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Migrative Type Functional Equations forTriangular Norms
Janos FODOR
Obuda University
FSTA 2012
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Acknowledgement
This talk is based on joint work and papers with
Imre J. Rudas,
Erich Peter Klement,
Radko Mesiar.
Their contribution is greatly acknowledged and appreciated.
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Outline
1 Introduction and background
2 Migrative t-normsMigrativity with respect to the minimumMigrativity with respect to strict t-normsMigrativity with respect to nilpotent t-normsMigrativity with respect to a continuous ordinal sum
3 Cross-migrative t-normsCross-migrativity with respect to the minimumCross-migrativity with respect to strict t-normsCross-migrativity with respect to nilpotent t-norms
4 Concluding remarks
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Introduction and background
INTRODUCTION AND BACKGROUND
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Introduction and background
Aim of this talk
To deliver results on (continuous) solutions of the following generalfunctional equation (x , y ∈ [0, 1], and α ∈ ]0, 1[ fixed)
T1(T2(α, x), y) = T3(x ,T4(α, y)),
where T1,T2,T3,T4 are triangular norms, in two particular cases:
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Introduction and background
Aim of this talk
To deliver results on (continuous) solutions of the following generalfunctional equation (x , y ∈ [0, 1], and α ∈ ]0, 1[ fixed)
T1(T2(α, x), y) = T3(x ,T4(α, y)),
where T1,T2,T3,T4 are triangular norms, in two particular cases:
T1(T2(α, x), y) = T1(x ,T2(α, y))
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Introduction and background
Aim of this talk
To deliver results on (continuous) solutions of the following generalfunctional equation (x , y ∈ [0, 1], and α ∈ ]0, 1[ fixed)
T1(T2(α, x), y) = T3(x ,T4(α, y)),
where T1,T2,T3,T4 are triangular norms, in two particular cases:
T1(T2(α, x), y) = T1(x ,T2(α, y))
T1(T2(α, x), y) = T2(x ,T1(α, y)).
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Introduction and background
Associativity
A function T : [0, 1]2 → [0, 1] is called associative if it satisfies
T (T (x , y), z) = T (x ,T (y , z)) for all x , y , z ∈ [0, 1].
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Introduction and background
Associativity
A function T : [0, 1]2 → [0, 1] is called associative if it satisfies
T (T (x , y), z) = T (x ,T (y , z)) for all x , y , z ∈ [0, 1].
An example is the product TP(x , y) = xy .
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Introduction and background
Associativity
A function T : [0, 1]2 → [0, 1] is called associative if it satisfies
T (T (x , y), z) = T (x ,T (y , z)) for all x , y , z ∈ [0, 1].
An example is the product TP(x , y) = xy .
Let us fix y = α ∈ ]0, 1[. Then we still have
TP(TP(x , α), z) = TP(x ,TP(α, z)) for all x , z ∈ [0, 1].
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Introduction and background
Associativity modified
Keep TP inside fixed, and consider a general T outside:
T (TP(x , α), z) = T (x ,TP(α, z)) for all x , z ∈ [0, 1].
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Introduction and background
Associativity modified
Keep TP inside fixed, and consider a general T outside:
T (TP(x , α), z) = T (x ,TP(α, z)) for all x , z ∈ [0, 1].
Keep TP outside fixed, and consider a general T inside:
TP(T (x , α), z) = TP(x ,T (α, z)) for all x , z ∈ [0, 1].
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Introduction and background
Associativity modified
Keep TP inside fixed, and consider a general T outside:
T (TP(x , α), z) = T (x ,TP(α, z)) for all x , z ∈ [0, 1].
Keep TP outside fixed, and consider a general T inside:
TP(T (x , α), z) = TP(x ,T (α, z)) for all x , z ∈ [0, 1].
Question 1: is there any solution T of the last equations that differsfrom TP?
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Introduction and background
Associativity modified
Keep TP inside fixed, and consider a general T outside:
T (TP(x , α), z) = T (x ,TP(α, z)) for all x , z ∈ [0, 1].
