Finitely generated T-fuzzy linear spaces

Fuzzy Scts and Systems 30 (1989) 69--81 North-Holland

FE~TELY GE~RATED YU

69

T - F U Z Z Y L I N E A R SPACES

Yandong

Deparunemof Mathematics, Yancheng Teache~ Cotiege, Ji~ga~ Frov~e, C~,~ Received August 1985

Abstract: Based on earlier papars by the author [4, 5] finitely generated T-fury Hnearspace, T being a general t-norm, are introduced and ~tudied. First, a few basic definitions and notafions~arepresented for reference purposes. Next, in Section 3, a definition of finitely generated T-f~zay linear spacesis Wopa~d and their s t ~ is n)u~hiy studied. Particularly, it is pointed out that if X is a finite dimensional real or cc~p~x Euclidean space, every Min-fuzzy li:lear space in X (i.e., fuzzy sub~ace ~f X in the sense of Katsaras and Liu [1]) is finitely generated. In Section 4, a definition is given of simply generated T-fuzay linear spaces and it is shown that every finitely generated T-fuzzy linear ~pace can be decomposed into a finite number of simply generated T-fury linear spaces. Finally, in S ~ ' ~ n 5, the concepts are introduced of T-equivalence and congruence for finite famil~ of fu~,y points and a su~:ient condition for them to be T-equivalent is given. Besides this, for the that T is a strict t-norm, it is shown that under an appropriate assumption, the condition is also

necessary. Keywords: Finitelygenerated ?'-fuzzylinear space; simply generated T-fury linea.-space; '/'-equivalence;~ngruence.

1. In~ducfion Fuzzy subspaces of a v e c t o r space were introduced by Katsaras and Liu [1]. Subsequently they were studied further by Lowen [2] and others in the context of convex fua~y sets. Up to now, it seems that the most important result in this field is Lowen's Representation Theorem (see [2]), which solves the problem of the structure of fuzzy subspaces of finite dimensional real Euclidean spaces. On the other hand, as a true extension of the concept of fuzzy subspaees, the concept of T-fuzzy linear spaces in a linear space was introduced by Yu [4] in the context of T-convex fuzzy sets, T being a general t-norm. The main reason for doing so was completely similar to that for modifying Rosenfeld's definition of fuzzy subgroups of a group (see [6, 7]). Unfortunately, if T # Min it is impossible to parallel Lowen's Representation Theorem to T-fu~.~ linear spaces. Thus it seems necessary to study the structures of T-fitzzy linear spaces in other ways. In this paper, using finite famiiies of fuzzy points (see [4, 5, 11]), we introduce and study finitely generated T-fuzzy linear spaces in a real or complex Euclidean space. One main reason for doing this is: the structures of these T-~zzy linear spaces are simple and interesting and it seems easy to find their practical app~.ications. 0165-0114189/$3.50 (~) 1989, Elsevier Science Publishers B.V. (North-Holland)

70

Yu Ya~fong

2. P r e ~ e s

For reference purposes we start with a presentation of a few basic d ~ i t i o n s and notations for t.nonns, fuzzy sets an~ T-fuzzy linear spaces. D e f 4 ~ o n 2.1. A t-norm is a two-place function T:[O, I] × [0, I]-->[0, I] satis~'i~g the following conditions (for any a, b, ¢ ~ [0, 1D: O) T(a, b) = T(b, a) (commutativity), (ii) T(T(a, b), c ) - T(a, T(b, c)) (associativity), (iii) T(a, b) ~ T(a, c) whenever b ~ c (monotonicity), (iv) T(a, 1 ) = a (boundary condition).

