A Characterization of the Ambiguity and Fuzziness by Means of Activity Orders on the Closed Interval in [0, 1].

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A characterization of the ambiguity and fuzziness by means of
activity orders on the closed intervals in [0,1]
C. Alcalde
Escuela Universitaria Polit´ ecnica.
Dep. de Matem´ atica Aplicada.
Universidad del Pa´ ıs Vasco.
Plaza de Europa 1,
20018 San Sebasti´ an, Spain.
c.alcalde@ehu.es
A. Burusco
Dep. Autom´ atica y Computaci´ on.
Universidad P´ ublica de Navarra,
Campus de Arrosad´ ıa,
31006 Pamplona, Spain.
burusco@si.unavarra.es
R. Fuentes-Gonz´ alez
Dep. Autom´ atica y Computaci´ on.
Universidad P´ ublica de Navarra,
Campus de Arrosad´ ıa,
31006 Pamplona, Spain.
rfuentes@si.unavarra.es
Abstract
We take as a departure point the definition
of activity orders, intending to measure the
ambiguity and fuzziness of any closed inter-
val in [0,1]. To do it, we define a family of
orders on the set of closed intervals of a dis-
tributive lattice L, that will allow us to set
up some preorders associated to the ambigu-
ity and fuzziness.
Keywords: activity orders, fuzziness and
ambiguity.
1Introduction. Activity orders
If J[L] represents the set of closed intervals in a dis-
tributive lattice L, then some order relations on J[L]
are defined to set up preorders associated to the fuzzi-
ness and ambiguity.
To do it, we will start from the concept of activity
order [10, 9, 11] that was introduced by Serra [10], in
the context of the morphologic operators on images of
Rnor Zn, as follows.
1.1 Definition
Let Ω = {Ψ/Ψ : ℘(R2) → ℘(R2)} be a set of functions
that transforms the images A of the plane R2with the
order (Ψ1 ≤ Ψ2) ⇐⇒ (Ψ1(A) ⊆ Ψ2(A), ∀A ∈ R2).
Let i ∈ Ω be the identity operator: i(A) = A, ∀A ∈
R2. The activity order between operators Ψ1,Ψ2 is
defined by:


Later, Heijmans and Keshet [8] extended this defini-
tion to any lattice, generating a family of orders that
are shown in the next:
Ψ1? Ψ2⇐⇒

(i ∧ Ψ1) ≥ (i ∧ Ψ2)
&
(i ∨ Ψ1) ≤ (i ∨ Ψ2)
1.2 Definition
Let (L,≤) be a lattice and ω0∈ L. The binary relation
?ω0is defined on L by the expression:


where ∨ and ∧ represent the joint and meet in L re-
spectively.
(x ?ω0y) ⇐⇒

(ω0∧ y) ≤ (ω0∧ x)
&
(ω0∨ y) ≥ (ω0∨ x)
(1)
1.3Proposition
The relation ?ω0is reflexive and transitive for each
ω0∈ L, that is:
(α ?ω0α), ∀α ∈ L
((α ?ω0β)&(β ?ω0γ)) =⇒ (α ?ω0γ).
It is not necessarily antisymmetric, since the lattice
in Figure 1 verifies: α ?ω0β , β ?ω0α and α ?= β.
Consequently, the binary relation ?ω0, in general, is
not an order relation.
An immediate consequence of the studied theory in [8]
is the following result that tells us when the previous
relation is an order.
1.4Theorem
If the ordered set (L,≤) is a distributive lattice
(L,∨,∧), it is verified:
(i) For all ω0∈ L, the relation ?ω0defined in (1) is
an order relation on L.
(ii) The partially ordered set (L,?ω0) associated to
ω0is a meet-semilattice (L,?ω0,0ω0) with a min-
imum element 0ω0= ω0, and where the meet op-
eration ?ω0is given, for all (α,β) ∈ L2, by