Keep TP outside fixed, and consider a general T inside:
TP(T (x , α), z) = TP(x ,T (α, z)) for all x , z ∈ [0, 1].
Question 1: is there any solution T of the last equations that differsfrom TP?
Question 2: what is the link between solutions of the two equations(if any)?
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Introduction and background
Example
Consider Tβ defined as follows:
Tβ(x , y) =
{
min(x , y) if max(x , y) = 1,βxy otherwise,
where β is an arbitrary number from [0, 1].
Tβ is a t-norm, and it satisfies Tβ(αx , y) = Tβ(x , αy) for(x , y) ∈ [0, 1[.
Notice the following particular cases:
if β = 0 then Tβ = TD the drastic t-norm;if β = 1 then Tβ = TP the product.
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Migrative t-norms
Migrative t-norms
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Migrative t-norms
Migrative t-norms
Definition
Let α ∈ ]0, 1[ and T1,T2 be t-norms. We say that the pair (T1,T2) isα-migrative (or, equivalently, that T1 is α-migrative with respect to T2, insymbols T1∼
αT2) if the following functional equation holds:
T1(T2(α, x), y) = T1(x ,T2(α, y)) for all (x , y) ∈ [0, 1]2.
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Migrative t-norms
Migrative t-norms
Definition
Let α ∈ ]0, 1[ and T1,T2 be t-norms. We say that the pair (T1,T2) isα-migrative (or, equivalently, that T1 is α-migrative with respect to T2, insymbols T1∼
αT2) if the following functional equation holds:
T1(T2(α, x), y) = T1(x ,T2(α, y)) for all (x , y) ∈ [0, 1]2.
Obviously, we have T1∼αT1 for any t-norm T and for each α ∈ ]0, 1[.
In other words: the relation ∼α
is reflexive on the set of all t-norms.
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Migrative t-norms
Equivalent forms of α-migrativity
Theorem
Let α be in ]0, 1[ and T1,T2 triangular norms. Then the following
statements are equivalent.
(i) (T1,T2) is α-migrative: T1(T2(α, x), y) = T1(x ,T2(α, y));
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Migrative t-norms
Equivalent forms of α-migrativity
Theorem
Let α be in ]0, 1[ and T1,T2 triangular norms. Then the following
statements are equivalent.
(i) (T1,T2) is α-migrative: T1(T2(α, x), y) = T1(x ,T2(α, y));
(ii) (T2,T1) is α-migrative: T2(T1(α, x), y) = T2(x ,T1(α, y));
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Migrative t-norms
Equivalent forms of α-migrativity
Theorem
Let α be in ]0, 1[ and T1,T2 triangular norms. Then the following
statements are equivalent.
(i) (T1,T2) is α-migrative: T1(T2(α, x), y) = T1(x ,T2(α, y));
(ii) (T2,T1) is α-migrative: T2(T1(α, x), y) = T2(x ,T1(α, y));
(iii) T1(α, x) = T2(α, x) for all x ∈ [0, 1].
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Migrative t-norms
Equivalent forms of α-migrativity
Theorem
Let α be in ]0, 1[ and T1,T2 triangular norms. Then the following
statements are equivalent.
(i) (T1,T2) is α-migrative: T1(T2(α, x), y) = T1(x ,T2(α, y));
(ii) (T2,T1) is α-migrative: T2(T1(α, x), y) = T2(x ,T1(α, y));
(iii) T1(α, x) = T2(α, x) for all x ∈ [0, 1].
Corollary
The relationα∼ is an equivalence relation on the set of all t-norms.
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Migrative t-norms
Further properties
If T1 ∼αT2 and ]a, b[ is a non-empty subinterval of [0, 1], and
α ∈ ]a, b[ then for the ordinal sums (〈a, b,T1〉) and (〈a, b,T2〉) wehave (〈a, b,T1〉)∼
γ(〈a, b,T2〉), where γ = α−a
b−a.
