719
The Topology of AFS Structure and AFS Algebras Liu Xiaodong Department of Basic Science, Dalian Maritime Uni¨ ersity, Dalian, 116026, People’s Republic of China Submitted by Ying-ming Liu Received December 16, 1996
In this paper, we will study the topological structure of the AFS structures published by the author; discuss the topological molecular lattice structures on EI, *EI, EII, *EII algebras published by the author; and give the relations of these topological structures. As applications, we study the topological produced by a family of fuzzy concepts in the AFS algebras and apply these to analyze relations among fuzzy concepts. Using these, we believe that we can study the law of human thinking. The most important fact is that all these can be operated by computers. We are also sure AFS theory will play an important role in information processing. Q 1998 Academic Press
Key Words: topological molecular lattice; AFS algebras; AFS structure.
1. TOPOLOGY ON *EI, *EII ALGEBRAS In this section, we discuss the topological structure of AFS structure on *EI, *EII algebras. We know that for any sets X and M there are two kinds of AFS algebras, which are EI, *EI, EII, *EII algebras Žrefer to w10x.. Although they have similar algebra structures, their topological structures are different. First, we consider the topological structure on *EI, *EII algebras, which, for any a, b g EM or EXM a k b s a) b and a n b s a q b. DEFINITION 1.1. Let X and M be sets, and Ž M, t , X . be an AFS structure. h : EM the *EI algebra over M, h is called a closed topology Žrefer to w11x. if f , M g h and h is closed under finite unions Žk or *. and arbitrary intersections Žn or q.. h is called a topological molecular lattice on *EI algebra over M of AFS structure Ž M, t , X ., denoted as Ž EM, h .. DEFINITION 1.2. Let X and M be sets, and Ž M, t , X . be an AFS structure. h is a topological molecular lattice on *EI algebra over M of 479 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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LIU XIAODONG
AFS structure Ž M, t , X .. For any x g X, Ý i g I A i g EM, and Ý i g I A i g h , we define NÝ i g I A iŽ x . s y ¬ t Ž x, y . G
½
Ý Ai igI
5.
This is called the neighborhood of x inducing by Ý i g I A i g h.
½
Nh Ž x . s NÝ i g I A iŽ x .
Ý Ai g h igI
5
is called the neighborhood of x inducing by h. PROPOSITION 1.1. Let X and M be sets and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. For any Ý i g I A i , Ý i g J Bi g h , and x g X, the following hold: Ž1.
Ý A i G Ý Bi , then NÝ
if
igI
Ž2. Ž3.
igJ
ig I
Ž x . : NÝ
Ai
ig J
Ž x.
Bi
NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x ..
Proof. Ž1. Suppose y g NÝ i g I A iŽ x .. There exists A k , k g I such that t Ž x, y . = A k . On the other hand, since Ý i g I A i G Ý i g J Bi , hence there exists Bj , j g J such that t Ž x, y . = A k = Bj . This implies y g NÝ i g J B iŽ x . and NÝ i g I A iŽ x . : NÝ i g J B iŽ x . . Ž2. For any y g NÝ i g I A iŽ x . l NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . and y g NÝ i g J B iŽ x . m t Ž x, y . G
t Ž x, y . G
Ý A i and igI
Ý Bi m t Ž x, y . G Ý A i ) Ý Bi m y g NÝ igJ
igI
igJ
ig I
A i ) Ý ig J Bi
This implies NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . .
Ž x. .
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TOPOLOGY OF AFS THEORY
Ž3. For any y g NÝ i g I A iŽ x . j NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . or y g NÝ i g J B iŽ x . m t Ž x, y . G
t Ž x, y . G
Ý Bi m t Ž x, y . G Ý A i n Ý Bi igJ
s
Ý A i q Ý Bi m y g NÝ igI
Ý A i or igI
igJ
igI ig I
A iqÝ i g J B i
igJ
Ž x. .
This implies NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x . . THEOREM 1.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. If
½
B s NÝ i g I A iŽ x . x g X ,
Ý Ai g h igI
5
,
then B is a base for some topology. Proof. First, because Ž M, t , X . is a strong relative AFS structure, for any x g X, t Ž x, x . s M. M is the maximal element of *EI algebra over M. This implies that for any Ý i g I A i g h , t Ž x, x . G Ý i g I A i so that x g NÝ i g I A iŽ x .. Therefore X s Db g B b . Second, suppose x g X, and U, V g B, for which x g U and V. We want to prove there exists W g B such that x g W : U l V. By the hypothesis, we know there exists Ý i g I A i , Ý i g J Bi g h such that U s NÝ i g I A iŽ u., V s NÝ i g J B iŽ ¨ . and 'l g I, 'k g J, t Ž u, x . = A l and t Ž ¨ , x . = Bk . For any y g NA lŽ x ., i.e., t Ž x, y . = A l , since AX 2, hence t Ž u, y . = t Ž u, x . l t Ž x, y . = A l and y g U. This implies NA lŽ x . : U. Similarly NB kŽ x . : V. Because NA lŽ x . l NB kŽ x . s NA l ) B kŽ x . , x g NA l ) B kŽ x . g B, we have W s NA l ) B kŽ x . : U l V . Now we prove that B is a base for some topology. The topological space Ž X, Th ., in which B is a base for Th , is called the topology induced by h.
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LIU XIAODONG
THEOREM 1.2. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. If
jh s
½Ý
ai A i
igI
Ý ai A i g EXM, ai g Th , ; i g I, Ý A i g h igI
igI
5
,
then jh is a topological molecular lattice on *EII algebra o¨ er M of AFS structure Ž M, t , X .. Proof. For any finite integer n, let A j s Ý i g I j a ji A ji g EXM, j s 1, 2, . . . , n. Because F1 F j F n a j f Ž j. g Th for any f g Ł 1 F j F n I j and Ý f g Ł 1 F j F n I j D1 F j F n A j f Ž j. g h ,
E
Aj s
Ý
F
D
a j f Ž j.
fgŁ 1F j F n I j 1FjFn
1FiFn
A j f Ž j. g jh .