As is well known, Min is an important t-norm. Besides Min, two t-n0rms which are frequently encountered are ~:,, and Prod given, respectively, by T~(a, b ) = Max(a + b - 1, 0)

and

Prod(a, b) = ab

for any a, b ~ [0, 1]. D e f ~ i o n 2.2. A t - n ¢ ~ T is said to be continuous if it is so a~ an ordinary real t~,o-place function in the usual sense; strict if it is continuous and strictly increasing in both p~ces, i.e., T(a, b) = T(b, a) < T(c, a) = T(a, c) whenever a > 0 and b < c (a, b, c ¢ [0, 1]). For example, Prod is a strict t-norm but both Min and Tm are not. Let T be a t-norm. We can consider T(~'l, a2 . . . . . a,), al, a~ . . . . . defined recursively by T(al, a,_, . . . , a~-l, an) = T(T(al, a2 ..... an-l), an),

an ¢ [0, 1],

n >~ 3.

Due to ~he associativity of T this definition makes sense. Additionally, fo~ the sequel, ~Te assume that T(a) = a for any a ¢ [0, 1]. For more d~tails about t-norms wc refer to [8-10]. From now on, unless otherwise stated, T will always denote any given t-norm and X any given linear spce over a real or complex field P. D e f i f i o n 2.3. A fuzzy set on X is a function A : X--l~ [0, 1]. The family of all fuzzy sets on X is denoted by [0, 1]X. D e f n ~ o ~ 2.~. Let A, B, C ~ [0, I] x. Then: A = B 0 A(~.) = B(x) for any x E X; A c B ¢~ A(x) <~B(x) for any x e X; C = A U B ¢~ C(x) = Max(A(x), B(x)) for any x ¢ X.

Generally, let {A~ I i ~ 1} be a subfamily of [0, 1]x and C ¢ [0, 1]x; then C = U A~ ¢-~ C(x) = Su~ A~(x) for any x ¢ X.

If I = {1, 2 . . . . .

n}, we write [_J~A~ = I.JT~ffi~A~.

Finitegy generated T.fuzzy linear spaces

71

Defini~on 2.5. Let A ~ [0, 1]x. If there is some xo ~ X and ~ e [0, 1] such that

A(x)={~

ifx=xo, otherwise,

for any x ¢ X, then A is called a fuzzy point on X and denoted by A = (xo, ~). Particularly, if ~ = 1, the fuo.y point (Xo, ~) is regarded to be identical with the crisp point Xo (see [4, 5, 11]). t The family of all fuzzy points on X is denoted by and its subfamily {(x, ~) ] x E X, ~ ~ (0, 1]} by ~÷. Definition 2.6. Let (x, ~.), (y, ~) e ~Yand a E P. Then

(x, ~) + (y, t+) = (x + y, r(~, ~)), a(x, ,t) = (a.~, ,~). Definition 2.7. A T-fuzzy linear space in X is a fuzzy set S on X satisfying the following conditions: O) S(x + y) >>-T(S(x), $(y)) for any x, y ~ X, (ii) S(ax) ~>S(x) for any x ~ X and any a E P. For more details about fuzzy sets and T-fuzzy linear spaces we refer to [4, 5,

111. 3. Finitely generated T - f e ~ T I~ear spaces D e f n d o n 3.1. Let b° be a subfamily of ~Y. The smallest T - f u r y linear space in X containing every fuzzy point (z, ~) in ,7' is called the T-fuzzy linear space generated by 5* and denoted by LT(Se). Particularly if b° is a finite family, say, ~0 = {(Xl ' ~"1), (X2, ~2) . . . . . (Xn, ~.,,)}, we can write LT(Y) = Lr((x,, ~.,),

(x~, a~) .....

(x., ;,,)).

Due to Proposition 3.3 in [5] this definition makes sense. D e f J f i o n 3.2. If a T-fuzzy finear space S i~ X can be generated by a finite family of fuzzy points on X, then we say that S ~s a finitely generated T-fuzzy linear space

inX. "l'heorem 3.1. For each finite subfamily of ~, (x,+, ~,,)}, we have

z,.(~e) =

U

5e= {(xl, ~-l), (x2, ~2). . . . .

(a~,(x,,, ,~,,,) + a,.(x,., ,~,.) +.-. + a,~(x,+, ~,+)).

ait,a~2.....aik e P

l ~it
Proof. Denote the right hand side by S. So, clearly S ¢ [0, 1]x. Now let us prove that LT(,7') = S in the following three steps. t To avoid typesetting proble!ns, we use (x, ~) instead e~ :ca to denote a fuzzy point.