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Migrative t-norms
Further properties
If T1 ∼αT2 and ]a, b[ is a non-empty subinterval of [0, 1], and
α ∈ ]a, b[ then for the ordinal sums (〈a, b,T1〉) and (〈a, b,T2〉) wehave (〈a, b,T1〉)∼
γ(〈a, b,T2〉), where γ = α−a
b−a.
Recall that for each t-norm T and for each strictly increasing bijectionϕ : [0, 1] → [0, 1] the function Tϕ : [0, 1]2 → [0, 1] defined by
Tϕ(x , y) = ϕ−1(T (ϕ(x), ϕ(y)))
is also a t-norm.
Let ϕ : [0, 1] → [0, 1] be a strictly increasing bijection, α ∈ ]0, 1[ andT1,T2 be two t-norms. If (T1,T2) is α-migrative then ((T1)ϕ, (T2)ϕ)is ϕ(α)-migrative.
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Migrative t-norms
Cases considered
We study three particular cases of t-norms that are α-migrative withrespect to a fixed T0:
T0 = TM,
T0 = TP,
T0 = TL.
Using these results, as a fourth case we study α-migrativity withrespect to arbitrary continuous t-norms:
T0 = (〈ai , bi ,Ti〉)i∈Γ .
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Migrative t-norms Migrativity with respect to the minimum
Migrativity with respect to the minimum
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Migrative t-norms Migrativity with respect to the minimum
Migrativity with respect to TM
Characterization
T (min(α, x), y) = T (x ,min(α, y))
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Migrative t-norms Migrativity with respect to the minimum
Migrativity with respect to TM
Characterization
T (min(α, x), y) = T (x ,min(α, y))
Theorem
A t-norm T is α-migrative with respect to TM if and only if there exist
two t-norms T1 and T2 such that T can be written in the following form:
T (x , y) =
αT1
( x
α,y
α
)
if x , y ∈ [0, α],
α+ (1 − α)T2
(
x − α
1 − α,y − α
1 − α
)
if x , y ∈ [α, 1],
min(x , y) otherwise.
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Migrative t-norms Migrativity with respect to the minimum
Migrativity with respect to TM
Illustration
alpha
alpha0
1
1
min
minT1
T2
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to strict t-norms
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
T (αx , y) = T (x , αy)
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
T (αx , y) = T (x , αy)
Historically, this is the notion introduced originally by Durante andSarkoczi (2008).
Rooted in an open problem of the 2nd FSTA, see Mesiar and Novak(1996).
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, necessary conditions
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, necessary conditions
Theorem
If a continuous t-norm T is α-migrative with respect to TP then T is
strict.
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, necessary conditions
Theorem
If a continuous t-norm T is α-migrative with respect to TP then T is
strict.
If t denotes an additive generator of an α-migrative continuous t-norm T
then t satisfies the following functional equation for all x ∈ [0, 1]:
t(αx) = t(α) + t(x). (1)
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, characterization and construction
Theorem
Suppose t is an additive generator of a strict t-norm T and α is in ]0, 1[.Then the following statements are equivalent:
(i) T is α-migrative with respect to TP;
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, characterization and construction
Theorem
Suppose t is an additive generator of a strict t-norm T and α is in ]0, 1[.Then the following statements are equivalent:
(i) T is α-migrative with respect to TP;
(ii) t satisfies the functional equation t(αx) = t(α) + t(x);
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Migrative t-norms Migrativity with respect to strict t-norms
Migrativity with respect to TP
Continuous case, characterization and construction
Theorem
Suppose t is an additive generator of a strict t-norm T and α is in ]0, 1[.Then the following statements are equivalent:
(i) T is α-migrative with respect to TP;
(ii) t satisfies the functional equation t(αx) = t(α) + t(x);
(iii) there exists a continuous, strictly decreasing function t0 from [α, 1] tothe non-negative reals with t0(α) < +∞ and t0(1) = 0 such that
t(x) = k · t0(α) + t0
( x
αk
)
if x ∈]
αk+1, αk]
, (2)
where k is any non-negative integer.