1FjFn
It is obvious that jh is closed under arbitrary intersection Žn or q.. Therefore jh is a topological molecular lattice. It is called the topological molecular lattice induced by h. In the following, we give an example of these topological structures. EXAMPLE 1.1. Let X s x 1 , x 2 , . . . , x 5 4 be a set of 5 persons. M s age, weight, height, salary, wealth, man, women4 . Suppose there exists the following table:
x1 x2 x3 x4 x5
age
weight
height
man
woman
21 30 27 60 45
50 52 65 63 54
1.69 1.62 1.80 1.50 1.71
yes no yes no yes
no yes no yes no
salary
wealth
0 120 100 80 140
0.000 200,000 40,000 324,000 486,940,000
We get the following table of structure t t
x1
x2
x3
x1 x2 x3 x4 x5
A, M, H, We, S4 A, W, We, S, Q4 A, M, H, We, S, Q4 A, W, We, Q4 A, M, S, Q4
M, H 4 A, W, H, We, S, Q4 M, H, We4 A, W, We, Q4 A, M, H, We, S, Q4
M4 A, W, S, Q4 A, M, H, We, S, Q4 A, W, Q4 A, M, S, Q4
TOPOLOGY OF AFS THEORY
t
x4
x5
x1 x2 x3 x4 x5
M, H 4 W, H, S 4 M, H, We, S4 A, W, We, H, S, Q4 M, H, S, Q4
M4 W 4 M, H, We4 A, W, We4 A, M, H, S, We, Q4
A, age; M, man; W, woman; H, height; We, weight; S, salary; Q, wealth. We can verify that t satisfies AX 1 and AX 2 and Ž M, t , X . is an AFS structure.
483
We consider the relations among age, height, and weight. Let h be the topological molecular lattice generated by A4 , H 4 , and We4 , which are elements in *EI algebra over M, i.e., h consists of B, A, M, W, H, We, S, Q4 and the following: A4 q H 4 q We4 , A4 q H 4 , A4 q We4 , H 4 q We4 A4 , H 4 , We4 , A, H 4 q A, We4 q We, H 4 , A, H 4 q A, We4 , A, H 4 q We, H 4 , A, We4 q We, H 4 , A, H 4 , A, We4 , We, H 4 , A4 q We, H 4 , H 4 q A, We4 , We4 q A, H 4 , A, H, We4 . Now we consider the base of the topology Th for x 1 , x 2 , x 3 , x 4 , x 54 : N A4q H 4qW e4Ž x 1 . s Ž x 1 , x 2 , x 4 4 N A4q H 4Ž x 1 . s x 1 , x 2 , x 4 4 , N A4qW e4Ž x 1 . s x 14 , Ž . 4 N H 4qW e4 x 1 s x 1 , x 2 , x 4 N A4Ž x 1 . s x 14 , N H 4Ž x 1 . s x 1 , x 2 , x 4 4 , NW e4Ž x 1 . s x 14 N A, H 4q A, W e4qW e, H 4Ž x 1 . s x 14 N A, H 4q A, W e4Ž x 1 . s x 14 , N A, H 4qW e, H 4Ž x 1 . s x 14 , N A, W e4qW e, H 4Ž x 1 . s x 14 N A, H 4Ž x 1 . s x 14 , N A, W e4Ž x 1 . s x 14 , NW e, H 4Ž x 1 . s x 14 N A4qW e, H 4Ž x 1 . s x 14 , N H 4q A, W e4Ž x 1 . s x 1 , x 2 , x 4 4 , Ž . 4 NW e4q A, H 4 x 1 s x 1 N A, H , W e4Ž x 1 . s x 14 . Therefore the neighborhoods of x 1 induced by h is Nh Ž x 1 . s x 1 , x 2 , x 4 4 , x 1 4 4 .
484
LIU XIAODONG
Similarly, we get the neighborhoods of x 2 induced by h Nh Ž x 2 . s x 1 , x 2 , x 3 , x 4 4 , x 1 , x 2 , x 4 4 , x 1 , x 2 , x 3 4 ,
x1 , x 2 4 , x 2 , x 4 4 , x 2 4 4 ; the neighborhoods of x 3 induced by h is Nh Ž x 3 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 3 4 4 ; the neighborhoods of x 4 induced by h is Nh Ž x 4 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 2 , x 4 , x 5 4 , x 4 4 4 ; the neighborhoods of x 5 induced by h is Nh Ž x 5 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 2 , x 3 , x 5 4 , x 2 , x 4 , x 5 4 , x 2 , x 5 4 4 . In the following, we discuss the real meaning of the neighborhoods of x i induced by h. First, from above, we know x 1 , x 2 , x 4 are discrete points for the topology Th . Coincidentally, they are the minimal points of concepts A4 , H 4 , We4 , respectively. For any i, we can prove that for any a g Nh Ž x i ., x 5 g a « x 2 g a. This implies that the degree of x 5 belongs to some concept which is always larger than that of x 2 . x 5 f x 1 , x 34 g Nh Ž x 3 ., x 3 f x 2 , x 54 g Nh Ž x 5 ., i.e., the separation property of topology Th . This implies that there exist two fuzzy concepts a, b g h such that x 5 , x 3 can be discriminated by fuzzy concept a, b. We should notice that the above computing process can be done by computer. THEOREM 1.3. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. jh is the topological molecular lattice induced by h. s is the Borel set corresponding to topological space Ž X, Th . and Ž M, t , X, s , m. is a semi-cogniti¨ e field; then Ž1. If Ý i g I A i g h and Ý j g J Bj G Ý i g I A i « Ý j g J Bj g h , then Ý i g U Ci is a measurable concept under s , for any Ý i g U Ci g h. Ž2. Ý i g I A i g h , I is any indexing set, D is a directed set, S: D ª X, S is con¨ erged to a g X under Th . S9: D ª EXM is defined as follows: for any d g D, S9Ž d . s Ý i g I A iŽ sŽ d .. A i , then S9 is con¨ erged to Ý i g I A iŽ a. A i g EXM under jh . Proof. Ž1. For any Ý i g I A i g h , x g X, since A i G Ý i g I A i , hence A i g h and A iŽ x . s NA i Ž x . g Th « NA i Ž x . g s , ; i g I. Therefore Ý i g I A i is a measurable concept under s .
