Download FUZZY SEMI-INNER-PRODUCT SPACE Eui-Whan Cho, Young

J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math.
2 (1995), no. 2, 163–171
FUZZY SEMI-INNER-PRODUCT SPACE
Eui-Whan Cho, Young-Key Kim and Chae-Seob Shin
Abstract. G.Lumer [8] introduced the concept of semi-product space. H.M.ElHamouly [7] introduced the concept of fuzzy inner product spaces.
In this paper, we defined fuzzy semi-inner-prodct space and investigated some
properties of fuzzy semi product space.
1. Preliminaries
Definition 1.1 Hutton [1]. A fuzzy real number ζ is a nonascending, left continuous function from R into I = [0, 1] with ζ(−∞+ ) = 1 and ζ(+∞− ) = 0. The
set of all fuzzy real numbers will be denoted by R(I). The partial ordering ≥ on
R(I) is the natural ordering of real functions. The set of all reals R is canonically
embedded in R(I) in the following fashion, for every r ∈ R, we associated the fuzzy
real number r̄ ∈ R(I) which is defined by
r=
1 if t ≤ r
0
if t > r
The set R∗ (I) of all nonnegative real numbers is defined by
R∗ (I) = {ζ ∈ R(I) : ζ ≥ 0̄}.
1991 Mathematics Subject Classification. Primary 46S40 (46C50).
Key words and phrases. Fuzzy semi-inner-product, Fuzzy Pseudo-norm.
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EUI-WHAN CHO, YOUNG-KEY KIM AND CHAE-SEOB SHIN
Definition 1.2 Grunsky [5]. Let ζ, ξ be two fuzzy real numbers in R(I), and let
s be any real number. Then
(i) Addition of fuzzy real numbers ⊕ is defined on R(I) by
(ξ ⊕ ζ)(s) = sup{ξ(t) ∧ ζ(s − t) : t ∈ R}.
(ii) Scalar multiplication by a nonnegative r ∈ R is defined on R(I) by
(rξ)(s) =
0̄
if r = 0
ξ( rs )
if r > 0
It is well known that the above two operations are well defined on R(I) and the
canonical embedding of R in R(I) preserves these two operations. The results in
the following proposition are well known.
Proposition 1.3 Morsi & Yehia [4].
(i) Addition and scalar multiplication preserve the order ≥ on R(I).
(ii) R(I) is closed under these two operations.
(iii) For η, ζ and ξ ∈ R(I), we have
η ⊕ ξ ≥ ζ ⊕ ξiffη ≥ ζ.
Definition 1.4 Rodabaugh [6]. Multiplication of two nonnegative fuzzy real
numbers η , ζ ∈ R∗ (I) is defined by
1
if s ≤ 0
(ηζ)(s) =
s
sup{η(b) ∧ ζ( b ) : b > 0} if s > 0,
where s ∈ R. it is shown in [6] that R∗ (I) is closed under multiplication.
Definition 1.5 Katsaras [2]. A fuzzy pseudo-norm on a real of complex space
X is a function k k : X → R∗ (I) which satisfies the following two conditions ; for
x, y ∈ X and s in the field
(i) ksxk = |s|kxk,
(ii) kx + yk ≤ kxk ⊕ kyk.
The algebraic properties of addition and nonnegative scalar multiplication on
R∗ (I) enable us to embed R∗ (I) in the smallest real vector space M (I) as follow.
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Definition 1.6 Mashhour & Morsi [3]. The set M (I) is the cartesian product
R∗ (I) × R∗ (I) modulo the equivalence relation ∼ defined by
(η, ζ) ∼ (ξ, λ) iff η ⊕ λ = ζ ⊕ ξ.
The partial order ≥ on M (I) is defined by
(η, ζ) ≥ (ξ, λ) iff η ⊕ λ ≥ ζ ⊕ ξ.