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Migrative t-norms Migrativity with respect to strict t-norms
Constructing an additive generatorAn example
Let α =3
4and
t0(x) = 4 − 4x for x ∈
[
3
4, 1
]
.
Then t
(
(
3
4
)k)
= k , and linear in between.
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Migrative t-norms Migrativity with respect to strict t-norms
Constructing an additive generatorA graphical illustration
t(x) = k · t0(α) + t0
( x
αk
)
if x ∈]
αk+1, αk]
1
1
2
3
4
5
3/4(3/4)2(3/4)3(3/4)4
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to nilpotent t-norms
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to TL
Continuous case, necessary condition
T (max(α + x − 1, 0), y) = T (x ,max(α + y − 1, 0))
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to TL
Continuous case, necessary condition
T (max(α + x − 1, 0), y) = T (x ,max(α + y − 1, 0))
Theorem
Assume that T is a continuous t-norm that is α-migrative with respect to
TL. Then there exists an automorphism ϕ of the unit interval such that
T = (TL)ϕ. That is, we have
T (x , y) = (TL)ϕ (x , y) = ϕ−1(max(ϕ(x)+ϕ(y)−1, 0)) for all x , y ∈ [0, 1].
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to TL
Continuous case, characterization and construction
Theorem
Let α be in ]0, 1[ and n = max{k ∈ N | 1 − k(1 − α) > 0}.
A t-norm T (x , y) = ϕ−1(max(ϕ(x) + ϕ(y) − 1, 0)) (x , y ∈ [0, 1]) is
α-migrative with respect to TL if and only if there exist an automorphism
ψ0 of the unit interval with
ψ0
(
n − (n + 1)α
1 − α
)
=n− (n + 1)α
1 − α, (3)
such that
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to TL
Continuous case, characterization and construction
Theorem (cont’d)
ϕ(x) = 1 − k(1 − α) + (1 − α)ψ0
(
x − 1 + k(1 − α)
1 − α
)
(4)
if x ∈ ]1 − k(1 − α), 1 − (k − 1)(1 − α)] and k ≤ n,
and
ϕ(x) = 1 − (n + 1)(1 − α) + (1 − α)ψ0
(
x − 1 + (n + 1)(1 − α)
1 − α
)
(5)
if x ∈ [0, 1 − n(1 − α)].
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Migrative t-norms Migrativity with respect to nilpotent t-norms
Migrativity with respect to TL
Graphical construction
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Migrativity with respect to a continuous ordinalsum
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Migrativity with respect to a continuousordinal sum
We study continuous t-norms T that are α-migrative with respect toa fixed continuous t-norm T0;
TCo: the set of all continuous t-norms;
TAr: the set of all continuous Archimedean t-norms;
T = (〈ai , bi ,Ti 〉)i∈Γ, T0 = (〈a0j , b0j ,T0j〉)j∈Γ0 ,
where Ti ,T0j ∈ TAr for all i ∈ Γ and j ∈ Γ0.
for any α ∈ ]0, 1[ there are two exhaustive and mutually exclusivecases:
α is an idempotent element of T0;
there exists a k ∈ Γ such that α ∈ ]a0k , b0k [.
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Migrativity with respect to a continuousordinal sumT0(α, α) = α, characterization
Theorem
Suppose T0 is a continuous t-norm and α ∈ ]0, 1[ is an idempotent
element of T0. Then the following statements are equivalent for a
continuous t-norm T:
(i) T is α-migrative with respect to T0;
(ii) T is α-migrative with respect to TM;
(iii) there exist continuous t-norms T1 and T2 such that T can be written
as T = (〈0, α,T1〉, 〈α, 1,T2〉).
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Migrativity with respect to a continuousordinal sumT0(α, α) < α, characterization
Theorem
Suppose T0 = (〈a0j , b0j ,T0j 〉)j∈Γ0 is a continuous t-norm and α ∈]a0k , b0k [for some k ∈ Γ0. Then the following statements are equivalent for a
continuous t-norm T.
