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Fuzzy linear transformations.

1 Introduction

In [5], A.R. Meenakshi defined fuzzy linear transformations on fuzzy vector spaces over fuzzy algebas and proved that the set of all fuzzy linear transformations between two fuzzy vector spaces forms a vector space. In this paper we introduce fuzzy linear transformation in a different direction. We introduce fuzzy linear transformations, fuzzy linear functionals and fuzzy linear operators based on the notions of fuzzy fields and fuzzy linear spaces introduced by Gu Wenxiang and Lu Tu [1]. Section 2 gives a brief summary of fuzzy fields, fuzzy linear spaces and some preliminaries. Fuzzy linear transformations, fuzzy linear functionals and fuzzy linear operators are introduced in section 3. Some fundamental properties of fuzzy linear transformations are discussed. Also, some results on images of a-cuts of fuzzy linear spaces and of convex fuzzy linear spaces are given. The algebra of fuzzy linear transformations is studied in section 4. It is shown that the set FL(V ~ of fuzzy linear transformations, if it is nonempty, is a linear space over [X.sub.v]. Moreover, the set FL(V) of fuzzy linear operators is an algebra.

2 Preliminaries

This section gives the basic definitions and results on fuzzy fields and fuzzy linear spaces over fuzzy fields.

Definition 1 [1] Let X be a field and F a fuzzy set in X with membership function [[mu].sub.F].. Suppose the following conditions hold:

(i) [[mu].sub.F] (x + y) [greater than or equal to] min {[[mu].sub.F](x), [[mu].sub.F](y)}, x, y [member of] X

(ii) [[mu].sub.F](x) [[mu].sub.F](-x), x [member of X

(iii) [[mu].sub.F](x) = [greater than or equal to] min {[[mu].sub.F](x), [[mu].sub.F](y)},x, y [member of] X

(iv) [[mu].sub.F](x) = [[mu].sub.F]([x.sup.-1]), x ([not equal to] 0) [member of] X.

Then F is called a fuzzy field in X and it is denoted by (F X). Also (F X) is called a fuzzy field of X.

Note that [[mu].sub.F] (0) [greater than or equal to] [[mu].sub.F](x) for all x [member of] X and [[mu].sub.F](1) [greater than or equal to] [[mu].sub.F] (x) for all x ([not equal to] 0) [member of] X.

Proposition 1 [1] Let X and Y be fields and f a homomorphism of X into Y. Suppose that

(F, X) is a fuzzy field of X and (G, Y) is a fuzzy field of Y. Then

(i) (f(F ), Y) is a fuzzy field of Y

(ii) ([f.sup.-1]) (G ), X) is a fuzzy field of X.

Definition 2 [1] Let X be a field and (F X) be a fuzzy field of X. Let Y be a linear space over X and V be a fuzzy set of Y with membership function [[mu].sub.v]. Suppose the following conditions hold:

(i) [[mu].sub.v](x - y) [greater than or equal to] min {[[mu].sub.v](x), [[mu].sub.v](y)}, x, y [member of] Y

(ii) [[mu].sub.v](x) = [[mu].sub.v](-x), x [member of] Y

(iii) [[mu].sub.v]([lambda]x) [greater than or equal to] min ([[mu].sub.F][lambda]), [[mu].sub.v](x)}, [lambda] [member of] X, x [member of] Y

(iv) [[mu].sub.F](1) [greater than or equal to] [[mu].sub.v](0).

Then (V, Y) is called a fuzzy linear space over (F X).

Note that [[mu].sub.v] (0) [greater than or equal to] [[mu].sub.v](x) for all x [member of] Y.

Proposition 2 [1] Let (F X) be a fuzzy field of X and Y a linear space over X Let V be a fuzzy set of Y. Then (V, Y) is a fuzzy linear space over (F X) if and only if

(i) [[mu].sub.v]([lambda]x - [[mu].sub.y]) [greater than or equal to] min {[[mu].sub.F]([lambda]) [congruent] [[mu].sub.v](x), [[mu].sub.F]([mu]) [congruent] [[mu].sub.v](y)}, [lambda], [mu] [member of] X and x, y [member of] Y

(ii) [[mu].sub.F](1) [greater than or equal to] [[mu].sub.v](x), x [member of] Y. Note 1 Condition (i) in proposition 2 can be restated as [[mu].sub.v]([lambda]x + [mu]y) [greater than or equal to] min {[[mu].sub.F] ([mu]), [[mu].sub.v](x), [[mu].sub.v](y)}.