The set M ∗ (I) is defined by
M ∗ (I) = {(η, ζ) ∈ M (I) : (η, ζ) ≥ 0̄}
= {(η, ζ) ∈ M (I) : η ≥ ζ}
R∗ (I) is canonically embedded in M (I) by representing each η ∈ R ∗ (I) as (η, 0̄) ∈
M (I). Also R is embedded in M (I) as follows : for r ∈ R, r is identified with
¯ ∈ M (I) if r < 0.
(r̄, 0̄) ∈ M (I) if r ≥ 0 and with (0̄, (−r))
Addition ⊕ and real scalar multiplication are defined on M (I) by:
(η, ζ) ⊕ (ξ, λ) = (η ⊕ ξ, ζ ⊕ λ),
(tη, tζ)
if t ≥ 0
t(η, ζ)
(|t|ζ, |t|η) if t < 0.
(i)
(ii)
Theorem 1.7 Mashhour & Morsi [3]. The above addition and scalar multiplication are well defined on M (I). Under these twooperations, M (I) is the smallest real
vector space including R∗ (I). In patricular, the canonical embeding of R ∗ (I) into
M (I) preserves addition and nonnegative scalar multiplication, while the canonical
embedding of R into M (I) is a vector space embedding.
Definition 1.8 Mashhour & Morsi [3]. The N -Euclidean norm on M (I) is the
fuzzy pseudo-norm |b c| defined by : for (η, ζ) ∈ M (I),
|b(η, ζ)c| = inf{ξ ∈ R∗ (I) : ξ ≥ (ζ, η) and ξ ≥ (zeta, η)}
= inf{ξ ∈ R∗ (I) : ξ ⊕ ζ ≥ η and ξ ⊕ η ≥ ζ},
where ξ = (ξ, 0̄) according to the embeding of R∗ (I) in M (I).
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EUI-WHAN CHO, YOUNG-KEY KIM AND CHAE-SEOB SHIN
Definition 1.9 Morsi & Yehia [4]. A real algebra X with a fuzzy pseudo-norm
k k on X will be called a fuzzy pseudo-norm algebra if for all x, y ∈ X,
kxyk ≤ kxkkyk,
where multiplication in the right-hand side is the fuzzy multiplication on R∗ (I).
Definition 1.10 Morsi & Yehia [4]. Multiplication on M (I) is defined by:
for (η, ζ), (ξ, λ) ∈ M (I),
(η, ζ)(ξ, λ) = (ηξ ⊕ ζλ, ηλ ⊕ ζξ).
Theorem 1.11 Morsi & Yehia [4].
(i) Multiplication on M (I) is well defined.
(ii) The canonical embeddings of R∗ (I) and R into M (I) preserve multiplication.
(iii) Underaddition, scalar multiplication, M (I) is a real associative and commutative algebra with unit element 1̄ = (1̄, 0̄).
(iv) M (I) is not an integral domain.
(v) (M (I), |b c|) is a fuzzy pseudo-normed vector space and is a fuzzy pseudonormed algebra under its multiplication.
Definition 1.12. Let U be a fuzzy subset of a universe X and let α ∈ I1 = [0, 1).
The α -cut of U is the crisp subset of X
U (α) = {x ∈ X : U (x) > α}.
Fuzzy real numbers in R∗ (I) can be considered as fuzzy subsets of the set R∗ of
all nonnegative reals. Therefore, for each η ∈ R(I), its α -cut η (α) = [0, t) or =
[0, t], where t = ∨{x ∈ R : η(x) > α} is uniquely identified with the number t. It is
obvious that α-cuts preserve the three operations on R∗ (I) and order on R∗ (I) in
the following sense : for every η, ζ ∈ R∗ (I), α ∈ I1 , and r ≥ 0 we have
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(η ⊕ ζ)(α) = η (α) + ζ (α)
(i)
(rη)(α) = rη (α)
(ii)
(ηζ)(α) = η (α) ζ (α)
(iii)
η ≤ ζ iff η (α) ≤ ζ (α) , ∀α ∈ I1 .