TOPOLOGY OF AFS THEORY
485
Ž2. Suppose Ý i g I A iŽ a. A i g Ý j g J pj Pj , Ý j g J pj Pj g h. This implies that there exists pl Pl such that ; i g I, either A iŽ a. W pl or pl W A i . First, we suppose ;k g I, Pl W A k . This implies for any d g D,
Ý A i Ž S Ž d . . A i g Ý pj Pj . igI
jgJ
Second, there exists k g I such that A kŽ a. W pl . Since a g A kŽ a. g s and S is converged to a g X under Th , hence there exists N g D such that for any n g D, n G N, SŽ n. g A kŽ a. W pl . For any y g A kŽ S Ž n.., i.e., t Ž SŽ n., y . = A k . Since SŽ n. g A kŽ a., i.e., t Ž a, SŽ n.. = A k and t is an AFS structure, hence t Ž a, y . = t Ž a, SŽ n.. l t Ž SŽ n., y . « y g A kŽ a. « A kŽ a. = A kŽ S Ž n... Since A iŽ a. W pl , hence A iŽ SŽ n... W pl . This implies that for any Ý j g J pj Pj g h and Ý i g I A iŽ a. A i g Ý j g J pj Pj , there exists N g D such that for any n g D, n G N, Ý i g I A iŽ S Ž n.. A i g Ý j g J pj Pj . Therefore S9 is converged to Ý i g I A iŽ a. A i g EXM under jh . 2. TOPOLOGY ON EI, EII ALGEBRAS In this section, we discuss the topological structure of AFS structure on EI, EII algebras. We consider the topological structure on EI, EII algebras, which, for any a, b g EM or EXM, a k b s a q b and a n b s a) b. DEFINITION 2.1. Let X and M be sets, and Ž M, t , X . be an AFS structure, h : EM the EI algebra over M, h is called a closed topology if f , M g h , and h is closed under finite unions Žk or q. and arbitrary intersections Žn or ).. h is called a topological molecular on EI algebra over M of AFS structure Ž M, t , X . denoted as Ž EM, h .. DEFINITION 2.2. Let X and M be sets, and Ž M, t , X . be an AFS structure. h is a topological molecular lattice on EI algebra over M of AFS structure Ž M, t , X .. For any x g X, Ý i g I A i g EM, and Ý i g I A i g h , we define NÝ i g I A iŽ x . s y N t Ž y, x . F Ý i g I A i 4 . It is called the neighborhood of x inducing by Ý i g I A i g h. Nh Ž x . s NÝ i g I A iŽ x . N Ý i g I A i g h 4 is called the neighborhood of x inducing by h. PROPOSITION 2.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on EI algebra o¨ er M of AFS structure Ž M, t , X .. For any Ý i g I A i , Ý i g J Bi g h , and x g X, the following hold: Ž1. Ž2. Ž3.
if Ý i g I A i G Ý i g J Bi , then NÝ i g I A iŽ x . = NÝ i g J B iŽ x . NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x .
486
LIU XIAODONG
Proof. Ž1. Suppose y g NÝ i g J B iŽ x .. This implies that 'k g J, Bk : t Ž y, x .. On the other hand, since Ý i g I A i G Ý i g J Bi , hence for Bk , there exists A l , l g I, such that A l : Bk : t Ž y, x .. This implies y g NÝ i g I A iŽ x . and NÝ i g I A iŽ x . = NÝ i g J B iŽ x .. Ž2. For any y g NÝ i g I A iŽ x . l NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . and y g NÝ i g J B iŽ x . m t Ž y, x . F
t Ž y, x . F
Ý Bi m t Ž y, x . G Ý A i ) Ý Bi igJ
s
igI
Ý A i n Ý Bi m y g NÝ igI
Ý A i and igI
igJ
ig I
igJ
A i ) Ý ig J Bi
Ž x.
This implies NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . . Ž3. For any y g NÝ i g I A iŽ x . j NÝ i g J B i Ž x . m y g NÝ i g I A i Ž x . or y g NÝ i g J B iŽ x . m t Ž y, x . . F
t Ž y, x . F
Ý Bi m t Ž y, x . F Ý A i k Ý Bi igJ
s
Ý A i q Ý Bi m y g NÝ igI
Ý A i or igI
igJ
igI ig I
A i ) Ý ig J Bi
igJ
Ž x. .
