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In-Depth Information
Chapter 5
T-Preorders and T-Indistinguishabilities
5.1 Which Is the Aim of This Section?
To characterize the T-Preorder and the T-indistinguishability relations by means
of particular classes of them, and showing ways of constructing T-preorders and
T-indistinguishabilities.
Given a fuzzy relation on
X
,
μ
:
X
×
X
ₒ[
0
,
1
]
, with the three properties
1. Reflexivity,
μ(
x
,
x
)
=
1, for all
x
∈
X
2. Symmetry,
μ(
x
,
y
)
=
μ(
y
,
x
)
, for all
(
x
,
y
)
∈
X
×
X
3. T-transitivity,
T
(μ(
x
,
y
), μ(
y
,
z
))
≤
μ(
x
,
z
)
, for all
x
,
y
,
z
in
X
, and a continuous
t-norm T,
it can be named
•
T-Preorders, those
μ
verifying 1 and 3.
•
T-Indistinguishabilities, those
μ
verifying 1, 2, and 3.
•
Similarities, those
verifying 1 and 2.
A particular class of T-Preorders is given by the known operators (R-implications)
μ
J
T
(
a
,
b
)
=
Sup
{
z
∈[
0
,
1
];
T
(
z
,
a
)
≤
b
}
,
and the corresponding class of T-indistinguishabilities is given by the operators
E
T
(
a
,
b
)
=
Min
(
J
T
(
a
,
b
),
J
T
(
b
,
a
)),
J
T
),
with
J
T
(
shortly written
E
T
=
min
(
J
T
,
a
,
b
)
=
J
T
(
b
,
a
)
. It is obvious that
J
T
(
a
,
a
)
=
1, and that
J
T
(
a
,
b
)
=
J
T
(
b
,
a
),
as well as that
E
T
(
a
,
a
)
=
1
,
and
E
T
(
a
,
b
)
=
E
T
(
b
,
a
).
What it is not so obvious is
that relations
J
T
are T-Transitive:
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