International Journal of Contemporary Mathematical Sciences
Vol. 11, 2016, no. 9, 425 - 436
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijcms.2016.6845
Fuzzy T -Ideals in BP-Algebras
Y. Christopher Jefferson
Dept. of Mathematics, Spicer Memorial College
Pune - 411 007, Maharashtra, India
M. Chandramouleeswaran
Dept. of Mathematics, S.B.K. College
Aruppukottai - 626101, Tamilnadu, India
c 2016 Y. Christopher Jefferson and M. Chandramouleeswaran. This article
Copyright is distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
Motivated by the study of fuzzy T −ideals in several algebras, that
arise from the propositional calculi, in this paper, we define the notion
of Fuzzy T-ideals and L-fuzzy T-ideals in BP-aglebras and discuss their
properties.
Mathematics Subject Classification: 03E72, 06F35, 03G25
Keywords: BP Algebras, L-BP Ideals, L-Fuzzy BP algebras, L-Fuzzy BP
Ideals
1
Introduction
Y.Imai and K.Iseki introduced two new classes of abstract algebra, BCK algebras and BCI algebras ( [8], [9], [10]). In 2002,J.Neggers and H.S. Kim [11]
introduced the notion of β−algebras. In [15], the authors introduced a new
class of algebras: TM-algebras and claimed that the class of TM-algebras is a
generalization of the classes of BCK and BCI algebras. However, in [2], the
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Y. Christopher Jefferson and M. Chandramouleeswaran
authors proved that the class of TM-algebras is not a generalization of the
class of BCK and BCI algebras, by giving several counter examples. In 2012
Sun Shin Ahn and Jeong Soon Han introduced the notion of BP-Algebras [14].
Lofti A. Zadeh [18] introduced the theory of fuzzy sets. Gogun extended
the notion of fuzzy sets to the notion of L-fuzzy sets [7]. The study of fuzzy
algebraic structures was started with the introduction of the concept of fuzzy
subgroups in 1977, by Rosenfeld [13]. O.G. Xi [17] applied the concept of fuzzy
sets to BCK algebras and got some results in 1991. In [?], we studied the fuzzy
subalgebras of a BP-algebras. In [16], the notion of fuzzy T −ideals in TMalgebras is discussed. In [1] and [12] Fuzzy T −ideals and L−fuzzy T −ideals
in β−algebras were discussed. Motivated by these works, in this paper, we
introduce the concept of Fuzzy T-ideals and L-fuzzy T-ideals in BP-Algebras
and discuss some of their properties.
2
Preliminares
In this section we recall some basic definitions that are needed for our work.
Definition 2.1 [14] A BP algebra (X,∗, 0) is a non-empty set X with a constant 0 and a binary operation ∗ satisfying the following conditions: for all x,
y, z ∈X,
1. x∗x = 0
2. x∗(x∗y) = y
3. (x∗z)∗(y∗z) = x∗y
Definition 2.2 [14] Let S be a non-empty subset of a BP-algebra X. Then S
is called a BP- sub algebra of X if x∗y ∈ S, for all x, y ∈ X.
Definition 2.3 [14] Let (X, ∗, 0) be a BP-Algebra. A non empty subset I of
X is called an ideal of X if it satisfies the following conditions:
1. 0 ∈ I
2. x∗y ∈ I and y ∈ I implies x ∈ I, ∀ x, y ∈ X.
Definition 2.4 An ideal I of a BP-ideal of X is said to be closed if 0 ∗ x ∈ I,
∀ x ∈ I.
Definition 2.5 Let (X,∗, 0) be a BP-algebra. A non-empty subset I of X is
called a T-ideal of X if it satisfies the following conditions
Fuzzy T -ideals in BP-algebras
427
1. 0 ∈ I
2. (x ∗ y) ∗ z ∈ I and y ∈ I implies x ∗ z ∈ I, ∀x, y, z ∈ X.
Definition 2.6 [3] A fuzzy sub set µ of a BP-algebra (X,∗ , 0) is called a fuzzy
BP sub algebra of X if, for all x,y∈X the following condition is satisfied
µ(x ∗ y) ≥ min{µ(x), µ(y)}, ∀x, y ∈ X.