(i) T is α-migrative with respect to T0;
(ii) There exist t-norms T1,T3 ∈ TCo and T2 ∈ TAr such that
(a) T = (〈0, a0k ,T1〉, 〈a0k , b0k ,T2〉, 〈b0k , 1,T3〉), and
(b) T2 is
(
α− a0k
b0k − a0k
)
-migrative with respect to T0k .
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Migrativity with respect to a continuousordinal sumRemarks
In the previous theorem a0k = 0 (or b0k = 1) is possible.
α-migrativity is restrictive on a continuous T mainly in aneighbourhood of α. This is just α itself if α is an idempotentelement of T0, and it is the square ]a0k , b0k [2 otherwise.
Outside this neighbourhood T can be defined arbitrarily in such a waythat the resulting t-norm be continuous.
The summand T2 ∈ TAr in the “middle” ofT = (〈0, a0k ,T1〉, 〈a0k , b0k ,T2〉, 〈b0k , 1,T3〉) can be determined bythe summand T0k (details in Fodor and Rudas, 2011).
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
A T0 for two examples
The ordinal sum T0 = (〈1/6, 1/3,TL〉, 〈1/3, 2/3,TP〉, 〈5/6, 1,TP〉).
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Example 1T0 with α = 2/3 (idempotent element)
The ordinal sum T0, and α = 2/3 (left). 2/3-migrative t-norm T withrespect to T0 (right).
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Migrative t-norms Migrativity with respect to a continuous ordinal sum
Example 2T0 with α = 5/12 (non-idempotent element)
The ordinal sum T0, and α = 5/12 (left). 5/12-migrative t-norm T withrespect to T0 (right).
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Cross-migrative t-norms
Cross-migrative t-norms
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Cross-migrative t-norms
Cross-migrative t-norms
Definition
Let α ∈ ]0, 1[ and T1,T2 be t-norms. We say that the pair (T1,T2) isα-cross-migrative (or, equivalently, that T1 is α-cross-migrative withrespect to T2, in symbols T1
α∼T2) if the following functional equation
holds:
T1(T2(α, x), y) = T2(x ,T1(α, y)) for all (x , y) ∈ [0, 1]2. (6)
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Cross-migrative t-norms
Cross-migrative t-norms
Definition
Let α ∈ ]0, 1[ and T1,T2 be t-norms. We say that the pair (T1,T2) isα-cross-migrative (or, equivalently, that T1 is α-cross-migrative withrespect to T2, in symbols T1
α∼T2) if the following functional equation
holds:
T1(T2(α, x), y) = T2(x ,T1(α, y)) for all (x , y) ∈ [0, 1]2. (6)
Theorem
The relationα∼ is reflexive and symmetric on the set of all t-norms.
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Cross-migrative t-norms
Cross-migrative t-normsFurther properties
For each α ∈ ]0, 1[ and for each t-norm T , (T ,TD) isα-cross-migrative.
The relationα∼ on the set of t-norms is not transitive and, therefore,
no equivalence relation: for each α ∈ ]0, 1[ we have TMα∼TD and
TDα∼TP, but we do not have TM
α∼TP.
For each t-norm T and for each c ∈ [0, 1], the functionT (c) : [0, 1]2 → [0, 1] defined by
T (c)(x , y) =
{
T (x , y , c) if (x , y) ∈ [0, 1[2 ,
T (x , y) otherwise,
is a t-norm (observe that T (0) = TD and T (1) = T ), and for eachα ∈ ]0, 1[ we have T (c) α
∼T .
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Cross-migrative t-norms
Cross-migrative t-normsFurther properties
If T1α∼T2 and ]a, b[ is a non-empty subinterval of [0, 1], then for the
ordinal sums (〈a, b,T1〉) and (〈a, b,T2〉) we have
(〈a, b,T1〉)γ∼(〈a, b,T2〉), where γ = α−a
b−a.
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Cross-migrative t-norms
Cross-migrative t-normsFurther properties
If T1α∼T2 and ]a, b[ is a non-empty subinterval of [0, 1], then for the
ordinal sums (〈a, b,T1〉) and (〈a, b,T2〉) we have
(〈a, b,T1〉)γ∼(〈a, b,T2〉), where γ = α−a
b−a.