Yu Yandong

(I) In this step we shall show that S is a T-fuzzy linear space in X. In order to do this let us choose any x, y e X and a ¢ P and consider S(x + y ) and S(ax): (i) If x ¢ L~xl, .:2 . . . . . x,,), 2 then clearly S ( x ) - O. If x ¢ L ( x t , x2 . . . . . x,,), then from the definition of S we can assert S(x) =

Sup

r(4,,, 4, 5. . . . .

4,,).

O~iJg/I+ a l 2 x t 2 +

•••+aikXlk ~Jt aq,al2 ..... alkeP

t ~it
Since the set {T(4~,, 4i2. . . . . 4sk)] l ~ i t < i 2 < . . . i : ~ < n } is a finite s o l we can asse~ ,*~,',,~r tha~ thel'e exi~ ito, i~.o. . . . . iko, 1 ~
(3.1)

+ a~koX~ko,

s ( x ) - T(4,,o, 4,~ . . . . . 4,.o) where ¢~to, arm, • • • , ai, o E P. Similarly, if y ¢ L ( x t , x2 . . . . . x,,), then S ( y ) = O; if y ¢ L ( x t , x2 . . . . . there exist jlo, i2o. . . . . jto, 1 ~
s ( y ) = r(4,,o, 4, . . . . . .

x , ) , then

(3.2)

+ bj, oxj,o,

4j,)

where hi, o, b j ~ , . . . , bj~ ¢ P. (ii) Consider S ( x + y). Clearl~ if S ( x ) = 0 or S ( y ) = 0, we have S ( x + y) ~>0 = T ( S ( x ) , S ( y ) ) . Now suppose that S ( x ) > 0 and S ( y ) > O. Then, without loss of generality, we assume that both (3.1) and (3.2) hold. So, we have x + y "-=(ajtoxito+ at~oxi2o + . . .

+ ai~oX~G)

+ (bj,oX~,o+ bj~xj~ + . - . + bj~x~). it follows tha~ among x~:~, xi2o, . . , x~,~, x~, o, x/~ . . . . . xi~o, there exist at most n distinct vectors, say, x~,, xs2. . . . . x~, 1 ~
=

where c~,, c~.. . . . .

c~tx~~ + cizx~ + • • • + Ci~:~

c~, ¢ P. So, by the definition ef S, it is easily seen that

s(x + y) ~> r(4,,, 4,z ..... 4~) ~> T(4,,o, 4 , . . . . . . = r(s(x), s(y)).

4,,o. 4~,o, 4 , . . . . . .

4~)

(iii) Consider S(ax). If a = O, clearly

s(ax) = s(ox) = s ( e) = Max(~t, 4~ . . . . .

4~) ~> S(x),

where 0 denotes the zero vector (hereinafter). If a ~ 0 , it is obvious that s(~z) = s(x). This proves that S is a T-fuzzy linear space in X. (H) Clearly S contains every fuzzy poim (x~, 4~) in 5e. 2L(.c~,x2 . . . . . xn) denotes the ordinat3, .';ubspace of X generated by the vector set {x~,z~ . . . . . x~}, hereinafter.

Finitelygener~edT-~zzy linear.~ces

73

(III) Let R be a T-fuzzy linear space in X containing every fuzzy point (x~, ~,~) in b°. Then it is easily seen that if at,, a4,. . . . . a~ ¢ P, 1 ~
4. Decomlm~flon t h r u m Defudfion 4.L A T-fuzzy linear space S in X is said to be simply generated if and only if its satisfies one of the following two conditions: (i) S = LT(~o), where b°o -- {(0, ~)~, 20 ~ [0, 1]. (ii) S = LT(S°), where .9' = {(0, ~o), (x~, X~), (Xz, X2). . . . . (xn, 4,,)} is a subfamily of go+; and x~, x2 . . . . . x~ are linearly independent. Of course, every simply generated T-fuzzy linear space is also finitely generated° Thgo~m 4.L Let S be a finitely generated T-fuzzy linear space in X. Then either S is itself simply generated or S can be decomposed into a finite number of simply generated T-fuzzy linear spaces in X, i.e., S can be expressed in the form

s=j~s~ where S~, ] = 1, 2 . . . . . m, are simply generated T-fuzzy linear spaces in X Proof. Since S is finitely generated~ we can assume that S = L~(~)

where b~ = {(x~, ~ ) , (x2, 3.z). . . . .