Proposition 3 [1] Let Y and Z be linear spaces over the field X and f a linear transformation of Y into Z. Let (F X) be a fuzzy field of X and (W, Z) be a fuzzy linear space over (F X). Then ([f.sup.-1] (W), Y) is a fuzzy linear space over (F, X).

Proposition 4 [1] Let Y and Z be linear spaces over the field X and f a linear transformation of Y into Z. Let (F, X) be a fuzzy field of X and (V, Y) be a fuzzy linear space over (F X). Then (f (V), Z) is a fuzzy linear space over (F X).

Definition 3 [6] If [A.sub.1], [A.sub.2] ..., [A.sub.n] are fuzzy sets of a set X, then [A.sub.1] x [A.sub.2] x ... x [A.sub.n] is a fuzzy set of [X.sup.n] with membership function given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular, if A is a fuzzy set of X, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a fuzzy set of [X.sup.n] with membership function given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 4 [4] Let Y [subset.bar] [R.sup.n], the n-dimensional Euclidean space. A fuzzy set V in Y is convex if [[mu].sub.v]([lambda][y.sub.1] + (1 - [lambda])[y.sub.2]) [greater than or equal to] min {[[mu].sub.v]([y.sub.1]), [[mu].sub.v]([y.sub.2])} for all [y.sub.1], [y.sub.2] [member of] Y and for all [lambda] [member of] [0, 1].

Definition 5[3] An algebra A over a field K is a vector space A over K such that for each ordered pair of elements x, y [member of] A, a unique product xy [member of] A is defined with properties:

(1) (xy)2 = x(yz)

(2a) x(y + z) = xy xz

(2b) (x + y)z = xz + yz

(3) [alpha](xy) = ([alpha]x)y = x([alpha]y)

for all x, y, z [member of] A and scalars [alpha].

We have the following propositions.

Proposition 5 Let (F X) be a fuzzy field of X. Let ([V.sub.1], [Y.sub.1]), ([V.sub.2], [Y.sub.2]), ..., ([V.sub.n], [Y.sub.n]) be fuzzy linear spaces over (F, X). Then ([V.sub.1] x [V.sub.2] x ... x [V.sub.1], [Y.sub.1] x [Y.sub.2] x ... [Y.sub.n] is a fuzzy linear space over (F, X).

Proof. Let V = [V.sub.1] x [V.sub.2] x ... x [V.sub.n].

Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]), y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [Y.sub.1] x [Y.sub.2] x ... x [Y.sub.N] and [alpha], [beta] [member of] X.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence ([V.sub.1] x [V.sub.2] x ... x [V.sub.x], [Y.sub.1] x [Y.sub.2] x ... x [Y.sub.n]) is a fuzzy linear space over (F, X).

Proposition 6 A fuzzy field (F X) is a fuzzy linear space over itself if and only if [[mu].sub.F](0) = [[mu].sub.F](1).

Proposition 7 If [[mu].sub.F] (0) = [[mu].sub.F](1), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a fuzzy linear space

over (FX)and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ..., n, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the standard basis for [X.sup.n].

Proposition 8 If (V, Y) is a fuzzy linear space over (F, X), then the nonempty set [X.sub.V] = {x [member of] X: [[mu].sub.F](x) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y} is a subfield of X. Also, (F, [X.sub.v]) is a fuzzy field of [X.sub.v] and (V, Y) is a fuzzy linear space over (F, [X.sub.v]).

Proof. Let [x.sub.1], [x.sub.2], [member of] [X.sub.v]. Then [[mu].sub.f]([x.sub.1] - [x.sub.2]) [greater than or equal to] min{[[mu].sub.f] ([x.,sub.1]), [[mu].sub.f] (x[.sub.2])} [greater than or equal to] [[mu].sub.v] (y) and for [x.sub.2] [not equal to] [[mu].sub.f]([x.sub.1][x.sup.-1.sub.2]) [greater than or equal to] min{[[mu].sub.f]([x.sub.1]), [[mu].sub.f]([x.sub.2])} [greater than or equal [[mu].sub.v](y) for all y [member of] Y. So [x.sub.1] - [x.sub.2] [member of] [X.sub.v] and [x.sub.1][x.sup.-1.sub.2] [member of] [X.sub.v] if [x.sub.2] [not equal to] 0. Therefore [X.sub.V] is a subfield of X.