Recall that for each t-norm T and for each strictly increasing bijectionϕ : [0, 1] → [0, 1] the function Tϕ : [0, 1]2 → [0, 1] defined by
Tϕ(x , y) = ϕ−1(T (ϕ(x), ϕ(y)))
is also a t-norm.
Let ϕ : [0, 1] → [0, 1] be a strictly increasing bijection, α ∈ ]0, 1[ andT1,T2 be two t-norms. If (T1,T2) is α-cross-migrative then((T1)ϕ, (T2)ϕ) is ϕ(α)-cross-migrative.
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Cross-migrative t-norms Cross-migrativity with respect to the minimum
Cross-migrativity with respect to the minimum
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Cross-migrative t-norms Cross-migrativity with respect to the minimum
Reminder
For a t-norm T and α ∈ ]0, 1[, (T ,TM) being α-cross-migrative meansthat for all (x , y) ∈ [0, 1]2
T (min(α, x), y) = min(x ,T (α, y)).
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Cross-migrative t-norms Cross-migrativity with respect to the minimum
Cross-migrativity with respect to TM
Characterization - general T
Theorem
Let α ∈ ]0, 1[ and T be a t-norm. Then (T ,TM) is α-cross-migrative if
and only if there is a β ∈ [0, α] and a t-norm T1 satisfying
(i) for all (x , y) ∈[
0, α−β1−β
]2: T1(x , y) = 0,
(ii) for all (x , y) with 0 ≤ x ≤ α−β1−β
≤ y ≤ 1:
T1(x , y) = min
(
x ,T1
(
α− β
1 − β, y
))
,
such that T can be written as
T = (〈β, 1,T1〉) . (7)
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Cross-migrative t-norms Cross-migrativity with respect to the minimum
Cross-migrativity with respect to TM
Consequences - general T
Let α ∈ ]0, 1[ and T be a t-norm. Then we have:
(i) T (α, α) = α and (T ,TM) is α-cross-migrative if and only if(〈α, 1,TD〉) ≤ T .
(ii) If T (α, α) = β < α and (T ,TM) is α-cross-migrative then
T(β)∗ ≤ T ≤ T ∗
(β), where the t-norms T(β)∗ and T ∗
(β) are defined,
respectively, by T(β)∗ = (〈β, 1,TD〉) and
T ∗
(β)(x , y) =
{
β if (x , y) ∈ [β, α]2 ,
TM(x , y) otherwise.
The converse is not true.
These boundaries are sharp because of T(β)∗
α∼TM and T ∗
(β)α∼TM.
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Cross-migrative t-norms Cross-migrativity with respect to the minimum
Cross-migrativity with respect to TM
Characterization - continuous T
Theorem
Let α ∈ ]0, 1[ and T be a continuous t-norm. Then the following are
equivalent:
(i) (T ,TM) is α-cross-migrative.
(ii) For all x ∈ [0, α] we have T (x , x) = x, i.e., T = (〈α, 1,T1〉) for some
continuous t-norm T1.
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Cross-migrativity with respect to strict t-norms
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Reminder
For a t-norm T and α ∈ ]0, 1[, (T ,TP) being α-cross-migrative meansthat for all (x , y) ∈ [0, 1]2
T (αx , y) = x T (α, y).
We restrict ourselves to continuous solutions only.
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Solution
Theorem
Let α ∈ ]0, 1[ and T be a continuous t-norm. Then (T ,TP) is
α-cross-migrative if and only if there exist
a β ∈ [α, 1],
a strict t-norm T1 with an additive generator t1 : [0, 1] → [0,∞]satisfying t1(x) = δ · (d − log x) for all x ∈
[
0, αβ
]
with some
constants δ ∈ ]0,∞[ and d ∈]
log αβ,∞[
, and
a continuous t-norm T2 such that
T = (〈0, β,T1〉, 〈β, 1,T2〉).