(iv)
Proposition 1.13 Morsi & Yehia [4].
(i) For (η, ζ) ∈ R∗ (I), η 2 < ζ 2 iff η < ζ.
(ii) For every η ∈ R∗ (I), there exists a unique square root ξ in R∗ (I) such that
ξ 2 = η.
(iii) For (η, ζ) ∈ M (I) we have (η, ζ)2 ∈ M ∗ (I).
(iv) For (η, ζ) ∈ M (I) we have |b(η, ζ)2 c| = |b(η, ζ)c|2 .
2. Fuzzy Semi-Inner-Product Space
In this section, we will define the fuzzy semi-inner-product and establish some
properties that goes with it. First we introduce the notation of the α-cuts of the
fuzzy real numbers to M (I).
Definition 2.1. Let (η, ζ) ∈ M (I) and α ∈ I1 . We define the α-cut of (η, ζ) to be
the real number
(η, ζ)(α) = η (α) − ζ (α) .
Proposition 2.2 El-abyad & El-Hamouly [7].
(i) The α-cut (η, ζ)(α) is well-defined on M (I).
(ii) (η, ζ) = (ξ, λ) in M (I) iff they have the same indexed family of α-cuts,
(iii) (η, ζ) ∈ M ∗ (I) iff ∀α ∈ I1 , (η, ζ)(α) ≥ 0.
Proposition 2.3 El-abyad & El-Hamouly [7]. For each fixed α ∈ I 1 , taking
α-cuts is an order preserving real algebra homomorphism from M (I) onto R.
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EUI-WHAN CHO, YOUNG-KEY KIM AND CHAE-SEOB SHIN
Definition 2.4. A fuzzy semi-inner-product on a unitary M (I)-modulo X is a
function · : X × X → M (I) which satisfies the following three axioms;
(F1 ) · is linear in one component only.
(F2 ) x · x > 0̄ for every nonzero x ∈ X.
(F3 ) |bx · yc|2 ≤ (x · x)(y · y) for every x, y ∈ X
The pair (X, ·) is called a fuzzy semi-inner-product space. The fuzzy pseudo-norm
k k associated with (X, ·) is the function k k : X → R∗ (I) defined for all x ∈ X by
1
kxk = |bx · xc| 2 with values in R∗ (I). We write (X, ·, k k) to show that the norm k
k is the function thus derived from the fuzzy semi-inner-product.
Let us new define the real quadratic form <, >α : X × X → R for every x, y ∈
X,α ∈ I1 fixed, by < x, y >α = (x · y)(α) .
Lemma 2.5. If for α ∈ I1 and x ∈ X, (x · x)(α) = 0, then (x, y)(α) = 0 for all
y ∈ X.
Proof. By (F3 ), we obtain | < x, y >α |2 ≤ < x, x >α < y, y >α . This yields
((x · y)(α) )2 ≤ (x · x)(α) (y · y)(α) . If (x · x)(α) = 0 for some α ∈ I1 , then (x · y)(α) = 0
for all y ∈ X.
Proposition 2.6. If (η, ζ) ≥ (ξ, λ) in M ∗ (I), then |b(ηζ)c| ≥ |b(ξ, λ)c| in R∗ (I).
Proof. Since (η, ζ) ∈ M ∗ (I), then θ ≥ (ζ, η) for all θ ∈ R∗ (I). From the properties
of the infimum and Definition 1.8, we have
|b(ηζ)c| = inf {θ ∈ R∗ (I) : θ ≥ (η, ζ)}
≥ inf{θ ∈ R∗ (I) : θ ≥ (ξ, λ)}
= |b(ξ, λ)c|.
In the following proposition, we will denote the elements of the ring M (I) by
just a single letter.
Proposition 2.7. Let x, y ∈ M ∗ (I) be such that y (α) = 0 whenever α ∈ I1 satisfies
x(α) = 0. Then for all z ∈ M ∗ (I) we have xz ≥ xy iff z ≥ y.