(xn, ~.,,)} is a sab~'amily of go.

Yu Yandong

74

Now

let us put ~o--Max(~,~2 ..... ~ )

and consider the vector set

{xl, x2 . . . . . xn}: If all x . i = I, 2 . . . . . n, are the zero vector O, clearly $ = LT(~o) where 5°o= {(0, 2o)}, and hence S is simply generated. Othe~vise, w¢ suppose ~hat the maximal linearly independent subsets of the vector set {xl, x~ . . . . . {x,.. % . . . . .

%},

/=1,2 .....

xn} are

m,

where 1 ~ i p
/ = I, 2 . . . . .

m.

So every ~ is a simply generated T-fuzzy linear space in X. Thus it suffices to show that S

--0 Sj. j=l

In fact, on the one hand, obviously every Sj is contained in S. Hence [.-~i=~S~ c S. On the other hand, in order to show that S = L.~=I S], let us choose any :c ~ X and consider S(x) and (I..J~=lS])(x): If S(x) = O, naturally S(x) <~ ( ~ = 1 ~)(x). If x = 0, clearly S(x) -- 20 = ( ~ = 1 S~)(x). Now suppose that x ~ 0 and S(x)>0. Then, from the proof of Theorem 3.1 we can assert that there are natural numbers i~, i2 . . . . . ip, 1 ~
s(~) T(~,,, ~,~. . . . . ~,,). =

F~rthermore, without loss of generality, we can assume that ~he vector set {.~;~a,x~2. . . . . x~,} is linearly independent. Thus there is some j a {1, 2 . . . . . m} such that

{J~i,, Xi 2.....

J~ip} = {Xiil, Xii2 .....

Xip)"

So, from the definition of S], we can assert that

S(x) = T(~,,, ~, ......

~,,,) <~SAx)

and hence

This shows that S c UT~ ~. Therefore S = L.~j=~S], which proves the theorem. R e m ~ k s . 1. The structures of simply generated T-fuzzy linear spaces are rather similar to the ones of o r d i n a ~ finite dimensional linear spaces. Theorem 4.1 leads us to turn our attention to the studying of the structures of simply generated T-fuz~/linear spaces.

Finitely generated T.fuzzy linear spm:¢~

75

2. ~ X is a finite dimensional l~nea~ space, Lowen's ~epresentatiou T h e o r e ~ hnpi~es that every M~n-fuzzy linear space in X is simply generated. 3. ff the d~mension of the l~near ~pace X is more than o,.,e and if T ~ Min, not all finitely generated T-fuzzy linear spaces in X are s ~ p l y ~3enera~ed.

$. T..~uivalen~ and c n g r a e n ~

De~on

5,~, Let S~I = ( ( ~ , 41), (x2, 42) . . . . .

(xm, 4m?) and ~2 = {(Yl, ~i),

(Y2, ~2) . . . . . (y,, ~,)} be two subfamilies of ~. Then 5PI and 5°2 are said to be T-equivalent if and only if L~.,.(,~I)- LT(5°2); congruent if and only if m = n and there are non-zero scalars a~,a2 . . . . . an e P such that $fz={al(xl, 41), a2(x2, ~ ) . . . . . an(x., 4.)}. Cle~.rly congruence implies T-equivalence° In addition, under the assumption that ~! and ,9'2 are subfamilies of ~÷, if the vectors xl, x2 . . . . . x~ are linearly indepen~e~:~, then the T-equivalence of ~ and ~a implies m--
~~on

s.l.. Let S~I = {(xl, 41), (x2, 42) . . . . . (x,,,, ~t.,)} and Y2-- {(Y,~ ~1), (Y2,/~2) . . . . . (Yn,/~)} be .ewe subfamilies of ~. Then £¢1 and £¢2 are Minequivalent if and only if for each 17~ (0, 1] the vector se~s {x, 14,~>~,i~ {1,2 . . . . .

