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μ
R
is T-transitive, and
T
1
is a t-norm such that
Remark 4.3.3
If the fuzzy relation
T
1
μ
R
is also
T
1
-transitive, since
T
, then
T
1
(μ
R
(
x
,
y
), μ
R
(
y
,
z
))
T
(μ
R
(
x
,
y
), μ
R
(
y
,
z
))
μ
R
(
x
,
z
),
for all
x
,
y
,
z
.
4.4 The Concept of T-State
Given a fuzzy relation
μ
:
X
×
X
ₒ[
0
,
1
]
and a continuous t-norm
T
, a fuzzy set
˃
:
X
ₒ[
0
,
1
]
is a
T
-state of
μ
,if
T
(˃(
x
), μ(
x
,
y
))
˃(
y
),
∀
(
x
,
y
)
∈
X
×
X
.
All constant fuzzy sets
μ
k
(
x
)
=
k
for all
x
∈
X
, and
k
∈[
0
,
1
]
, are T-states of
any fuzzy relation
μ
:
X
×
X
ₒ[
0
,
1
]
:
T
(μ
k
(
x
), μ(
x
,
y
))
μ
k
(
x
)
=
k
=
μ
k
(
y
)
.
For example,
μ
0
=
μ
∅
and
μ
1
=
μ
X
, are always
T
-states. Hence, the set
T
(μ)
of
all
T
-states
˃
of
μ
is never empty. From now on, in general we will only refer to non
constant
T
-states
˃
.
μ
:
×
ₒ[
,
]
∈
Given a fuzzy relation
X
Y
0
1
, once
y
Y
is fixed, we can define
μ
y
:
×
ₒ[
,
]
the fuzzy set
X
X
0
1
, defined by,
μ
y
(
x
)
=
μ(
x
,
y
),
for all
x
∈
X
.
When
X
,
Y
are finite sets,
μ
y
is the y-column of the matrix
[
μ
]
.
If
μ
:
X
×
X
ₒ[
0
,
1
]
is a symmetric and
T
-transitive fuzzy relation, from
T
(μ(
x
,
y
), μ(
y
,
z
))
μ(
x
,
z
),
for all
x
,
y
,
z
in
X
,
follows
T
(μ(
y
,
x
), μ(
y
,
z
))
μ(
z
,
x
)
,or
T
(μ
x
(
y
), μ(
y
,
z
))
μ
x
(
z
),
that is,
μ
x
is a
T
-state of
μ
.
For example, with
X
={
x
1
,
x
2
,
x
3
}
and
μ
:
X
×
X
ₒ[
0
,
1
]
given by
⊛
⊞
11
/
82
/
8
⊝
⊠
[
μ
]=
1
/
813
/
8
2
/
83
/
81
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