This implies NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x . . THEOREM 2.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice of AFS structure Ž M, t , X .. If B s NÝ i g I A iŽ x . ¬ x g X, Ý i g I A i g h 4 , then B is a base for some topology. Proof. First, because Ž M, t , X . be a strong relative AFS structure, for any x g X, t Ž x, x . s M. M is the maximal element of EI algebra over M. This implies that for any Ý i g I A i g h , t Ž x, x . F Ý i g I A i so that x g NÝ i g I A iŽ x .. Therefore X s Db g B b . Second, suppose x g X, and U, V g B, for which x g U and V. We want to prove that there exists W g B such that x g W : U l V. By the
487
TOPOLOGY OF AFS THEORY
hypothesis, we know there exists Ý i g I A i , Ý i g J Bi g h such that U s NÝ i g I A iŽ u., V s NÝ i g J B iŽ ¨ . and 'l g I, 'k g J, t Ž u, x . = A l and t Ž ¨ , x . = Bk . For any y g NA lŽ x ., i.e., t Ž x, y . = A l , since AX 2, hence t Ž u, y . = t Ž u, x . l t Ž x, y . = A l and y g U. This implies NA lŽ x . : U. Similarly NB kŽ x . : V. Because NA lŽ x . l NB kŽ x . s NA l ) B kŽ x . , x g NA l ) B kŽ x . g B, we have W s NA l ) B kŽ x . : U l V . Now we prove that B is a base for some topology. The topological space Ž X, Th ., in which B is a base for Th , is called the topology induced by h. THEOREM 2.2. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on EI algebra o¨ er M of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. If
jh s
½Ý
igI
ai A i
Ý ai A i g EXM, ai g Th , ; i g I, Ý A i g h igI
igI
5
,
and X is a finite set, then jh is a topological molecular lattice on EII algebra o¨ er M of AFS structure Ž M, t , X .. Proof. It is obvious that h is closed under finite unions Žk or q. and arbitrary intersections Žn or ).. Therefore jh is a topological molecular lattice. It is called the topological molecular lattice induced by h. For any infinite indexing set J, let A j s Ý i g I j a ji A ji g EXM, j g J because Fj g J a j f Ž j. g Th can not always hold, h is not closed under arbitrary intersections Žn or ).. Therefore X in Theorem 2.2 must be finite set. In the following, we continue to discuss Example 1.1 using this topological structure. If we consider the relations among age, height, and weight. Let h be the topological molecular lattice generated by A4 , H 4 and We4 , which are elements in EI algebra over M, i.e., B, A, M, W, H, We, S, Q4 g h and the following: A4 q H 4 q We4 A4 q H 4 , A4 q We4 , H 4 q We4
488
LIU XIAODONG
A4 , H 4 , We4 A, H 4 q A, We4 q We, H 4 A, H 4 q A, We4 , A, H 4 q We, H 4 , A, We4 q We, H 4 A, H 4 , A, We4 , W, H 4 A4 q We, H 4 , H 4 q A, We4 , We4 q A, H 4 A, H, We4 . Let us consider the base of the topology Th for x 1 , x 2 , x 3 , x 4 , x 54 . Therefore the neighborhoods of x 1 , x 2 , x 3 , x 4 , x 5 induced by h are Nh Ž x 1 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 1 , x 2 , x 3 , x 4 4 , x 1 , x 3 , x 54 , x 1 x 344 ; Nh Ž x 2 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 2 , x 3 , x 4 , x 54 , x 1 , x 2 , x 3 , x 4 4 , x 2 , x 4 , x 54 , x 2 , x 3 , x 54 , x 2 , x 544 ; Nh Ž x 3 . s x 2 , x 3 , x 4 , x 54 , x 344 ; Nh Ž x 4 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 3 , x 4 4 , x 4 44 ; Nh Ž x 5 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 3 , x 4 , x 54 , x 4 , x 5 4 , x 3 , x 54 , x 544 . In the following, we discuss the real meaning of the neighborhoods of x i induced by h. First, from above, we know x 3 , x 4 , x 5 are discrete points for the topology Th . Coincidentally, they are the maximal points of concepts A4 , H 4 , We4 , respectively. For any i, we can prove that for any a g Nh Ž x 1 ., x 1 g a « x 3 g a. This implies that the degree of x 3 belongs to some concept which is always larger than that of x 1. x 2 f x 1 , x 3 4 g Nh Ž x 1 ., x 1 f x 2 , x 54 g Nh Ž x 5 ., i.e., the separation property of topology Th . This implies that there exist two fuzzy concepts a, b g h such that x 2 , x 1 can be discriminated by fuzzy concept a, b. We should notice that the above computing process can be done by computer.
REFERENCES 1. Liu Xiaodong, The structure of fuzzy matrices, J. Fuzzy Math. 2 Ž1994., 311]325. 2. Liu Xiaodong, The solution of fuzzy equations, J. Fuzzy Math. 2 Ž1994., 327]333. 3. Liu Xiaodong, A new mathematical axiomatic system of fuzzy set and systems, J. Fuzzy Math. 3 Ž1995., 559]560. 4. Liu Xiaodong, Two algebraic structures of AFS structure, J. Fuzzy Math. 3 Ž1995., 561]562. 5. Liu Xiaodong, The fuzzy theory based on AFS structure, EI algebra and EII algebra, Fuzzy Sets Syst., in press. 6. Liu Xiaodong, Pattern recognition and hitch diagnosis of complex systems, Fuzzy Sets Syst., in press. 7. Liu Xiaodong, On a new mathematical axiomatic system of fuzzy set and systems, J. Dalian Maritime Uni¨ ersity 22 Ž1996., 96]99.
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8. Liu Xiaodong, On a new mathematical axiomatic system of fuzzy set and systems, in ‘‘Proceedings of the Second Academic Congress of Liao Ning Province, 1995,’’ Dalian Technology University Publishing House, Dalian, China, 1995. 9. Liu Xiaodong, The structure of Boolean matrices, in ‘‘Proceedings of ICIK’1995,’’ Dalian Maritime University Publishing House, Dalian, China, 1995. 10. Liu Xiaodong, The fuzzy theory based on AFS algebras and AFS structure, J. Math. Anal. Appl., in press. 11. Wang Guo-jun, Theory of topological molecular lattices, Fuzzy Sets Syst. 47 Ž1992., 351]376. 12. N. Jacobson, ‘‘Basic Algebra I,’’ 2nd ed., Freeman, New York, 1985. 13. J. E. Graver and M. E. Watkins, ‘‘Combinatorics with Emphasis on the Theory of Graphs’’ Springer-Verlag, New York, 1977. 14. Zou Kaiqi and Liu Xiaodong, ‘‘CF Algebra,’’ Hebei Education Publishing House, Shijiazhuang, 1993.