m}}

and

{ y j l , ~ > ~ , / e { 1 , 2 . . . . . n}} are equivalent, i.e., generate the same subspace of X. ~ e e f , Trivial.

T h ~ r e m $.2. Let ~1 = {~:1, 41), (x2, 42) . . . . . (x,. 4~)} and ~2 = {(Yl,/~1), (Y2,/~2) . . . . . (y,,,/~,,)} be two subfamilies of ~C+.Suppose that both the vector sets {z~, x2 . . . . . xn} and {y~, Y2. . . . . y,,} are linearly independent. Then a sufficient condition for ~ and 5¢2 to be T-equivalent is that the following properties are fulfilled: ' (i) The vector sets ~ = {~ ] (x. 4~) ~ S~, ~ = 1} and 62 = {~ I (Yp ~) ~ Y2, p~ = 1} are equivalent. (ii) There exist vectors zi in L(G~) and non-zero scalars a~ in P such that

{(y~. ,~) e s~21 ,~ < ~} -- {(~. ~) + a&~. 4~) I (x. 4,) ~ se,, 4, < ~). Proof. Suppose that properties (i) and (ii) are fulfilled. We shall show that ~1 and ~2 are T-equivalent. Case 1 : G a = { x ~ , x 2 . . . . . xn}. In this case, property (~) says that the ve~or sets (xt,x2 . . . . . xn} and G2 are equivalent. Since the vector sets

76

Y~ Y ~ n 8

{~!, X~. . . . . ~ } and {Yl, Y~. . . . . y~} are linearly i n d ~ n ~ a n t , we ~ n assert that G~ = {3,1, y~. . . . . y,,}. It follows that 3"1 and 3"~ are T-equivaI~at. Ca~'e 2 : G 1 = ~. In this case L(G~) = {0}. From property 0~) we can asser~ that there e ~ t non-zero scalars ~ in P such that

Thus, o b s e ~ n g the facts that both the famflfies {a~(xl,~,l),a2(x2,~.z), . . . . a,,(x,, ~.,)} and 3'2 are families of n distinct fuzzy points, respectively, and {(y~, ~ ) ¢ 3,~.I ~ < I} = 3,~, we a s ~ r t that Y'I and Y2 are cong~er~t and 3,~= {a~@~,/~), a2(.~z, ~a), .- • t~,(x,,/~,)}. There.~ore ff~ and 3,2 are T-equivalent. Case 3: T ~ remaining case. We cat~ a~sc,~ne without loss of generality that .c/1 and 5~2 have respectively the following forms:

~, = {(xl, I), (x2, ~) . . . . .

(x., I), (x.+1, ~..,) . . . . .

~ = {(yl, ~), (y~ ~) . . . . .

( y . ~),

~x~, ~:,)),

(z,.l, ~)+ ~,÷~(x,.l, ~,.1) . . . . . (z.. ~)+ a.(x., ~.)} v,.htr¢ l ~ r < n ; G l = ( x l , x~ . . . . . x,}; G z = { y t , y2. . . . . . y,); L ( G 1 ) = L ( G 2 ) ; z,+~ . . . . . z~ ¢ L(G1); and ~,.1 . . . . . an are non-zero scalars in P. Put $1 = LT(3,1) and ~ - Lr(3,2). We shall finish our proof by showing that S~ = ~ . Le~ x ¢ X be Exed. (1) If x ~ L ( x l, x2 . . . . . x~), clearly S~.(x) = Sz(x ) -- O. (2) If x e L(G1), it is easily seen that Sl(x) -- S2(x) = 1. (3) If x e L(x.+l . . . . . x,) and x ~ O, then x can be expressed uniquely as.

x=~,~,+~+...+~ where r + 1 ~ h <]2 < " " <]q ~ n, and hi,, bi~. . . . . So, from the definition of $1 it follows that

b]q are non-zero scalars in P.