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r
r
rrr
J
J
J
J
J
J
J
J
?
J
J
J
J
J
J
J
J
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
ω0∨ α = ω0∨ β
ω0∧ α = ω0∧ β
αω0
β
Figure 1: ?ω0is not an order
(α ?ω0β) = ξ, where
ξ = [ω0∧(α∨β)]∨(α∧β) = [ω0∨(α∧β)]∧(α∨β).
Hence, we will work with the distributive lattices and
with the following family of orders.
1.5Definition
If (L,≤) is a distributive lattice and ω0∈ L, then the
order relation ?ω0defined in (1), is said to be the
ω0-activity order.
1.6Example
In Figure 2, we represent some examples of the family
of orders (?ω)ω∈[0,1]obtained in the chain ([0,1],≤).
r
?
r
r
r
?0=≤
0
α
β
1
r ?
?α
?
?
?
?
?
?
?
?
?
?
?
?
??
r
r
E
E
E
E
E
E
EE
r
0
α
β
1
r ?
?β
?
?
?
?
?
??
r
r
E
E
E
E
E
E
E
E
E
E
E
E
E
E
EE
r
1
β
α
0
r
r
r
r
?1=≥
1
β
α
0
Figure 2: Orders of the family (?ω)ω∈[0,1]
Note that ?op
ω0, the dual order of ?ω0, is given by:


Therefore, (L,?op
(α?op
(α ?op
ω0β) ⇐⇒

(ω0∧ α) ≤ (ω0∧ β)
&
(ω0∨ α) ≥ (ω0∨ β)
ω0) is a joint-semilattice where
ω0β) = [ω0∧(α∨β)]∨(α∧β) = [ω0∨(α∧β)]∧(α∨β),
and ω0is the maximum element of (L,?op
ω0).
Note that, for every distributive lattice (L,≤) with a
minimum element 0, it is verified: (?0) = (≤). If
there is a maximum element 1, then the relation ?1is
≥, (opposite to ≤).
Let us see a particular case of this order relation that
has already been defined.
1.7Proposition
The ”sharpened” order ≤s[5, 13] used in fuzziness
degree measures [2] and in the theory of fuzzy entropies
[12], is a particular case of the orders ?ω. Specifically,
for L = [0,1], ω =1
2:
(α ≤sβ) ⇐⇒ (α ?op


1/2β) ⇐⇒
2,β)

min(1
2,α) ≤ min(1
&
max(1
2,α) ≥ max(1
2,β)
If we consider now an interval [ω0] represented by
[ω0] = [ω0,ω0] ∈ J[L], with ω0 ≤ ω0, we can char-
acterize the family of orders that we have defined in
the sets of intervals (J[L],≤).
1.8 Proposition
If (L,≤) is a distributive lattice and (J[L],≤) is the
distributive lattice of the closed intervals in L with the
order
([α,α] ≤ [β,β]) ⇐⇒ (α ≤ β)&(α ≤ β),
then
([α,α] ?[ω0][β,β]) ⇐⇒ (α ?ω0β)&(α ?ω0β)
Proof Given the interval [ω0] = [ω0,ω0], the ω0-
activity order associated to the element ω0is


(α ?ω0β) ⇐⇒

(ω0∧ α) ≥ (ω0∧ β)
&
(ω0∨ α) ≤ (ω0∨ β)

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On the other hand, if we take the order relation defined
on the set of intervals
([α,α] ?ω0[β,β]) ⇐⇒
([ω0,ω0] ∧ [α,α]) ≥ ([ω0,ω0] ∧ [β,β])
&
([ω0,ω0] ∨ [α,α]) ≤ ([ω0,ω0] ∨ [β,β])
Then, it follows that
⇐⇒



([α,α] ?ω0[β,β]) ⇐⇒
[ω0∧ α,ω0∧ α] ≥ [ω0∧ β,ω0∧ β]
&
[ω0∨ α,ω0∨ α] ≤ [ω0∨ β,ω0∨ β]




⇐⇒













⇐⇒















ω0∧ α ≥ ω0∧ β
&
ω0∧ α ≥ ω0∧ β
&
ω0∨ α ≤ ω0∨ β
&
ω0∨ α ≤ ω0∨ β
⇐⇒





α ?ω0β
&
α ?ω0β
?
1.9 Example
If L = [0,1], then we obtain:
([α] ?[0,1][β]) ⇐⇒ (α ?0β)&(α ?1β),
that is
([α] ?[0,1][β]) ⇐⇒ (α ≤ β)&(α ≥ β)
It proves that ?[0,1]is the inclusion order ⊇ between
intervals:
([α] ?[0,1][β]) ⇐⇒ ([α] ⊇ [β]).
Notation. Given a distributive lattice (L,≤), for ev-
ery element ω0∈ L, the order relation dual of ?ω0will
be denoted by ≤ω0. That is, ≤ω0=?op
Therefore, we will have that:


As we said before, the set L with this order relation is
a joint-semilattice where the joint of any two elements
is given by
ω0.
α ≤ω0β ⇐⇒



ω0∧ α ≤ ω0∧ β
&
ω0∨ α ≥ ω0∨ β
α ∨ω0β = {ω0∧ (α ∨ β)} ∨ (α ∧ β)
and where ω0is the maximum element.
1.10Example
In particular, if L = [0,1] and (J[L],≤) is the dis-
tributive lattice of the closed intervals in L with the
order
[a,b] ≤ [c,d]
then


⇐⇒ [a,b] ≤ [c,d]
that is, the order ≤[1,1]is the usual order between in-
tervals ≤.
⇐⇒
a ≤ c
&
b ≤ d
[a,b] ≤[1,1][c,d] ⇐⇒



[a,b] ∧ [1,1] ≤ [c,d] ∧ [1,1]
&
[a,b] ∨ [1,1] ≥ [c,d] ∨ [1,1]
2Preorder relation associated to the
fuzziness of the intervals
In the Fuzzy sets theory, diverse magnitudes have been
defined in order to measure the fuzziness [6, 7]. Similar
measures have also been considered for interval-valued
fuzzy sets [3].
From a different point of view, using the order relations
≤ω0, we will introduce in this section some preorder
relations on the set of closed intervals in L = [0,1],
that later we will allow us to define other measures of
fuzziness and ambiguity on the interval-valued fuzzy
sets.
In order to set up comparisons among the fuzziness of
the elements of the set of closed intervals in L = [0,1]
we will remind the relation ’at least as fuzzy as’ defined
by De Luca and Termini in [6].
2.1 Definition
Let (L,≤) be a distributive lattice endowed with a
negation?. Given two elements x,y ∈ L, y is said to
be at least as fuzzy as x, and we will represent by


The intuitive idea of this definition is that y and y?are
closer in L than x and x?.
x ≺fuzy ⇐⇒



x ∧ x?≤ y ∧ y?
&
x ∨ x?≥ y ∨ y?
2.2 Proposition
The relation ≺fuzis a preorder relation on L.
Proof The relation ≺fuzis reflexive:
x ∧ x?≤ x ∧ x?
&
x ∨ x?≥ x ∨ x?





⇐⇒ x ≺fuzx, ∀x ∈ L

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It is also transitive:
x ≺fuzy ⇒ x ∧ x?≤ y ∧ y?& x ∨ x?≥ y ∨ y?
y ≺fuzz ⇒ y ∧ y?≤ z ∧ z?& y ∨ y?≥ z ∨ z?





=⇒
x ≺fuzz
Therefore, the relation ≺fuzis a preorder relation.
Remark. The previous relation is not an order rela-
tion because it is not an antisymetrical relation:
?
x ≺fuzy ⇒ x ∧ x?≤ y ∧ y?& x ∨ x?≥ y ∨ y?
y ≺fuzx ⇒ y ∧ y?≤ x ∧ x?& y ∨ y?≥ x ∨ x?





=⇒
x ∧ x?= y ∧ y?
&
x ∨ x?= y ∨ y?





and this does not imply that, necessarily, x = y.
2.3Proposition
If L = [0,1], it is proved in [6] that
x ≺fuzy ⇐⇒ x≤sy or x≤sy?
where ≤sis the sharpened order relation used in the
Proposition 1.7.
Remark. It is immediate to prove that
1. The element 0.5 is the maximum element for this
relation.
x ≺fuz0.5, ∀x ∈ L
2. The elements 0 and 1 are the minimal elements.
∀x ∈ L,
0 ≺fuzx, 1 ≺fuzx
3. The following equivalences are verified
x ≺fuz y ⇐⇒ x ≺fuz y?⇐⇒ x?≺fuz y?⇐⇒
x?≺fuzy
In order to characterize the relation ’to be at least as
fuzzy as’ in the set of closed intervals in L = [0,1], we
define on J[L] the following order relation.
2.4 Definition
Given two intervals [a,b],[c,d] ∈ J[L],
[a,b] ≤[0.5,0.5][c,d] ⇐⇒