The second part is trivial.

Proposition 9 If (V, Y) is a fuzzy linear space over (F X) and if T : Y [right arrow] Z is a linear transformation, then (T(V), T(Y)) is a fuzzy linear space over (F X).

3 Fuzzy Linear Transformations

We now introduce fuzzy linear transformations, fuzzy linear functionals and fuzzy linear operators on fuzzy linear spaces.

Definition 6 Let (V, Y) and (W, Z) be fuzzy linear spaces over the fuzzy field (F, X) and T: Y [right arrow] Z be a linear transformation. If [[mu].sub.w](Ty) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y, then (T, (V, Y), (W, Z), (F X)) is said to be a fuzzy linear transformation.

Definition 7 Let (V, Y) be a fuzzy linear space over the fuzzy field (F X) with [[mu].sub.F] (0) = [[mu].sub.F](1) and T be a linear functional on Y. If [[mu].sub.F](Ty) [greater than or equal to [[mu].sub.v](y) for all [member of] Y, then (T, (V, Y), (F X), (F, X)) is said to be a fuzzy linear functional.

If Ty [member of] [X.sub.v] for all y [member of] Y, then (T, (V, Y), (F X), (F, X)) is a fuzzy linear functional.

Definition 8 Let (V, Y) be a fuzzy linear space over the fuzzy field (F, X) and T be a linear operator on Y. If : [[mu].sub.v](Ty) [greater than or equal to](y) for all y [member of] Y, then (T, (V, Y), (V, Y), (F, X)) is said to be a fuzzy linear operator.

Example 1 Let (V, Y) be a fuzzy linear space over (F X) and Z a linear space over X. For any linear transformation T : Y [right arrow] Z, (T, (V, Y), (T(V), Z), (F, X)) is a fuzzy linear transformation since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 9 Let (V, Y) be a fuzzy linear space over the fuzzy field (F, X), Z a linear space over X and T: Y [right arrow] Z be a linear transformation. Then T is said to be a fuzzy linear transformation on (V, Y) if [[mu].sub.T(v)] (Ty)= [[mu].sub.v] (y) for all [member of] Y.

Example 2 If (V, Y) is a fuzzy linear space over (F, X) and T is the zero operator on Y, then (T, (V, Y), (V, Y), (F, X)) is a fuzzy linear operator.

Example 3 (I, (V, Y), (V, Y), (F, X)), where I is the identity operator on Y, is a fuzzy linear operator.

Example 4 If (F, R) is a fuzzy field of R with [[mu].sub.F](0) = [[mu].sub.F](1) and T is the linear operator on [R.sup.2] defined by T([x.sub.1],[x.sub.2]} = ([x.sub.2], [x.sub.1]), then (T, (F x F, [R.sup.2]), (F x F, [R.sup.2]), (F, R)) is a fuzzy linear operator.

Example 5 If (F, R) is a fuzzy field of R with [[mu].sub.F] (0) = [[mu].sub.F](1) and if [T.sub.1], [T.sub.2]: [R.sup.3] [right arrow] [R.sup.2] are the linear transformations defined by [T.sub.1]([x.sub.1], [x.sub.2],[x.sub.3]) = ([x.sub.2], [x.sub.3]), [T.sub.2]([x.sub.1],[x.sub.2],[x.sub.3]) = ([x.sub.1] - [x.sub.2], [x.sub.1] - [x.sub.3]), then ([T.sub.2], (F x F x [??] [R.sup.3]), (F x F, [R.sup.2]), (F, R))(for i = 1, 2) are fuzzy linear transformations since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 6 If (F, R) is a fuzzy field of R with [[mu].sub.F](0) = [[mu].sub.F](1) and T is the linear operator on [R.sup.2] defined by T ([x.sub.1], [x.sub.2]) = ([x.sub.1] - [x.sub.2], 0), then (T, (F x F, [R.sup.2]), (F x F, [R.sup.2]), (F, R)) is a fuzzy linear operator since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 10 1. If (T, (V, Y), (W, Z), (F, X)) is a fuzzy linear transformation, then

(i) [[mu].sub.W](0) [greater than or equal to](y) for all y [member of] Y

(ii) [[mu].sub.W](Ty) [greater than or equal to] [[mu].sub.W](0) for all y [member of] K or T.