217, 479]489 Ž1998.
AY975719
The Topology of AFS Structure and AFS Algebras Liu Xiaodong Department of Basic Science, Dalian Maritime Uni¨ ersity, Dalian, 116026, People’s Republic of China Submitted by Ying-ming Liu Received December 16, 1996
In this paper, we will study the topological structure of the AFS structures published by the author; discuss the topological molecular lattice structures on EI, *EI, EII, *EII algebras published by the author; and give the relations of these topological structures. As applications, we study the topological produced by a family of fuzzy concepts in the AFS algebras and apply these to analyze relations among fuzzy concepts. Using these, we believe that we can study the law of human thinking. The most important fact is that all these can be operated by computers. We are also sure AFS theory will play an important role in information processing. Q 1998 Academic Press
Key Words: topological molecular lattice; AFS algebras; AFS structure.
1. TOPOLOGY ON *EI, *EII ALGEBRAS In this section, we discuss the topological structure of AFS structure on *EI, *EII algebras. We know that for any sets X and M there are two kinds of AFS algebras, which are EI, *EI, EII, *EII algebras Žrefer to w10x.. Although they have similar algebra structures, their topological structures are different. First, we consider the topological structure on *EI, *EII algebras, which, for any a, b g EM or EXM a k b s a) b and a n b s a q b. DEFINITION 1.1. Let X and M be sets, and Ž M, t , X . be an AFS structure. h : EM the *EI algebra over M, h is called a closed topology Žrefer to w11x. if f , M g h and h is closed under finite unions Žk or *. and arbitrary intersections Žn or q.. h is called a topological molecular lattice on *EI algebra over M of AFS structure Ž M, t , X ., denoted as Ž EM, h .. DEFINITION 1.2. Let X and M be sets, and Ž M, t , X . be an AFS structure. h is a topological molecular lattice on *EI algebra over M of 479 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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LIU XIAODONG
AFS structure Ž M, t , X .. For any x g X, Ý i g I A i g EM, and Ý i g I A i g h , we define NÝ i g I A iŽ x . s y ¬ t Ž x, y . G
½
Ý Ai igI
5.
This is called the neighborhood of x inducing by Ý i g I A i g h.
½
Nh Ž x . s NÝ i g I A iŽ x .
Ý Ai g h igI
5
is called the neighborhood of x inducing by h. PROPOSITION 1.1. Let X and M be sets and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. For any Ý i g I A i , Ý i g J Bi g h , and x g X, the following hold: Ž1.
Ý A i G Ý Bi , then NÝ
if
igI
Ž2. Ž3.
igJ
ig I
Ž x . : NÝ
Ai
ig J
Ž x.
Bi
NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x ..
Proof. Ž1. Suppose y g NÝ i g I A iŽ x .. There exists A k , k g I such that t Ž x, y . = A k . On the other hand, since Ý i g I A i G Ý i g J Bi , hence there exists Bj , j g J such that t Ž x, y . = A k = Bj . This implies y g NÝ i g J B iŽ x . and NÝ i g I A iŽ x . : NÝ i g J B iŽ x . . Ž2. For any y g NÝ i g I A iŽ x . l NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . and y g NÝ i g J B iŽ x . m t Ž x, y . G
t Ž x, y . G
Ý A i and igI
Ý Bi m t Ž x, y . G Ý A i ) Ý Bi m y g NÝ igJ
igI
igJ
ig I
A i ) Ý ig J Bi
This implies NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . .
Ž x. .
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TOPOLOGY OF AFS THEORY
Ž3. For any y g NÝ i g I A iŽ x . j NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . or y g NÝ i g J B iŽ x . m t Ž x, y . G
t Ž x, y . G
Ý Bi m t Ž x, y . G Ý A i n Ý Bi igJ
s
Ý A i q Ý Bi m y g NÝ igI
Ý A i or igI
igJ
igI ig I
A iqÝ i g J B i
igJ
Ž x. .
This implies NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x . . THEOREM 1.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. If
½
B s NÝ i g I A iŽ x . x g X ,
Ý Ai g h igI
5
,
then B is a base for some topology. Proof. First, because Ž M, t , X . is a strong relative AFS structure, for any x g X, t Ž x, x . s M. M is the maximal element of *EI algebra over M. This implies that for any Ý i g I A i g h , t Ž x, x . G Ý i g I A i so that x g NÝ i g I A iŽ x .. Therefore X s Db g B b . Second, suppose x g X, and U, V g B, for which x g U and V. We want to prove there exists W g B such that x g W : U l V. By the hypothesis, we know there exists Ý i g I A i , Ý i g J Bi g h such that U s NÝ i g I A iŽ u., V s NÝ i g J B iŽ ¨ . and 'l g I, 'k g J, t Ž u, x . = A l and t Ž ¨ , x . = Bk . For any y g NA lŽ x ., i.e., t Ž x, y . = A l , since AX 2, hence t Ž u, y . = t Ž u, x . l t Ž x, y . = A l and y g U. This implies NA lŽ x . : U. Similarly NB kŽ x . : V. Because NA lŽ x . l NB kŽ x . s NA l ) B kŽ x . , x g NA l ) B kŽ x . g B, we have W s NA l ) B kŽ x . : U l V . Now we prove that B is a base for some topology. The topological space Ž X, Th ., in which B is a base for Th , is called the topology induced by h.