Definition 2.7 [5] A fuzzy sub set µ of a BP-algebra (X,∗ , 0) is called a Lfuzzy BP sub algebra of X if, for all x,y∈X the following condition is satisfied
µ(x ∗ y) ≥ {µ(x) ∧ µ(y)}, ∀x, y ∈ X.
3
Fuzzy T-ideals in BP-Algebras
In this section, we introduce the notion of fuzzy T-ideals and prove some simple
results.
Definition 3.1 [4] A fuzzy subset µ in a BP-algebra X is called a fuzzy ideal
of X if it satisfies the following conditions:
1. µ(0) ≥ µ(x)
2. µ(x) ≥ min { µ(x∗y), µ(y) } ∀ x, y ∈ X.
Definition 3.2 A fuzzy subset µ in a BP-algebra X is called a fuzzy T-ideal
of X if it satisfies the following conditions:
1. µ(0) ≥ µ(x)
2. µ(x∗ z) ≥ min { (µ(x∗y) ∗z), µ(y) }, ∀ x, y, z ∈ X.
Theorem 3.3 A fuzzy set µ of a BP-algebra X is a fuzzy subalgebra if and
only if for every t ∈ [0,1], µt is either empty or a subalgebra of X.
Proof: Assume that µ is a fuzzy subalgebra of X and µt 6=φ. Then for any x,
y ∈ µt , we have,
µ(x∗y) ≥ min { µ(x), µ(y) }≥ t.
Therefore x∗y ∈ µt .
Hence µt is a subalgebra of X.
Conversely, Let µt be a subalgebra of X.
Let x, y ∈ X. Take t = min{ µ(x), µ(y) }
Then by assumption µt is a subalgebra of X implies x∗y ∈ µt
Therefore µ(x∗y) ≥ min { µ(x), µ(y) }≥ t.
Hence µ is a subalgebra of X.
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Y. Christopher Jefferson and M. Chandramouleeswaran
Theorem 3.4 Any subalgebra of a BP-algebra X can be realized as a level
subalgebra of some fuzzy subnalgrbra of X.
Proof: Let µ be a subalgebra of a given BP-algebra X and let µ be a fuzzy
set in X defined by
t if x ∈ A
µ(x) =
0 t∈
/A
where t ∈ (0,1) is fixed. It is clear that µt = A.
Let x, y ∈ X. If x, y ∈ A then x∗y ∈A also,
Hence µ(x) = µ(y) = µ(x∗y) = t and µ(x∗y) ≥ min { µ(x), µ(y) },
If x, y A then µ(x) = µ(y) = 0 implies µ(x∗y) ≥ min { µ(x), µ(y) } = 0.
If at most one of x and y belongs to A, then at least one of µ(x) and µ(y) is
equal to 0.
Therefore, min { µ(x), µ(y) } = 0 so that µ(x∗y) ≥ 0.
Theorem 3.5 Two level subalgebras µs , µt (s < t) of a fuzzy subalgebra are
equal if and only if there is no x ∈ X such that s ≤ µ(x) < t.
Proof: Let µs = µt for some s < t. If there exists x ∈ X, such that s ≤ µ(x)
< t. If x ∈ µs , then µ(x) ≥ s and µ(x) ≥ t, since µ(x) does not lie between s
and t. Thus x ∈ µt , which implies µs µt .
Also µt µs .
Therefore µs = µt
Theorem 3.6 Every fuzzy T-ideal µ of a BP-algebra X is order reversing, that
is if x ≤ y then: µ(x) ≥ µ(y) ∀ x, y ∈ X.
Proof: Let x, y ∈ X such that x ≤ y.
Therefore x∗y = 0
Now,
µ(x) =
≥
=
=
=
µ(x ∗ 0)
min{(µ(x ∗ y) ∗ 0), µ(y)}
min{(µ(0 ∗ 0), µ(y)}
min{(µ(0), µ(y)}
µ(y)
Theorem 3.7 A fuzzy set µ in a BP-algebra X is a fuzzy T-ideal if and only
if it is a fuzzy ideal of X.
Proof: Let µ be a fuzzy T-ideal of X
µ(0) ≥ µ(x)
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Fuzzy T -ideals in BP-algebras
µ(x∗ z) ≥ min { (µ(x∗y) ∗z), µ(y) }, ∀ x, y, z ∈ X.