On the other hand, we have xffi -('b-hzi'+b~t~z'z+",a,, a,2 '÷~

Since z~,. ~:.. . . . . ai,

zb)

zj, e L(G~) = L(G2), we can write ai~

a~ zi~. = c~yl t czyz + . . . . + c~y,

g c n e ~ d T-fuzzy ~

where c~, c2 . . . . .

c~ ¢ P. $ o w©

spaces

77

have

x =c~y~ + c~y2 + . • • + c , y ,

(z~, 1 ) + aj,(xj~, A~) are r + q distinct fuzzy points in ~ ; ~gz= L r ( ~ ) ; and the vectors y~. y2 . . . . . y . z~+~ + a.+ix..~ . . . . . z~. + a~x. ate linearly independent, we can asse~ that L~(x) = T(~,, . ~ . . . . . ~ ) ~se~ Step (l)(~) in the proof of Theorem 3.~). '~ Therefore $~(x) = ~(x). r ( 4 ) If x ¢ L(x~, x2 . . . . . x , ) \ ( L ( G ~ ) LJ L(x,+~ . . . . . ~,)), then x ~An be ~ presseduniquelyas x = c~,x~ + c~zx~z + . • • + % x ~ + d#~x~,+ dhx h + . • • + d#,x#,~

where l ~ i~ < i2 < . . . < ~p ~ r; r + I ~ h < j~ < .... < j e a n ; d~, d h . . . . . dj, ate non-zero scalar in F. From the definition of S~ we have

s,(~) = r ( ~ , , ~h . . . . .

v.md c~,, c~ . . . . .

%,

~.).

O n . ~ e other hand, we have x = -

zj, +

aj,

zj~ + . . . +

+ (ca,:~,, + c~2xiz + . "

%

%

+ c~,)

+ . (zj, + aj,xj,) + ~ah (zj, + %xj,~ + .... + ~. tzjo+ aj xj,). Since z~,, z h . . . . .

zj,, x,,, x, 2. . . . . . x~o ¢ L ( G , ) = L(G2), we c~n write

x = e t y x + e2y2 + • • - + e , y ,

+ ( da~s~~z" + a~,xj,) + a,z d-h (zh + ahxh) + o . . + d'~ (z'~ + ,j, xj,)/

where et, e2 . . . . .

e, ¢ P. As in (3) it follows

"[his shows Sl(x) = S2(x) for all x ¢ A, i.e., $1 = Sz. T h e o ~ m $.3. Under the assumption o f Tkeorem 5.2, if the t-norm T is strict, t h ~ the condition is also necessary f o r ff~ and ~'2 to be T-equivalent.

Pn)ef. Assume that the t-norm T is strict ~nd suppose that ffl and ,9'z ate T-equiw~ent. Put Sl = Lr(ffl) and $2 - Lr(ffz). We distinguish three cases.

?~

Yu Yo~long

Case 1: G~ = {x~, x2 . . . . . hand, we have s~(x) = &(x) = o

xn} (see Theorem 5.2), In this c~se, on the one for any x E X \ L ( x ~ , x2 . . . . .

x.)

and 51(z) = ~ ( x ) -- 1 for any z ¢ L(xl, :<2. . . . .

x,);

on tile other hand, it is easily ~ e n that

S1(yj)=&(yj)-::~:>0

foralljE {I, 2 ..... ,z),

which implies that Sl(yj)=S2(y~)= 1 f o r a l l j G {1, 2 . . . . .

n},

and hence y~, Y2. . . . . y,, ~ L(GO. Using the fact that yt, Y2. . . . . y, are linea~iy independent, we got L(G1)= L(G2), i.e., property (i) is fulfilled. Next, because

property (ii) is trivially fulfilled. Case 2: G~ = #. In this case, property (i) is obviously fulfi]16d. In order to show that property (ii) is fulfilled, we only need to prove that ff~ and ~2 are congruent. This is done in two steps. Step I. In this step, we shall show that after an apI:.repfiate exchanging of the. names of fuzzy points in 5°, and ff~ (if-necessary), ~ and if2 can be rewritten, respectively, as: Y~ = {(x~, ~tO, (x2, ~ ) . . . . . , ~,x,~,A,,)), ~2 = {a~(x~, ~t~), (>'z, ~ ) . . . . .