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LIU XIAODONG
THEOREM 1.2. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on *EI algebra o¨ er M of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. If
jh s
½Ý
ai A i
igI
Ý ai A i g EXM, ai g Th , ; i g I, Ý A i g h igI
igI
5
,
then jh is a topological molecular lattice on *EII algebra o¨ er M of AFS structure Ž M, t , X .. Proof. For any finite integer n, let A j s Ý i g I j a ji A ji g EXM, j s 1, 2, . . . , n. Because F1 F j F n a j f Ž j. g Th for any f g Ł 1 F j F n I j and Ý f g Ł 1 F j F n I j D1 F j F n A j f Ž j. g h ,
E
Aj s
Ý
F
D
a j f Ž j.
fgŁ 1F j F n I j 1FjFn
1FiFn
A j f Ž j. g jh .
1FjFn
It is obvious that jh is closed under arbitrary intersection Žn or q.. Therefore jh is a topological molecular lattice. It is called the topological molecular lattice induced by h. In the following, we give an example of these topological structures. EXAMPLE 1.1. Let X s x 1 , x 2 , . . . , x 5 4 be a set of 5 persons. M s age, weight, height, salary, wealth, man, women4 . Suppose there exists the following table:
x1 x2 x3 x4 x5
age
weight
height
man
woman
21 30 27 60 45
50 52 65 63 54
1.69 1.62 1.80 1.50 1.71
yes no yes no yes
no yes no yes no
salary
wealth
0 120 100 80 140
0.000 200,000 40,000 324,000 486,940,000
We get the following table of structure t t
x1
x2
x3
x1 x2 x3 x4 x5
A, M, H, We, S4 A, W, We, S, Q4 A, M, H, We, S, Q4 A, W, We, Q4 A, M, S, Q4
M, H 4 A, W, H, We, S, Q4 M, H, We4 A, W, We, Q4 A, M, H, We, S, Q4
M4 A, W, S, Q4 A, M, H, We, S, Q4 A, W, Q4 A, M, S, Q4
TOPOLOGY OF AFS THEORY
t
x4
x5
x1 x2 x3 x4 x5
M, H 4 W, H, S 4 M, H, We, S4 A, W, We, H, S, Q4 M, H, S, Q4
M4 W 4 M, H, We4 A, W, We4 A, M, H, S, We, Q4
A, age; M, man; W, woman; H, height; We, weight; S, salary; Q, wealth. We can verify that t satisfies AX 1 and AX 2 and Ž M, t , X . is an AFS structure.
483
We consider the relations among age, height, and weight. Let h be the topological molecular lattice generated by A4 , H 4 , and We4 , which are elements in *EI algebra over M, i.e., h consists of B, A, M, W, H, We, S, Q4 and the following: A4 q H 4 q We4 , A4 q H 4 , A4 q We4 , H 4 q We4 A4 , H 4 , We4 , A, H 4 q A, We4 q We, H 4 , A, H 4 q A, We4 , A, H 4 q We, H 4 , A, We4 q We, H 4 , A, H 4 , A, We4 , We, H 4 , A4 q We, H 4 , H 4 q A, We4 , We4 q A, H 4 , A, H, We4 . Now we consider the base of the topology Th for x 1 , x 2 , x 3 , x 4 , x 54 : N A4q H 4qW e4Ž x 1 . s Ž x 1 , x 2 , x 4 4 N A4q H 4Ž x 1 . s x 1 , x 2 , x 4 4 , N A4qW e4Ž x 1 . s x 14 , Ž . 4 N H 4qW e4 x 1 s x 1 , x 2 , x 4 N A4Ž x 1 . s x 14 , N H 4Ž x 1 . s x 1 , x 2 , x 4 4 , NW e4Ž x 1 . s x 14 N A, H 4q A, W e4qW e, H 4Ž x 1 . s x 14 N A, H 4q A, W e4Ž x 1 . s x 14 , N A, H 4qW e, H 4Ž x 1 . s x 14 , N A, W e4qW e, H 4Ž x 1 . s x 14 N A, H 4Ž x 1 . s x 14 , N A, W e4Ž x 1 . s x 14 , NW e, H 4Ž x 1 . s x 14 N A4qW e, H 4Ž x 1 . s x 14 , N H 4q A, W e4Ž x 1 . s x 1 , x 2 , x 4 4 , Ž . 4 NW e4q A, H 4 x 1 s x 1 N A, H , W e4Ž x 1 . s x 14 . Therefore the neighborhoods of x 1 induced by h is Nh Ž x 1 . s x 1 , x 2 , x 4 4 , x 1 4 4 .
484
LIU XIAODONG
Similarly, we get the neighborhoods of x 2 induced by h Nh Ž x 2 . s x 1 , x 2 , x 3 , x 4 4 , x 1 , x 2 , x 4 4 , x 1 , x 2 , x 3 4 ,
x1 , x 2 4 , x 2 , x 4 4 , x 2 4 4 ; the neighborhoods of x 3 induced by h is Nh Ž x 3 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 3 4 4 ; the neighborhoods of x 4 induced by h is Nh Ž x 4 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 2 , x 4 , x 5 4 , x 4 4 4 ; the neighborhoods of x 5 induced by h is Nh Ž x 5 . s x 1 , x 2 , x 3 , x 4 , x 5 4 , x 1 , x 2 , x 3 , x 5 4 , x 2 , x 4 , x 5 4 , x 2 , x 5 4 4 . In the following, we discuss the real meaning of the neighborhoods of x i induced by h. First, from above, we know x 1 , x 2 , x 4 are discrete points for the topology Th . Coincidentally, they are the minimal points of concepts A4 , H 4 , We4 , respectively. For any i, we can prove that for any a g Nh Ž x i ., x 5 g a « x 2 g a. This implies that the degree of x 5 belongs to some concept which is always larger than that of x 2 . x 5 f x 1 , x 34 g Nh Ž x 3 ., x 3 f x 2 , x 54 g Nh Ž x 5 ., i.e., the separation property of topology Th . This implies that there exist two fuzzy concepts a, b g h such that x 5 , x 3 can be discriminated by fuzzy concept a, b. We should notice that the above computing process can be done by computer. THEOREM 1.3. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. jh is the topological molecular lattice induced by h. s is the Borel set corresponding to topological space Ž X, Th . and Ž M, t , X, s , m. is a semi-cogniti¨ e field; then Ž1. If Ý i g I A i g h and Ý j g J Bj G Ý i g I A i « Ý j g J Bj g h , then Ý i g U Ci is a measurable concept under s , for any Ý i g U Ci g h. Ž2. Ý i g I A i g h , I is any indexing set, D is a directed set, S: D ª X, S is con¨ erged to a g X under Th . S9: D ª EXM is defined as follows: for any d g D, S9Ž d . s Ý i g I A iŽ sŽ d .. A i , then S9 is con¨ erged to Ý i g I A iŽ a. A i g EXM under jh . Proof. Ž1. For any Ý i g I A i g h , x g X, since A i G Ý i g I A i , hence A i g h and A iŽ x . s NA i Ž x . g Th « NA i Ž x . g s , ; i g I. Therefore Ý i g I A i is a measurable concept under s .