By putting z = 0 in (ii) we have µ(x) ≥ min { µ(x∗y) , µ(y) }
Hence µ is a fuzzy ideal of X.
Conversely, µ is a fuzzy ideal of X.
T hen, µ(x ∗ z) ≥ min{(µ(x ∗ z) ∗ y), µ(y)}, ∀x, y, z ∈ X.
= min{(µ(x ∗ y) ∗ z), µ(y)}, ∀x, y, z ∈ X, whichprovestheresult.
Theorem 3.8 Let µ be a fuzzy set in a BP-algebra X and let t ∈ Im(µ). Then
µ is a fuzzy T-ideal of X if and only if the level subset, µt = { x ∈ X / µ(x)
≥ t } is a T-ideal of X, which is called a level T-ideal of µ.
Proof: Assume that µ is a fuzzy T-ideal of X.
Clearly, 0 ∈ µt
Let (x∗y)∗z ∈ µt and y ∈ µt
T hen µ((x ∗ y) ∗ z) ≥ t and µ(y) ≥ t.
N ow, µ(x ∗ z) ≥ min{(µ(x ∗ y) ∗ z), µ(y)}
≥ {t, t} = t
Hence µt is T-ideal of X.
Conversely, Let µt be T-ideal of X for any t ∈[0,1]. Suppose, assume that there
exist some xo ∈ X such that µ(0) < µ(xo ).
T ake s =
1
[µ(0) + µ(xo )]
2
⇒ s¡ µ (xo ) and 0 ≤ µ(0) < s < 1.
⇒ xo ∈ µs and 0 ∈
/ µs , which is a contradiction, since µs is a T-ideal of X.
Therefore, µ(0) ≥ µ(x) ∀ x ∈ X.
If possible, assume that xo , yo , zo ∈ X such that
µ(xo ∗ zo ) ≥ min { (µ(xo ∗yo ) ∗zo ), µ(yo ) }
1
[µ(xo ∗ zo ) + min{(µ(xo ∗ yo ) ∗ zo ), µ(yo )}]
2
⇒ s > µ(xo ∗ zo )
and s < min{(µ(xo ∗ yo ) ∗ zo ), µ(yo )}
⇒ s > µ(xo ∗ zo ), s < {(µ(xo ∗ yo ) ∗ zo )ands < µ(yo )
T akes =
⇒ xo ∗ zo ∈
/ µs , which is a contradiction, since µs is a T-ideal of X.
Therefore, µ(x∗ z) ≥ min { (µ(x∗y) ∗z), µ(y) } ∀ x, y, z ∈ X.
Definition 3.9 Let X and Y be BP-algebras. A mapping f : X→Y is said to
be a homomorphism if it satisfies: f(x∗y) = f(x)∗ f(y), for all x, y∈ X .
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Y. Christopher Jefferson and M. Chandramouleeswaran
Definition 3.10 Let f: X → X be an endomorphism and µ a fuzzy set in X.
We define a new fuzzy set in X by µf in X by µf (x) = µ(f(x)), for all x in X.
Theorem 3.11 Let f be an endomorphism of a BP- algebra X. If µ is a fuzzy
T-ideal of X, then so is µf .
Proof:
µf (x) = µ(f (x)) ≤ µ(0)
Let x, y, z ∈ X. T hen µf (x ∗ z) = µ(f (x ∗ z))
= µ(f (x) ∗ f (z))
≥ min{µ((f (x) ∗ f (y)) ∗ f (z)), µ(f (y))}
= min{µ((f (x ∗ y)) ∗ f (z)), µ(f (y))}
= min{µ(f ((x ∗ y) ∗ z)), µ(f (y))}
= min{µf ((x ∗ y) ∗ z)), µf (y))}.
Hence µf is a fuzzy T-ideal of X.
4
Cartesian product of fuzzy t-ideals of bpalgebras
In this section, we introduce the notion of Cartesian product of two fuzzy
T-ideals and study its properties.
Definition 4.1 Let µ and λ be the fuzzy set in a set X. the Cartesian product
µ×λ: X×X →[0, 1] is defined by (µ×λ)(x,y) = min {µ(x),λ(y)} ∀x, y∈ X.
Theorem 4.2 If µ and λ are fuzzy T-ideal in a BP algebra of X, then µ × λ
is a fuzzy T-ideal in X × X.