(S.I)

(y., ~ . ) ) ,

where a~ ~s a non-zero scalar in P; ~ ~ ~ ~>. • • ~ A, a~d A1 ~ ~2 ~ " • • ~>/~. In fact, it is obvious that after appropriate exchanging of the ~ames of fuzzy points b°~ and b"~ (if necessary), b~ and 5°~ can be rew~tten, respect~;~ly, as: ~, = {(x~, ~), (~.~,~t~)..... (~, ~t,)}.

(5.2)

s~2= {(yl,m), (y~,~) ..... (y~,~)), where Ax ~>~ >~'-" ~-"~.,, a n d / ~ ~-/~ ~>.-- ~>~ . From (5.2) it is easily seen that Sz(x) as a function defined on X reaches its maximum :,~ at the point x =y~. On the other hand, since ~,t, ~ . . . . . ~,, ¢ (0, 1), x~, x:~. . . . . xn are linearly ~ndependent, and T ~s strict, w~ can assert that S~(x) as a function on X reaches its -

. . . . . .

~__

~.

~.~.~

*

where i ¢ {1, 2 . . . . . n} such that ~.: = ~ , and ~ ¢ P. Since S~ = S.z, ~here exist i e {1, 2 . . . . . n} and a~ e P ~,uch that (y~,/~) -- a~(:~, ~} and ~ ffi ~.~ffi/~. So, by

Finitely generated T-~uzzy li~ear s ~ ¢ s

79

exchanging the names of the fuzzy points (x~, 4~) and (xa 4~) (if i ~ 1), we can get

(5.~). Step 2. Assume that after successive exchanging the names of fuzzy points in ~ and if2, ~'~ and b°z have been rewritten, respectively, as:

s~ = {(x,, 4~), (x~, 4z) . . . . . (~,, 4,), (x,÷~, 4,÷~) . . . . . (x~, 4.)}, se, = {a~(~, 4~), a~(:~, 4~)m . . . . . a,(~,, ~,), (y,÷~, ~,÷,) . . . . . (y., ~.)},

(5.3)

where 4~-0>42~...->-4,-->4,÷~>-...-->4.; 4,~>~,÷~>...~>~.; aa, a2 . . . . . a, are non-zero scalar in P; and r < n. Now let us show that by exchanging the names of t~/o of (x,÷~, 4,÷~)..... (x,, 4,), or the names of two of (y,÷~,~,÷~ ..... (y,, ~,) in (5.3), ~ and ,7'z can be rewritten, respectively, as:

,~? = {(XI, 41), (X2, 42) . . . . .

(Xr÷I, 4,÷1), (Xr÷2, 4,÷2) . . . . .

~. = { ~(~, 40, ~2(~,, 4~)..... a,÷~(~,÷~,4,~,), (y,÷~, ~,,~)..... (y., ~.)},

(~n, 4n)},

(5.4)

where A~ ~ 4z ~>-.. ~>A,+~ ~ 4,+z ~>4,; 4,+~ ~ ~u,+2~>... >~/~n; and a~, a2 . . . . . a,÷~ are non-zero scalars in P. In fact, from (5.3) it is easily seen that S~(y,+~) = S~(y,+~) = ~u,+~> 0 and hence y , ~ e L(z~, x2 ....:, x,). On the other hand, we can write y,÷~ = o~,~',,+ a,..x~~... + a~ .r~,, S~(y,+,) = T(4,,, 4,, . . . . . 4 0 ,