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Solution rewritten
Theorem
Let α ∈ ]0, 1[, T0 be a strict t-norm with additive generator
t0 : [0, 1] → [0,∞], and T be a continuous t-norm. Then (T ,T0) is
α-cross-migrative if and only if
T = (〈0, β,T1〉, 〈β, 1,T2〉),
where β ∈ [α, 1], T2 is an arbitrary continuous t-norm and T1 is a strict
t-norm with an additive generator t1 : [0, 1] → [0,∞] such that there are
constants d ∈]
−t0(αβ
),∞[
and δ ∈ ]0,∞[ and we have
t1(x) = δ · (t0(x) + d) for all x ∈[
0, αβ
]
.
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Solution in another form
Theorem
Let α ∈ ]0, 1[, T0 be a strict t-norm with additive generator
t0 : [0, 1] → [0,∞], and T be a strict t-norm. Then the following are
equivalent:
(i) (T ,T0) is α-cross-migrative.
(ii) The function t : [0, 1] → [0,∞] defined by
t(x) =
{
t0(x) + c if x ∈ [0, α],
t1(x) otherwise,(8)
where c ∈ ]−t0(α),∞[ and t1 : [α, 1] → [0,∞] is a continuous,
strictly decreasing function satisfying t1(1) = 0 and
t1(α) = t0(α) + c, is an additive generator of T .
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Consequences for all strict t-norms
(i) For each α ∈ ]0, 1[, the relationα∼ is transitive on the class of all
strict t-norms, i.e.,α∼ is an equivalence relation on the class of all
strict t-norms.
(ii) For all α, β ∈ ]0, 1[ with β ≤ α and for all strict t-norms T1 and T2
we have that T1α∼T2 implies T1
β∼T2, i.e., the partition of the class
of strict t-norms induced by the equivalence relationα∼ is a
refinement of the partition induced byβ∼.
(iii) For a fixed α ∈ ]0, 1[ and a fixed strict t-norm T0 the equivalenceclass (with respect to
α∼) {T | T is a strict t-norm and T
α∼T0}
consists of all strict t-norms satisfying, for some constantsη, ϑ ∈ ]0, 1] and for all (x , y) ∈ [0, α]2, the equalityT (x , y , η) = T (x , y , ϑ).
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Example 1
Consider the family of Dubois-Prade t-norms (TDPλ )λ∈[0,1] given by
TDPλ = (〈0, λ,TP〉).
Evidently, for each α ∈ ]0, λ] we have TDPλ
α∼TP.
Observe that TDPλ (x , y , λ2) = TP(x , y , 1) for all (x , y) ∈ [0, α]2.
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Example 2
Consider the Hamacher product TH which is generated by the additivegenerator tH(x) = 1
x− 1:
TH(x , y) =x · y
x + y − x · y
for all (x , y) ∈ [0, 1]2 \ {(0, 0)}).
Define the function t : [0, 1] → [0,∞] by
t(x) =
{
1x− 1 if x ∈
[
0, 12]
,
2 − 2x otherwise.
Then t is an additive generator of the strict t-norm T given by
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Cross-migrative t-norms Cross-migrativity with respect to strict t-norms
Example 2 (cont.)
T (x , y) =
TH(x , y) if (x , y) ∈[
0, 12]2,
x1+2x−2x ·y if (x , y) ∈
[
0, 12[
×]
12 , 1]
,y
1+2y−2x ·y if (x , y) ∈]
12 , 1]
×[
0, 12[
,1
5−2x−2y if (x , y) ∈]
12 , 1]2
and x + y < 32 ,
x + y − 1 otherwise,
and we have T12∼TH. Obviously, T (x , y , 1) = TH(x , y , 1) for all
(x , y) ∈[
0, 12]2
.
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Cross-migrativity with respect to nilpotent t-norms
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Reminder
For a t-norm T and α ∈ ]0, 1[, (T ,TL) being α-cross-migrative meansthat for all (x , y) ∈ [0, 1]2
T (max(x + α− 1, 0), y) = max(x + T (α, y) − 1, 0).