Proof: For any (x, y) ∈ X × X, we have,
(µ × λ)(0, 0) = min{µ(0), λ(0)}
≥ min{µ(x), λ(y)}
= (µ × λ)(x, y)
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Fuzzy T -ideals in BP-algebras
Let (x1 , x2 ), (y1 , y2 ) and (z1 , z2 ) ∈ X×X.
(µ × λ)((x1 , x2 ) × (z1 , z2 ))
=
(µ × λ)((x1 ∗ z1 , x2 × z2 ))
= min{µ(x1 × z1 ), λ(x2 × z2 )}
≥ min{min{µ(x1 × y1 ) × z1 ), µ(y1 )}, min{λ(x2 × y2 ) × z2), λ(y2 )}}
= min{min{µ(x1 × y1 ) × z1 ), λ(x2 × y2 ) × z2 )}, min{µ(y1 ), λ(y2 )}}
= min{(µ × λ)((x1 × y1 ) × z1 ), (x2 × y2 ) × z2 )}, (µ × λ)(y1 , y2 )}
= min{(µ × λ)((x1 × y1 , x2 × y2 ) × (z1 , z2 )), (µ × λ)(y1 , y2 )}
= min{(µ × λ)((x1 , x1 ) × (y1 × y2 ) × (z1 , z2 )), (µ × λ)(y1 , y2 )}
Hence µ × λ is a fuzzy T-ideal in X × X.
Theorem 4.3 Let µ and λ be fuzzy sets in a BP-algebra X such that µ×λ is
a is a fuzzy T-ideal of fuzzy T-ideal of X×X. Then
1. Either µ(0) ≥ µ(x) or λ(0) ≥ λ(x) for all x ∈X
2. If µ(0)≥ µ (x) for all x ∈X, then either λ(0) ≥ µ(x) or λ(0) ≥ λ(x)
3. If λ(0) ≥ λ(x) for all x ∈X, then either µ(0) ≥ µ(x) or µ(0) ≥ λ(x)
4. Either µ or λ is a fuzzy T-ideal of X.
Proof: µ ×λ is a fuzzy p-ideal of X×X. Therefore
(µ × λ)(0, 0) ≥ (µ × λ)(x, y) f or all (x, y) ∈ X × X
And
(µ×λ)((x1 , x2 ), (z1 , z2 )) ≥ min(µ × λ)((x1 , x2 ) ∗ (z1 , z2 )) ∗ ((y1 , y2 ) ∗ (z1 , z2 )), (µ × λ)(y1 , y2 )
for all (x1 , x2 ), (y1 , y2 )and(z1 , z2 ) ∈ X × X.
Suppose that µ (0) < µ (x) and λ (0) < λ (y) for some x,y∈X.
T hen : (µ × λ)(x, y) = min{µ(x), λ(y)}
> min{µ(0), λ(0)}
= (µ × λ)(0, 0). which is a contradiction.
Therefore either µ(0) ≥ µ (x) or λ (0) ≥ λ (x) for all x∈ X.
Assume that there exist x, y ∈ X such that: λ(0) < µ(x) and λ(0) < λ(y).
Then
(µ × λ)(0, 0) =
=
(µ × λ)(x, y) =
=
min{µ(0), λ(0)}
λ(0) and hence
min{µ(x), λ(y) > λ(0)
(µ × λ)(0, 0), a contradiction.
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Y. Christopher Jefferson and M. Chandramouleeswaran
Hence if µ(0) ≥ µ(x) ∀x ∈ X, then either λ(0) ≥ µ(x) or λ(0) ≥ λ(x)
Similarly we can prove that if λ(0) ≥ λ(x) for all x∈ X,
then either µ (0) ≥ µ (x) or µ (0) ≥ λ(x).
First we prove that λis a fuzzy T-ideal of X.
Since ,by (i),either µ(0) ≥ µ(x) or λ(0) ≥ λ(x) for all x∈ X.
Assume that λ(0) ≥ λ(x) for all x∈ X .
It follows from (iii) that either µ (0) ≥ µ (x) or µ(0) ≥ λ(x).