⇐⇒

⇐⇒



[a,b] ∧ [0.5,0.5] ≤ [c,d] ∧ [0.5,0.5]
&
[a,b] ∨ [0.5,0.5] ≥ [c,d] ∨ [0.5,0.5]
[a,b] ≤ [c,d]
a ≤ c & b ≥ d
[a,b] ≥ [c,d]




if[a,b],[c,d] ≤ [0.5,0.5]
a,c ≤ 0.5 & b,d ≥ 0.5
[a,b],[c,d] ≥ [0.5,0.5]
if
if
2.5Proposition
The order relation ≤[0.5,0.5]is an extension to the set
of closed intervals in L of the ’sharpened order’ ≤s
defined on L.
Proof The order relation ≤[0.5,0.5]is the dual order of
the ω0-activity order ?[0.5,0.5]defined on set of inter-
vals J[L] as
[a,b] ?[0.5,0.5][c,d] ⇐⇒


As we proved in the Proposition 1.8, the ω0-activity
order between intervals ?[0.5,0.5]is an extension of the
order ?0.5
([a,b] ?[0.5,0.5][c,d]) ⇐⇒ (a ?0.5c)&(b ?0.5d)
Considering the dual orders and taking into account
that ?0.5is the dual order of the ’sharpened order’ ≤s
defined on L = [0,1], we obtain:



[a,b] ∧ [0.5,0.5] ≥ [c,d] ∧ [0.5,0.5]
&
[a,b] ∨ [0.5,0.5] ≤ [c,d] ∨ [0.5,0.5]
([a,b] ≤[0.5,0.5][c,d]) ⇐⇒ (a≤sc)&(b≤sd)
That is, the order relation ≤[0.5,0.5]is an extension to
the set of closed intervals in L of the ’sharpened order’
≤s.
?
2.6Example
In Figure 3 we have represented the semilattice
(J[C],≤[0.5,0,.5] ) of the closed intervals in the chain
C = {0,0.1,0.2,0.3,0.4,0.5,0.6,0.7, 0.8,0.9,1}.
The extension to the set of closed intervals in L of the
order relation ≤sallows us to extend the relation ’to
be at least as fuzzy as’ in the following way:

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? ?? ?? ?? ??
? ?? ?? ?? ??
? ? ? ?? ?? ? ?
? ?? ? ? ?? ??
? ? ? ?? ?? ??
Figure 3: Semilattice (J[C],≤[0.5,0.5])
2.7Definition
Given two intervals [a,b],[c,d] ∈ J[L], we will say that
the interval [c,d] is at least as fuzzy as the interval
[a,b], and we will denote it by:


that is:







Remarks.
[a,b] ≺fuz[c,d] ⇐⇒



[a,b] ≤[0.5,0.5][c,d]
or
[a,b] ≤[0.5,0.5][1 − d,1 − c]
[a,b] ≺fuz[c,d] ⇐⇒






































[a,b] ≤ [c,d] ≤ [0.5,0.5]
or
a ≤ c ≤ 0.5 & b ≥ d ≥ 0.5
or
[a,b] ≥ [c,d] ≥ [0.5,0.5]
or
[a,b] ≤ [1−d,1 − c] ≤ [0.5,0.5]
or
a ≤ 1 − d ≤0.5 & b ≥ 1 − c ≥ 0.5
or
[a,b] ≥ [1−d,1 − c] ≥ [0.5,0.5]
1. The interval [0.5,0.5] is at least as fuzzy as any
other element of the set J[L].
[a,b] ≺fuz[0.5,0.5], ∀[a,b] ∈ J[L]
Therefore, it will be the interval that represents
the greatest fuzziness.
2. The intervals [0,0], [0,1] and [1,1] are the min-
imal elements. However, it is not verified that
any element is at least as fuzzy as the minimal
elements, since they can be not comparable el-
ements.For example, [0,0] ⊀fuz [0,0.8] and
[0,0.8] ⊀fuz[0,0].
In next section we will try to do a similar study in
order to be able to compare the ambiguity of the closed
intervals in L.
3Preorder relation associated to the
ambiguity of the intervals
Intuitively, we associate the amplitude of the intervals
to the ambiguity. The interval [0,1] is the one that has
the biggest amplitude and so, following this idea, it will
be the one that would represent the greatest ambigu-
ity. For this reason, we will do a development similar
to the one done in the previous section to compare the
fuzziness between too intervals using an order relation
for which the interval [0,1] is the maximum element.
3.1Definition
Given two intervals [a,b],[c,d] ∈ J[L], the order rela-
tion ≤[0,1]is defined as:


⇐⇒

The order relation ≤[0,1]agrees with the inclusion ⊆
between intervals:
[a,b] ≤[0,1][c,d] ⇐⇒



[a,b] ∧ [0,1] ≤ [c,d] ∧ [0,1]
&
[a,b] ∨ [0,1] ≥ [c,d] ∨ [0,1]
[0,b] ≤ [0,d]
&
[a,1] ≥ [c,1]
⇐⇒




⇐⇒





b ≤ d
&
a ≥ c
[a,b] ≤[0,1][c,d] ⇐⇒ [a,b] ⊆ [c,d]
Remark. The order relation ≤[0,1]is the order rela-
tion dual of the ω0-activity order ?[0,1], which, as we
saw before, is the order of inclusion ⊇ between inter-
vals.
3.2Example
Let
vals in the chain C
0.6,0.7,0.8,0.9,1}, then the semilattice (J[C],≤[0,1])
can be represented by the graphic of Figure 4.
(J[C],≤)be thelattice
{0,0.1,0.2,0.3,0.4,0.5,
ofclosed inter-
=
Using this order relation defined on the set of in-
tervals, we will define the relation ’to be at least as
ambiguous as’.
3.3 Definition
Given two intervals [a,b],[c,d] ∈ J[L], we will say that
the interval [c,d] is at least as ambiguous as the inter-

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Figure 4: Semilattice (J[C],≤[0,1])
val [a,b] and we will denote it by:
[a,b] ≺amb[c,d] ⇐⇒





[a,b] ≤[0,1][c,d]
or
[a,b] ≤[0,1][1 − d,1 − c]
or what is the same:
[a,b] ≺amb[c,d] ⇐⇒





[a,b] ⊆[c,d]
or
[a,b] ⊆[1 − d,1 − c]
Remarks.
1. The interval [0,1] is at least as ambiguous as any
other element of the set J[L].
[a,b] ≺amb[0,1], ∀[a,b] ∈ J[L]
This interval represents the greatest ambiguity.
2. The intervals of the type [a,a] are the minimal el-
ements and, therefore, there are the less ambigu-
ous of the set of intervals. We will consider that
the intervals of the type [a,a] have the smallest
ambiguity.
3. As it happened with the previous relation, also
in this case we can find some elements that are
not comparable. For example, [0.2,0.3] ⊀amb
[0.5,0.7] and [0.5,0.7] ⊀amb[0.2,0.3].
4. It is immediate to prove that any pair of intervals
verifies that:
[a,b] ≺amb[c,d] ⇔ [a,b] ≺amb[1−d,1−c] ⇔ [1−
b,1−a] ≺amb[1−d,1−c] ⇔ [1−b,1−a] ≺amb[c,d]
4 Future work
We are interested in being able to compare the am-
biguity and the fuzziness of any pair of intervals and,
hence, of an interval-valued L-Fuzzy set in order to
work with the interval-valued L-Fuzzy contexts [4, 1].
However, that is not possible if we only use the pre-
order relations, as we remarked before. So, we will
try to define an ambiguity and fuzziness degree of a
closed interval in [0,1]. Also, in future works, we will
use these degrees in the interval-valued contexts area
[4, 1] in order to mesure the ambiguity and fuzziness
of an L-Fuzzy concept.
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