FUZZY SEMI-INNER-PRODUCT SPACE
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proof. xz ≥ xy iff (xz)(α) ≥ (x, y)(α) , ∀α ∈ I1 . Using the properties of α-cuts on
M (I), we have x(α) , y (α) ,z (α) ≥ 0, and x(α) z (α) ≥ x(α) y (α) . If x(α) 6= 0, then
z (α) ≥ y (α) . If x(α) = 0, then y (α) = 0, and hence z (α) ≥ y (α) . Since this holds for
all α ∈ I1 , then we conclude thar z ≥ y.
Theorem 2.8. Let (x, ·k k) be a fuzzy semi-inner-product space. Then, considering
X as a real vector space, k k is indeed a fuzzy pseudo-norm on X. It also satisfies :
(i) kxk > 0̄ for every nonzero x ∈ X, and
(ii) k(η, ζ)xk ≤ |b(η, ζ)c| |xk for all x ∈ X and (η, ζ) ∈ M (I).
Proof. ktxk2 = |btx · txc| = t|bx · txc| ≤ |t| |xk |txk. Thus ktxk ≤ |t| |xk. For λ 6= 0,
kxk = k 1t txk ≤
1
|t| ktxk,
|t| |xk ≤ ktxk. Therefore ktxk = |t| |xk,
∀x ∈ X, t ∈ R. Also,
kx + yk2 = |b(x + y) · (x + y)c|
= |bx · (x + y)c| + |by · (x + y)c|
≤ kxkkx + yk + kykkx + yk
= (kxk + kyk)kx + yk.
Then kx + yk ≤ kxk + kyk, ∀x, y ∈ X
Hence k k is a fuzzy pseudo-norm on X.
To prove (i), let x ∈ X be a nonzero element in X. Then x · x > 0̄ in M (I) and
1
hence kxk = |bx · xc| 2 > 0̄ in R∗ (I).
To prove (ii), let (η, ζ) ∈ M (I), then
k(η, ζ)xk2 = |b(η, ζ)x · (η, ζ)xc|
= |b(η, ζ)2 (x, x)c|.
But, since (M (I), |b c|) is a fuzzy pseudo-normed algebra(Theorem 1.11(v)),
then
|b(η, ζ)2 (x · x)c| ≤ |b(η, ζ)2 c||b(x · x)c|
= |b(η, ζ)c|2 |b(x · x)c|
= |b(η, ζ)c|2 kxk2 ,
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EUI-WHAN CHO, YOUNG-KEY KIM AND CHAE-SEOB SHIN
where the first equality is true by Proposition 1.13 (iv).
Due to the fact that the square roots exist and are unique in R∗ (I), we obtain
by Proposition 1.13 (i),
|b(η, ζ)xc| ≤ |b(η, ζ)c| |xk.
Proposition 2.9. Let {(Xi , ·i , k ki ) : i = 1, 2, · · ·, n} be a finite indexed family
Q
of fuzzy semi-inner-product spaces, and let X =
Xi = {(x1 , x2 , · · ·, xn ) : x1 ∈
X1 , x2 ∈ X2 , · · ·, xn ∈ Xn } be the product module of the Xi ’s(under coordinate-wise
operations). Define the function · : X × X → M (I) by for x = (xi ) and y = (yi ) in
X,
x·y =
n
M
xi · y i .
i=1
Then this function · is a fuzzy semi-inner-product on X.
Proof. The proof is straightforward from the properties of each fuzzy semi-innerproduct ·i and the properties of fuzzy summation.
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(Eui-Whan Cho) Department of Mathematics, MyongJi University, Samga-dong
470,Yong-In, Gyeonggi-do 449-728, Korea
(Young-Key Kim) Department of Mathematics, MyongJi University, Samga-dong
470, Yong-In, Gyeonggi-do 449-728, Korea
(Chae-Seob Shin) Department of Mathematics, MyongJi University, Samga-dong
470, Yong-In, Gyeonggi-do 449-728, Korea