If µ(0) ≥ λ(x) for any x∈ X, then:
λ(x) =
=
=
=
≥
=
=
=
=
min{µ(0), λ(x)}
(µ × λ)(0, x).
min{µ(0), λ(x)}
(µ × λ)(0, x)
min{(µ × λ)((0, x) ∗ (0, z)) ∗ ((0 ∗ y) ∗ (0, z)), (µ × λ)(0, y)}
min{(µ × λ)(((0 ∗ 0), (x ∗ z)) ∗ ((0 ∗ 0), (y ∗ z))), (µ × λ)(0, y)}
min{(µ × λ)(((0 ∗ 0) ∗ (0 ∗ 0)), ((x ∗ z) ∗ (y ∗ z))), (µ × λ)(0, y)}
min{(µ × λ)(0, ((x ∗ z) ∗ (y ∗ z))), (µ × λ)(0, y)}λ(x)
min{λ((x ∗ z) ∗ (y ∗ z)), λ(y)}.
Hence λis a fuzzy T-ideal of X.
Now we will prove that µis a fuzzy T-ideal of X.
Let µ (0) ≥ µ (x).
By (ii) either λ(0) ≥ µ (x) or λ(0) ≥ λ(x).
Assume that λ(0) ≥ µ (x),
T hen µ(x) =
=
=
=
≥
=
=
=
min{µ(x), λ(0)}
(µ × λ)(x, 0).µ(x)
min{µ(x), λ(0)}
(µ × λ)(x, 0)
min{(µ × λ)(((x, 0) ∗ (z, 0)) ∗ ((y ∗ 0) ∗ (z, 0))), (µλ)(y, 0)}
min{(µ × λ)(((x ∗ z), (0 ∗ 0)) ∗ ((y ∗ z), (0 ∗ 0))), (µλ)(y, 0)}
min{(µ × λ)(((x ∗ z) ∗ (y ∗ z)), ((0 ∗ 0) ∗ (0 ∗ 0))), (µλ)(y, 0)}
min{µ((x ∗ z) ∗ (y ∗ z)), µ(y)}
Hence µ is a fuzzy T-ideal of X.
5
L-Fuzzy T-ideals in BP-Algebras
Analogous to the notion of fuzzy T-ideals in section 3, in this section we study
L-fuzzy T-ideals.
Fuzzy T -ideals in BP-algebras
433
Definition 5.1 [6] A L-fuzzy subset µ in a BP-algebra X is called a L- fuzzy
ideal of X if it satisfies the following conditions:
1. µ(0) ≥ µ(x)
2. µ(x∗z) ≥ { µ(x∗y)∧ µ(y) } ∀ x, y ∈ X
Definition 5.2 A fuzzy subset µ in a BP-algebra X is called a L-fuzzy T-ideal
of X if it satisfies the following conditions:
1. µ(0) ≥ µ(x)
2. µ(x∗ z) ≥ { ((µ(x∗y) ∗z) ∧ µ(y) }, ∀ x, y, z ∈ X.
Theorem 5.3 Every L- fuzzy T-ideal µ of a BP-algebra X is order reversing
if x ≤ y then µ(x) ≥ µ(y) ∀ x, y ∈ X.
Proof: Let x, y ∈ X such that x ≤ y.
T heref ore x ∗ y = 0
N ow, µ(x) = µ(x ∗ 0)
≥ {(µ(x ∗ y) ∗ 0) ∧ µ(y)}
= {(µ(0 ∗ 0) ∧ µ(y)}
= {(µ(0) ∧ µ(y)}
= µ(y)
Theorem 5.4 A L-fuzzy set µ in a BP-algebra X is a L-fuzzy T-ideal if and
only if it is a L-fuzzy ideal of X.
Proof: Let µ be a fuzzy T-ideal of X
µ(0) ≥ µ(x)
µ(x ∗ z) ≥ {(µ(x ∗ y) ∗ z) ∧ µ(y)}, ∀x, y, z ∈ X.
By putting z = 0 in (ii) we have µ(x) ≥ { µ(x∗y) ∧ µ(y) }
Hence µ is a fuzzy ideal of X.
Conversely, µ is a fuzzy ideal of X.
T hen, µ(x ∗ z) ≥ {(µ(x ∗ z) ∗ y) ∧ µ(y)}, ∀x, y, z ∈ X.
= {(µ(x ∗ y) ∗ z) ∧ µ(y)}, ∀x, y, z ∈ X, which proves the result.