2. If T is a fuzzy linear transformation on (V, Y), then

(i) [[mu].sub.T(v)](0) = [[mu].sub.v](0) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y

(ii) [[mu].sub.v](r) = [[mu].sub.v](0) [greater than or equal to] [[mu].sub.v](y) for all [??] [member of] Ker T and for all y [member of] Y.

Proof.

1. (i) [[mu].sub.W](0) = [[mu].sub.W](T0)[greater than or equal to] [[mu].sub.v](0) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y. (ii) is trivial.

2. (i) [[mu].sub.F(V)](0) = [[mu].sub.T(V)](T0) = [[mu].sub.v](0) [greater than or equal to](y) for all y [member of] Y. (ii) For all r [member of] Ker T, [[mu].sub.v][R] = [[mu].sub.T(V)](Tr) = [[mu].sub.T(V)](0) = [[mu].sub.v](0) [greater than or equal to] [[mu].sub.v](y).

Proposition 11 Let (V, Y) be a fuzzy linear space over the fuzzy field (F, X), Z a linear space over X and T : Y [right arrow] Z a linear transformation. If T is injective, then T is a fuzzy linear transformation on (V, Y).

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 12 Let (V, Y) be a fuzzy linear space over the fuzzy field (F, X), Z a linear space over X and T : Y [right arrow] Z a linear transformation. Then the following statements are equivalent.

(i) T is a fuzzy linear transformation on (V, Y)

(ii) [y.sub.1], [y.sub.2] [member of] Y and T[y.sub.1] = T[y.sub.2] [right arrow] [[mu].sub.v]([y.sub.1]) = [[mu].sub.v]([y.sub.2])

Proof.

(i) [right arrow] (ii). If T is a fuzzy linear transformation on (V, Y) and T[y.sub.1] = T[y.sub.2], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, [[mu].sub.v]([y.sub.2]) [greater than or equal to] [[mu].sub.v]([y.sub.1]). So [[mu].sub.v]([y.sub.1]) = [[mu].sub.v]([y.sub.2]).

(ii) [right arrow] (i). If (ii) holds, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 13 Let Y be a linear space over X, (W, Z) a fuzzy linear space over (F, X) and T : Y [right arrow] Z a linear transformation. Then (T, ([T.sup.- 1](W),Y), (W, Z), (F, X))is a fuzzy linear transformation.

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 14 If (T, (V, Y), (W, Z), (F, X)) is a fuzzy linear transformation and [alpha] [member of] [0, 1], then T([sup.[alpha]V]) [sup.[alpha]W].

Proof. If Ty [member of] T([sup.[alpha]V]), then [[mu].sub.v](y) [greater than or equal to] [alpha], so [[mu].sub.W](Ty) [greater than or equal to] [alpha], so [[mu].sub.W](Ty) [greater than or equal to] [[mu].sub.v][greater than or equal to] [alpha]. Therefore Ty [member of] [sup.[alpha]W].

If A and B are arbitrary crisp sets and f : A [right arrow] B is an arbitrary crisp function, then for any V [member of], F(A) and all [alpha] [member of] [0, 1], f ([sup.[alpha]V]) [sup.[alpha]][f(V)]. In general, f([sup.[alpha]V]) [not equal to] [sup.[alpha][f(V)]. But equality holds in the case of fuzzy linear transformations on fuzzy linear spaces.

Proposition 15 If T is a fuzzy linear transformation on the fuzzy linear space (V, Y) over (F X), then

(i) { y [member of] Y : [[mu].sub.T(V)](Ty) [greater than or equal to] [alpha]} = [sup.[alpha]V]

(ii) T ([sup.[alpha]]V) = [sup.[alpha]][T(V)] in T(Y).

Proof. (i) is trivial.

(ii) In T(Y),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 16 If a linear transformation T of Y onto Z is a fuzzy linear transformation on the fuzzy linear space (V, Y), then T ([sup.[alpha]]V) = [sup.[alpha]][T(V)] in Z.