(5.5)

where 1 ~-.-4: . T ~ , : , using the facts that 4~, Aa. . . . . 4, e (0, 1) and that T is strict, we can ass¢~i. ~hat ff F > l, then s~(y.÷.) = s.(y,÷~)= T(4i.. 4~ ..... 4 0 < 4,~. i.e., ~,+i < 4~+i. ~ i l a r l y , we can write

x~., = bj, yj, + bi2yj2 + . . . + b~qyiq, &(x.+.) = r(~j~. ,j~. . . . . ~,,),

(5.6)

where i ~ j l < h < . . . 1, then s,(x,+~) = &(x,÷O = T(~j,, ~h . . . . .

~ , ) < ~,÷~.

~

Yu Yandong

From these discussions on x,÷l and on y,+l, it is easily seen that either p = I in (5.5) or q = 1 in (5.6).

I f p = 1 in (5.5), th~n we bare y~+l~=a~,xi, and

~,+l=~i,;

(5.7)

"where r + 1 ~ il ~
=

bj~yj, and ~,+~=

~j,

(5.8)

where r + 1 ~<] ~ A , ÷ , ~--.- ~>~, > 0 and 1 >~,÷~ ~>--. ~>~, > 0 . l[a a w~y ~imilar to tl~e one in Ca~e 2, the proof can be completed. F~om Theorem 5.2 and the proof of Theorem 5.3 we immediately get! Theo~e~ ~,4. Suppose that T is a strict t-norm. Let $~ = {(x~, ;~), (x2, ~2). . . . . (x~, ~ ) } and ,~°2 = {(y., ~'1), "~Y2, ~A2)..... (yn, t~.~)} be two subfamilies o f $~. Suppose that the vector sets {x~, x~ . . . . . x~} a~e ]ine~rly independent ~nd s~tppose ~ t ~ , ~2 . . . . . :~; ~ , !~ . . . . . !~ < 1. T~zen ~ and ~#z are T-equivalent if and only if they are congruent. Ac~w|~g~eut Th~ a~thor wou~d like to thank ~ofessoz R. Lowen for his great help in modffyh~g an earlier manuscript of t~e paper. [~efe~e~es [1] A.K. Ka~aras ap~ D.B. Lh~.,Fuzzy,vector spaces a~d fu~" topologlca~vector spaces, I. Math. Anal. A,opL 58 (1977) ~35--IA6.

F ~ e l y g e n t l e d T-f~.zy t ~ r s ~ e s

8!

[2] R. Lowen, Cenvex fury sets, Fuzzy Sets and Systems 3 (19g0) 291-310. [3] L.-W. Pei, C~,nvexf ~ sets (|), J. W ~ Univ. (Nat. $cLEd.) $ (1984) 13-.22. [4] Y.-D. Yu, On the convex filzzysets (1), Fuzzy Math. 4 (2) (I~,A) 29-40. [5] Y.-D. Yu, Ftu~y Hnear spaces redefined, Fuzzy M~h. 4 (3) (1984) 59-62. [6] A. R~nfe|d, Fu~,y groups, J. M~h. An~d. AFF/. 35 (1971) 512-517. [7] J.M. Anthony and H. Sherwood, Fuzzy groups redefined, J. Math. Anat. A?p/. 69 (1979) 124-130. [8] B. Schwe~er and A. Sklar, Associative functions and statistic! t~angle inequities.;, Pub/. Math. Debrecen $ (1961) 169-186. [9] B. SchweW.er and A. Sklar, Associat[v© functions and abstract sen~-grou~, Pub/. Mat~ Debrecen 10 (1963) 69-61. [10] Y.-D. Yu, Triangular norms and TNF-sigma-algcbr:,% Fuzzy Sets end Systems 16 (1985) 251-264. [11] B.-M. Pu and Y.-M. Liu, Fuzzy topelogy I, N¢;gh~orhood a t ~ c t u r e of a fu~: point and Moore-Smith convergent, I. Math. Anal. App/. "i~ (~980) 571-599.