Theorem 5.5 Let µ be a fuzzy set in a BP-algebra X and let t ∈ Im(µ). Then
µ is a fuzzy T-ideal of X if and only if the level subset, µt = { x ∈ X / µ(x)
≥ t } is a T-ideal of X, which is called a level T-ideal of µ.
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Y. Christopher Jefferson and M. Chandramouleeswaran
Proof: Assume that µ is a fuzzy T-ideal of X.
Clearly, 0 ∈ µt
Let (x∗y)∗z ∈ µt and y ∈ µt
T hen µ((x ∗ y) ∗ z) ≥ t and µ(y) ≥ t.
N ow, µ(x ∗ z) ≥ {(µ(x ∗ y) ∗ z) ∧ µ(y)}
≥ {t, t}
= t
Hence µt is T-ideal of X.
Conversely, Let µt be T-ideal of X for any t ∈[0,1]. Suppose, assume that there
exist some xo ∈ X such that µ(0) < µ(xo ).
1
[µ(0) + µ(xo )]
2
⇒ s < µ(xo ) and 0 ≤ µ(0) < s < 1.
⇒ xo ∈ µs and 0 ∈
/ µs , which is a contradiction, since µs is a T − ideal of X.
T akes =
Therefore, µ(0) ≥ µ(x) ∀ x ∈ X.
If possible, assume that xo , yo , zo ∈ X such that
µ(xo ∗ zo ) ≥ {(µ(xo ∗ yo ) ∗ zo ) ∧ µ(yo )}
1
T akes =
[µ(xo ∗ zo ) + {(µ(xo ∗ yo ) ∗ zo ) ∧ µ(yo )}]
2
⇒ s > µ(xo ∗ zo ) and s < {(µ(xo ∗ yo ) ∗ zo ) ∧ µ(yo )}
⇒ s > µ(xo ∗ zo ), s < {(µ(xo ∗ yo ) ∗ zo ) and s < µ(yo )
⇒ xo ∗ zo ∈
/ µs , which is a contradiction, since µs is a T-ideal of X.
Therefore, µ(x∗ z) ≥ { (µ(x∗y) ∗z) ∧ µ(y) } ∀ x, y, z ∈ X.
Definition 5.6 Let µ and λ be the fuzzy set in a set X. the Cartesian product
µ×λ : X×X →[0, 1] is defined by (µ×λ)(x,y) = min { µ(x), λ (x)} ∀x, y ∈
X.
Theorem 5.7 If µ and λ are fuzzy T-ideal in a BP algebra of X, then µ × λ
is a fuzzy T-ideal in X × X.
Proof: For any (x, y) ∈ X × X, we have,
µ × λ(0, 0)) = {µ(0) ∧ λ(0)}
≥ {µ(x) ∧ λ(y)}
= (µ × λ)(x, y)
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Fuzzy T -ideals in BP-algebras
Let (x1 , x2 ), (y1 , y2 ) and (z1 , z2 ) ∈ X × X.
(µ × λ)((x1 , x2 ) ∗ (z1, z2)) =
=
≥
=
=
=
=
(µ × λ)((x1 ∗ z1 , x2 ∗ z2 ))
{µ(x1 ∗ z1 ) ∧ λ(x2 ∗ z2 )}
{{µ(x1 ∗ y1 ) ∗ z1 ) ∧ µ(y1 )} ∧ {λ(x2 ∗ y2 ) ∗ z2 ) ∧ λ(y2 )}}
{{µ(x1 ∗ y1 ) ∗ z1 ) ∧ λ(x2 ∗ y2 ) ∗ z2 )} ∧ {µ(y1 ) ∧ λ(y2 )}}
{(µ × λ)((x1 ∗ y1 ) ∗ z1 ) ∧ (x2 ∗ y2 ) ∗ z2 )} ∧ (µ × λ)(y1 , y2 )}
{(µ × λ)((x1 ∗ y1 , x2 ∗ y2 ) ∗ (z1 , z2 )) ∧ (µ × λ)(y1 , y2 )}
{(µ × λ)((x1 , x1 ) ∗ (y1 ∗ y2 ) ∗ (z1 , z2 )) ∧ (µ × λ)(y1 , y2 )}
Hence µ × λ is a fuzzy T-ideal in X × X.
Acknowledgements. The authors thank the referee for valuable suggestion to modify the paper.
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Received: August 28, 2016; Published: November 6, 2016