TOPOLOGY OF AFS THEORY
485
Ž2. Suppose Ý i g I A iŽ a. A i g Ý j g J pj Pj , Ý j g J pj Pj g h. This implies that there exists pl Pl such that ; i g I, either A iŽ a. W pl or pl W A i . First, we suppose ;k g I, Pl W A k . This implies for any d g D,
Ý A i Ž S Ž d . . A i g Ý pj Pj . igI
jgJ
Second, there exists k g I such that A kŽ a. W pl . Since a g A kŽ a. g s and S is converged to a g X under Th , hence there exists N g D such that for any n g D, n G N, SŽ n. g A kŽ a. W pl . For any y g A kŽ S Ž n.., i.e., t Ž SŽ n., y . = A k . Since SŽ n. g A kŽ a., i.e., t Ž a, SŽ n.. = A k and t is an AFS structure, hence t Ž a, y . = t Ž a, SŽ n.. l t Ž SŽ n., y . « y g A kŽ a. « A kŽ a. = A kŽ S Ž n... Since A iŽ a. W pl , hence A iŽ SŽ n... W pl . This implies that for any Ý j g J pj Pj g h and Ý i g I A iŽ a. A i g Ý j g J pj Pj , there exists N g D such that for any n g D, n G N, Ý i g I A iŽ S Ž n.. A i g Ý j g J pj Pj . Therefore S9 is converged to Ý i g I A iŽ a. A i g EXM under jh . 2. TOPOLOGY ON EI, EII ALGEBRAS In this section, we discuss the topological structure of AFS structure on EI, EII algebras. We consider the topological structure on EI, EII algebras, which, for any a, b g EM or EXM, a k b s a q b and a n b s a) b. DEFINITION 2.1. Let X and M be sets, and Ž M, t , X . be an AFS structure, h : EM the EI algebra over M, h is called a closed topology if f , M g h , and h is closed under finite unions Žk or q. and arbitrary intersections Žn or ).. h is called a topological molecular on EI algebra over M of AFS structure Ž M, t , X . denoted as Ž EM, h .. DEFINITION 2.2. Let X and M be sets, and Ž M, t , X . be an AFS structure. h is a topological molecular lattice on EI algebra over M of AFS structure Ž M, t , X .. For any x g X, Ý i g I A i g EM, and Ý i g I A i g h , we define NÝ i g I A iŽ x . s y N t Ž y, x . F Ý i g I A i 4 . It is called the neighborhood of x inducing by Ý i g I A i g h. Nh Ž x . s NÝ i g I A iŽ x . N Ý i g I A i g h 4 is called the neighborhood of x inducing by h. PROPOSITION 2.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on EI algebra o¨ er M of AFS structure Ž M, t , X .. For any Ý i g I A i , Ý i g J Bi g h , and x g X, the following hold: Ž1. Ž2. Ž3.
if Ý i g I A i G Ý i g J Bi , then NÝ i g I A iŽ x . = NÝ i g J B iŽ x . NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x .
486
LIU XIAODONG
Proof. Ž1. Suppose y g NÝ i g J B iŽ x .. This implies that 'k g J, Bk : t Ž y, x .. On the other hand, since Ý i g I A i G Ý i g J Bi , hence for Bk , there exists A l , l g I, such that A l : Bk : t Ž y, x .. This implies y g NÝ i g I A iŽ x . and NÝ i g I A iŽ x . = NÝ i g J B iŽ x .. Ž2. For any y g NÝ i g I A iŽ x . l NÝ i g J B iŽ x . m y g NÝ i g I A iŽ x . and y g NÝ i g J B iŽ x . m t Ž y, x . F
t Ž y, x . F
Ý Bi m t Ž y, x . G Ý A i ) Ý Bi igJ
s
igI
Ý A i n Ý Bi m y g NÝ igI
Ý A i and igI
igJ
ig I
igJ
A i ) Ý ig J Bi
Ž x.
This implies NÝ i g I A iŽ x . l NÝ i g J B iŽ x . s NÝ i g I A i ) Ý i g J B iŽ x . . Ž3. For any y g NÝ i g I A iŽ x . j NÝ i g J B i Ž x . m y g NÝ i g I A i Ž x . or y g NÝ i g J B iŽ x . m t Ž y, x . . F
t Ž y, x . F
Ý Bi m t Ž y, x . F Ý A i k Ý Bi igJ
s
Ý A i q Ý Bi m y g NÝ igI
Ý A i or igI
igJ
igI ig I
A i ) Ý ig J Bi
igJ
Ž x. .