Proof. T ([sup.[alpha]]V) = {Ty : y [member of] [sup.[alpha]]V} = {Ty: [[mu].sub.v](y) [greater than or equal to] [alpha]} = {Ty: [[mu].sub.T(V)](Ty) [greater than or equal to] [alpha]} = [sup.[alpha]][T(V)].

Proposition 17 If (F X) is a fuzzy field of X with [[mu].sub.F](0) = [[mu].sub.F](1) and T is a fuzzy linear transformation on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1, 2, ..., n, where {[[??].sub.1], [[??].sub.2], ..., [[??].sub.n]} is the standard basis for [X.sup.n].

Proposition 18 Let Y be a linear space in [R.sup.n] and Z a linear space in [R.sup.m]. Let V be a convex fuzzy set in Y and (V, Y) be a fuzzy linear space over (F,R). If a linear transformation T of Y onto Z is a fuzzy linear transformation on (V, Y), then T(V) is a convex fuzzy set in Z.

Proof. Let 0 [greater than or equal to] [lambda] [greater than or equal to] and T[y.sub.1], T[y.sub.2] [member of] Z.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So T(V) is a convex fuzzy set in Z.

Proposition 19 Let ([T.sub.i], ([V.sub.i], [Y.sub.i]), ([W.sub.i], [Z.sub.i]), (F, X)) (for I = 1, 2, ..., n) be fuzzy linear transformations, Y = [Y.sub.1] x [Y.sub.2] x ... x [Y.sub.n], Z = [Z.sub.1] x [Z.sub.2] x ... x [Z.sub.n], V = [V.sub.1] [V.sub.2] x ... x [V.sub.x], W = [W.sub.1], x [W.sub.2] x ... x [W.sub.1] and T: Y [right arrow] Z be the linear transformation defined by T([y.sub.1], [y.sub.2], ..., [y.sub.n]) = ([T.sub.1][y.sub.1], [T.sub.2][y.sub.2], ..., [T.sub.n][y.sub.n]). Then (T, (V, Y), (W, Z), (F, X) is a fuzzy linear transformation.

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4 Algebra of Fuzzy Linear Tranformations

Proposition 20 If ([T.sub.i], (V, Y), (W, Z), (F, X)) (for i = 1, 2) are fuzzy linear transformations, then ([T.sub.1] [??] [T.sub.2], (V, Y), (W, Z), (F, X)) are fuzzy linear transformations.

Proof. [[mu].sub.W](([T.sub.1] [??] [T.sub.2])(y)) [greater than or equal to] min{[[mu].sub.W]([T.sub.1]y), [[mu].sub.W]([T.sub.2]y)} [greater than or equal to] min{[[mu].sub.v](y), [[mu].sub.v](y)} = [[mu].sub.v](y) for all y [member of] Y.

Proposition 21 Let FL(V,W) = {T : (T, (V, Y), (W, Z), (F, X)) is a fuzzy linear transformation}. Then the following statements are equivalent.

(i) FL(V,N~ is nonempty.

(ii) O [member of] FL(V,W), where O is the zero linear transformation of Y into Z.

(iii) [[mu].sub.W]([0.sub.z]) [greater than or equal to] [[mu].sub.v]([0.sub.y]), where [0.sub.y], [0.sub.z] are respectively the zero vectors in Y and Z.

Proof. (i) [right arrow] (u) since if T [member of] FL(V,N~, then O = T - T FL(V,W).

(ii) [right arrow] (iii) since O [member of] FL(V,W) [right arrow][[mu].sub.W](Oy) [greater than or equal to](y) for all y [member of] Y [right arrow] [[mu].sub.W]([0.sub.z]) [greater than or equal to] [[mu].sub.v]([0.sub.y]).

(iii) [right arrow] (i) since [[mu].sub.W](Oy) = [[mu].sub.W]([0.sub.z]) [greater than or equal to] [[mu].sub.v]([0.sub.y]) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y [right arrow] O [member of] FL(V,W).

Proposition 22 If FL(V,W) is nonempty, then FL(V,W) is an abelian group.

Proof. If [T.usb.1], [T.sub.2] [member of] FL(V,W), then [T.sub.1] - [T.sub.2] [member of] FL(V,W).