We restrict ourselves to continuous solutions only.
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Solution
Theorem
Let α ∈ ]0, 1[ and T be a continuous t-norm. Then (T ,TL) is
α-cross-migrative if and only if there exist
a β ∈ [α, 1],
a nilpotent t-norm T1 whose normed additive generator
t1 : [0, 1] → [0,∞] satisfies t1(x) = 1 − c · x for some constant
c ∈ ]0,∞[ and all x ∈[
0, αβ
]
, and
a continuous t-norm T2 such that
T = (〈0, β,T1〉, 〈β, 1,T2〉).
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Strict and nilpotent cases differ
For the product t-norm TP, a continuous t-norm T satisfies Tα∼TP if
and only if there is a c ∈]
0, 1α
[
such that T (x , y) = c · x · y for all
(x , y) ∈ [0, α]2.
For the Lukasiewicz t-norm TL, if a continuous t-norm T satisfiesT
α∼TL then there is a constant c ∈ [α, 1] such that
T (x , y) = max(x + y − c , 0) for all (x , y) ∈ [0, α]2.
The opposite implication may not hold: for the Yager t-norm TY2
given by TY2 (x , y) = max(1 −
√
(1 − x)2 + (1 − y)2, 0) we have
TY2 (x , y) = 0 = max(x + y − 1, 0) for all (x , y) ∈
[
0, 15]2
, but TY2 is
not 15 -cross-migrative with respect to TL.
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Characterization
Theorem
Let α ∈ ]0, 1[, T0 be a nilpotent t-norm with additive generator
t0 : [0, 1] → [0,∞], and T be a continuous t-norm. Then (T ,T0) is
α-cross-migrative if and only if
T = (〈0, β,T1〉, 〈β, 1,T2〉),
where β ∈ [α, 1], T2 is an arbitrary continuous t-norm and T1 is a
nilpotent t-norm with an additive generator t1 : [0, 1] → [0,∞] such that
there are constants d ∈]
−t0(αβ
),∞[
and δ ∈ ]0,∞[ and we have
t1(x) = δ · (t0(x) + d) for all x ∈[
0, αβ
]
.
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Cross-migrative t-norms Cross-migrativity with respect to nilpotent t-norms
Consequences
For each α ∈ ]0, 1[, the relationα∼ is transitive on the class of all
nilpotent t-norms, i.e.,α∼ is an equivalence relation on the class of all
nilpotent t-norms.
For all α, β ∈ ]0, 1[ with β ≤ α and for all nilpotent t-norms T1 and
T2 we have that T1α∼T2 implies T1
β∼T2, i.e., the partition of the
class of nilpotent t-norms induced by the equivalence relationα∼ is a
refinement of the partition induced byβ∼.
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Concluding remarks
Concluding remarks
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Concluding remarks
Remarks 1
Migrativity (T1(T2(α, x), y) = T1(x ,T2(α, y))) and cross-migrativity(T1(T2(α, x), y) = T2(x ,T1(α, y))) of t-norms are interestingproperties expressed in the form of functional equations.
We have given characterizations for the basic continuousArchimedean t-norms and for the minimum.
Constructions could be illustrated graphically. This also supports theterm “migrative” (characterized by migration; undergoing periodicmigration).
While migrativity defines an equivalence relation on the set oft-norms, cross-migrativity implies an equivalence relation only in theclasses of strict and nilpotent t-norms.
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Concluding remarks
Remarks 2
The α-cross-migrativity can be seen as a special kind of commuting
of t-norms T1 and T2, if we rewrite the equation into the equivalentform
T1(T2(x , α),T2(1, y)) = T2(T1(x , 1),T1(α, y)).
It seems to be interesting to study the functional equation (valid forall (x , y) ∈ [0, 1]2)
T1(T2(x , α),T2(β, y)) = T2(T1(x , β),T1(α, y)).
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Concluding remarks
THANK YOU FOR YOUR ATTENTION!
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