This implies NÝ i g I A iŽ x . j NÝ i g J B iŽ x . s NÝ i g I A iqÝ i g J B iŽ x . . THEOREM 2.1. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice of AFS structure Ž M, t , X .. If B s NÝ i g I A iŽ x . ¬ x g X, Ý i g I A i g h 4 , then B is a base for some topology. Proof. First, because Ž M, t , X . be a strong relative AFS structure, for any x g X, t Ž x, x . s M. M is the maximal element of EI algebra over M. This implies that for any Ý i g I A i g h , t Ž x, x . F Ý i g I A i so that x g NÝ i g I A iŽ x .. Therefore X s Db g B b . Second, suppose x g X, and U, V g B, for which x g U and V. We want to prove that there exists W g B such that x g W : U l V. By the
487
TOPOLOGY OF AFS THEORY
hypothesis, we know there exists Ý i g I A i , Ý i g J Bi g h such that U s NÝ i g I A iŽ u., V s NÝ i g J B iŽ ¨ . and 'l g I, 'k g J, t Ž u, x . = A l and t Ž ¨ , x . = Bk . For any y g NA lŽ x ., i.e., t Ž x, y . = A l , since AX 2, hence t Ž u, y . = t Ž u, x . l t Ž x, y . = A l and y g U. This implies NA lŽ x . : U. Similarly NB kŽ x . : V. Because NA lŽ x . l NB kŽ x . s NA l ) B kŽ x . , x g NA l ) B kŽ x . g B, we have W s NA l ) B kŽ x . : U l V . Now we prove that B is a base for some topology. The topological space Ž X, Th ., in which B is a base for Th , is called the topology induced by h. THEOREM 2.2. Let X and M be sets, and Ž M, t , X . be a strong relati¨ e AFS structure. h is a topological molecular lattice on EI algebra o¨ er M of AFS structure Ž M, t , X .. Topological space Ž X, Th . is the topology induced by h. If
jh s
½Ý
igI
ai A i
Ý ai A i g EXM, ai g Th , ; i g I, Ý A i g h igI
igI
5
,
and X is a finite set, then jh is a topological molecular lattice on EII algebra o¨ er M of AFS structure Ž M, t , X .. Proof. It is obvious that h is closed under finite unions Žk or q. and arbitrary intersections Žn or ).. Therefore jh is a topological molecular lattice. It is called the topological molecular lattice induced by h. For any infinite indexing set J, let A j s Ý i g I j a ji A ji g EXM, j g J because Fj g J a j f Ž j. g Th can not always hold, h is not closed under arbitrary intersections Žn or ).. Therefore X in Theorem 2.2 must be finite set. In the following, we continue to discuss Example 1.1 using this topological structure. If we consider the relations among age, height, and weight. Let h be the topological molecular lattice generated by A4 , H 4 and We4 , which are elements in EI algebra over M, i.e., B, A, M, W, H, We, S, Q4 g h and the following: A4 q H 4 q We4 A4 q H 4 , A4 q We4 , H 4 q We4
488
LIU XIAODONG
A4 , H 4 , We4 A, H 4 q A, We4 q We, H 4 A, H 4 q A, We4 , A, H 4 q We, H 4 , A, We4 q We, H 4 A, H 4 , A, We4 , W, H 4 A4 q We, H 4 , H 4 q A, We4 , We4 q A, H 4 A, H, We4 . Let us consider the base of the topology Th for x 1 , x 2 , x 3 , x 4 , x 54 . Therefore the neighborhoods of x 1 , x 2 , x 3 , x 4 , x 5 induced by h are Nh Ž x 1 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 1 , x 2 , x 3 , x 4 4 , x 1 , x 3 , x 54 , x 1 x 344 ; Nh Ž x 2 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 2 , x 3 , x 4 , x 54 , x 1 , x 2 , x 3 , x 4 4 , x 2 , x 4 , x 54 , x 2 , x 3 , x 54 , x 2 , x 544 ; Nh Ž x 3 . s x 2 , x 3 , x 4 , x 54 , x 344 ; Nh Ž x 4 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 3 , x 4 4 , x 4 44 ; Nh Ž x 5 . s x 1 , x 2 , x 3 , x 4 , x 54 , x 3 , x 4 , x 54 , x 4 , x 5 4 , x 3 , x 54 , x 544 . In the following, we discuss the real meaning of the neighborhoods of x i induced by h. First, from above, we know x 3 , x 4 , x 5 are discrete points for the topology Th . Coincidentally, they are the maximal points of concepts A4 , H 4 , We4 , respectively. For any i, we can prove that for any a g Nh Ž x 1 ., x 1 g a « x 3 g a. This implies that the degree of x 3 belongs to some concept which is always larger than that of x 1. x 2 f x 1 , x 3 4 g Nh Ž x 1 ., x 1 f x 2 , x 54 g Nh Ž x 5 ., i.e., the separation property of topology Th . This implies that there exist two fuzzy concepts a, b g h such that x 2 , x 1 can be discriminated by fuzzy concept a, b. We should notice that the above computing process can be done by computer.
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8. Liu Xiaodong, On a new mathematical axiomatic system of fuzzy set and systems, in ‘‘Proceedings of the Second Academic Congress of Liao Ning Province, 1995,’’ Dalian Technology University Publishing House, Dalian, China, 1995. 9. Liu Xiaodong, The structure of Boolean matrices, in ‘‘Proceedings of ICIK’1995,’’ Dalian Maritime University Publishing House, Dalian, China, 1995. 10. Liu Xiaodong, The fuzzy theory based on AFS algebras and AFS structure, J. Math. Anal. Appl., in press. 11. Wang Guo-jun, Theory of topological molecular lattices, Fuzzy Sets Syst. 47 Ž1992., 351]376. 12. N. Jacobson, ‘‘Basic Algebra I,’’ 2nd ed., Freeman, New York, 1985. 13. J. E. Graver and M. E. Watkins, ‘‘Combinatorics with Emphasis on the Theory of Graphs’’ Springer-Verlag, New York, 1977. 14. Zou Kaiqi and Liu Xiaodong, ‘‘CF Algebra,’’ Hebei Education Publishing House, Shijiazhuang, 1993.