O [member of] FL(V,W)

If t [member of] FL(V,W), then - T [member of] FL(V,W).

Remark 1 If [[mu].sub.F](0) = [[mu].sub.F] (1) and FL(V,F) = {T : (T, (V, Y), (F X), (F, X)) is a fuzzy linear functional}, then FL(V,F) is nonempty since the zero linear functional on V is in FL(V,F). Hence FL(V,F) is an abelian group.

Remark 2 If FL(V) = {T : (T, (V, Y), (V, Y), (F X)) is a fuzzy linear operator}, then FL(V) is nonempty since it contains the zero linear operator on Y. So FL(V) is an abelian group.

Proposition 23 If FL(V,W) is nonempty, then it is a linear space over [X.sub.V]. In particular, FL(V) is a linear space over [X.sub.V]. Also, if [[mu].sub.F] (0) = [[mu].sub.F](1), then FL(V,F) is a linear space over [X.sub.V].

Proof. [X.sub.V] = {[alpha] [member of] X: [[mu].sub.F]([alpha]) [greater than or equal to] [[mu].sub.v](y) for all y [member of] Y} and (FL(V,W), +) is an abelian group.

If [alpha] [member of] [X.sub.V] and T [member of] FL(V,W), then

[[mu].sub.W](([alpha]T)(y)) = [[mu].sub.W]([alpha]Ty) [greater than or equal to] min {[[mu].sub.F]([alpha]), [[mu].sub.W](Ty)} [greater than or equal to] min {[[mu].sub.v](y), [[mu].sub.v](y)} = [[mu].sub.v](y) for all y [member of] Y.

[??] [alpha]T [member of] FL(V,W).

Hence FL(V,W) is a linear space over [X.sub.V].

Proposition 24 If ([T.sub.1], ([V.sub.1], [Y.sub.1]), ([V.sub.2], [Y.sub.2]), (F, X)) and ([T.sub.2], ([V.sub.2], [Y.sub.2]), ([V.sub.3], [Y.sub.3]), (F, X)) are fuzzy linear transformations, then ([T.sub.2][T.sub.1], ([V.sub.1], [Y.sub.1]), ([V.sub.3], [Y.sub.3]), (F, X)) is a fuzzy linear transformation, the multiplication being composition of linear transformations.

In particular, if ([T.sub.1], (V, Y), (V, Y), (F X)) ( for i = 1, 2) are fuzzy linear operators, then ([T.sub.1][T.sub.2], (V,Y), (V, Y), (F, X)) is a fuzzy linear operator, the multiplication being composition of linear operators.

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [??] ([T.sub.2][T.sub.1], ([V.sub.1], [Y.sub.1]), ([V.sub.3], [Y.sub.3]), (F, X)) is a fuzzy linear transformation.

Proposition 25 FL(V) is an algebra with identity.

Proof. The linear space FL(V) is an algebra with identity I (the identity operator on Y), the multiplication being composition of operators.

Acknowledgement

The authors wish to record their sincere gratitude to Prof. T. Thrivikraman for his valuable discussions and help in the preparation of this paper.

References

[1] Gu Wenxiang and Lu Tu, 1992, "Fuzzy Linear Spaces", Fuzzy Sets and Systems, 49, pp. 377-380.

[2] Klir, G.J., and Yuan, B., 2002, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall of India Private Limited, New Delhi.

[3] Kreyszig, E., 2005, Introductory Functional Analysis with Applications, John Wiley & Sons, New York.

[4] Lee, K.H., 2005, First Course on Fuzzy Theory and Applications, Springer-Verlag, Heidelberg.

[5] Meenakshi, A.R., 2008, Fuzzy Matrix: Theory and Applications, MJP Publishers, Chennai.

[6] Zimmermann, H.J., 1996, Fuzzy Set Theory-And Its Applications (Second revised edition), Allied Publishers Limited, New Delhi.

T. V. Ramakrishnan and C. P. Santhosh

Department of Mathematical Sciences, Kannur University Mangattuparamba,

Kannur Kerala, India. 670 567

E-mail: santhoshcpchu@yahoo.co.in
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Author:Ramakrishnan, T.V.; Santhosh, C.P.
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:9INDI
Date:Apr 1, 2009
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