- Email: [email protected]
Fibred triangular norms
•
t,
FUllY
sets and systems ELSEVIER
Fuzzy Sets and Systems 103 (1999) 67-82
Fibred triangular n o r m s 1 S~mdor Jenei* Department of ComputerScience, E6tv6sLor&ndUniversity,Mtbeum krt. 6-8, 1-1-1088Budapest, Hungary Received May 1996; received in revised form May 1997
Abstract
In this paper, we deal with the formula g [T(f(x),f(y))] where f, g are functions mapping from the unit interval to itself and T is a triangular norm. We are interested in the conditions under which this formula yields a triangular norm. The case, when f is continuous is investigated in detail. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. These new triangular norms are the so-called 'fibred triangular norms' which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. We investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, some examples are given for the case when f is not continuous. © 1999 Elsevier Science B.V. All rights reserved.
Keywords: Oper; Ana; Triangular norm; Homomorphism; Semigroup; Additive generator function
1. I n t r o d u c t i o n
A triangular n o r m (t-norm for short) is a function T from [0, 1] 2 to [0, 1] being commutative, associative, non-decreasing in each place and T (1, x) = x holds for all x e [0, 1]. A t - n o r m T is said to be continuous if it is continuous as a two-place function. A continuous t - n o r m T is called Archimedean if T (x, x) < x is true for all x e (0, 1). Since we will deal with additive generators in this paper, we need the following representation theorem of continuous Archimedean t-norms [5]. Theorem 1. A t-norm T is continuous and Archimedean if and only if there exists a strictly decreasing continuous function f: [0, 1] ~ [0, oo] with f(1) = 0 such that
T(x, y) = f ( - 1)(f(x) + f(y)), * E-mail: [email protected].
1Supported in part by OTKA (National Scientific Research Fund, Hungary) 1/6-14144. 0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 1 65-0 1 14(97)00 1 97-8
(1)
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
68
where ft-1) is the pseudo-inverse of f defined by f~_i)(x )
=
~ ( - I(X) (O
/f X ~f(O),
otherwise.
Moreover, representation (1) is unique up to a positive muhiplicative constant. If a t-norm T admits representation (1) then the function f is called an additive generator of T. There are several methods in the literature concerning the construction of new triangular norms from a given t-norm T. We cite here some of them. (1) Suppose that { [ a , bi]}i~r is a countable family of non-overlapping, closed, proper subintervals of [0, 1], denoted by J . Notice that either K = N or there exists ne N such that K = {1, ..., n}. With each [ a , b~] e J associate a t-norm T~. Let T be a function defined on [0, 1] 2 by am
+
I x--am
(bin
a")T~bm~-a,,'b,. - a,./
T(x, y) = min (x, y)
if (x,y)e[am, b,,] 2,
(2)
otherwise.
In this case T is called the ordinal sum of {([ai, bi], T~)}~r and each T~ is called a summand. For more details see e.g. [5]. (2) If q~is an increasing bijection of the closed unit interval then the following formula defines the so called q)-transform of T:
T~o(x,y) = (o-l(T((o(x), (o(y))), x, y~[0, 1]. Then q~ is an isomorphism between the semigroups ([0, 1], T) and ([0, 1], T¢) from the algebraic point of view. (3) Let (0 be an increasing bijection from [0, 1] to [a, 1] (ae [0, 1)). Let
To(x, y)= qo(-1)(T(q)(x), (o(y))), x, y e [ 0 , 1], where q¢-i)(x) = ~0-i (max(a, x)) for x • [0, 1] and called the pseudo-inverse of ~o. Then we obtain a semi9roup deformation of T. For more details see [7, 8]. (4) and (5) Recently, Demant [ 1] suggested two new constructions with the help of an increasing bijection ¢ from [0, 1] to [0, a] (a ~ (0, 1]). (Further investigation can be found in [6].) The first transformation is called contraction, the second one is dilatation and are defined as follows, (x, y e [0, 1]):
T~)(x, Y)
f,p~-l)(T(,p(x),
~0(y))) if max(x, y) < 1,
%
(rain (x, y)
otherwise,
fq~(T(~0t-1)(x), cpt-1)(y)))
T (~)(x, y) = [ T(x, y)
if T(x, y) < a, otherwise,
where (o(- i)(x) : = ¢p- 1(rain(a, x)). (6) Let n be a strong negation (i.e., an involutive order reversing bijection of the closed unit interval). Define the following operator:
T~,,(x,y)={:(x,y)
ifx>n(Y),otherwise.
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The question is whether Tt,) is a t-norm or not for a given t-norm T. In [3] a sufficient and necessary condition was established for that. An important example the 'nilpotent minimum' was found (T = min, n is an arbitrary strong negation). Another example is given in [4], the nilpotent ordinal sum family. However, the complete characterization of the problem is still unsolved. In this paper a new construction which produces new triangular norm from an arbitrary triangular norm will be defined and investigated. We deal with the formula 9 [T (f(x), f(Y))l, where f, 9 are functions mapping from the unit interval to itself and T is a triangular norm. So, we generalize the o-transform, which was mentioned above in point (2). After this introduction, we give a brief preliminary on pseudo-inverses of monoton functions in Section 2. In Section 3 we are interested in the conditions under which the above-mentioned formula yields a triangular norm. The case, when f is continuous is investigated in detail in Sections 4 and 5. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. Applying this method we obtain a new class of discontinuous t-norms. These new triangular norms are the so called 'fibred triangular norms' which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. In Section 6 we investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, an open problem is formulated and some examples are given for the case when f is not continuous in Section 7.
2. Pseudo-inverses We call co a disjoint interval system if
o9 = {(ak, bk] C [0, 1 ] t k e O ,
(ak, bk]n(at, bt] = 0 if k # l}.
Let 0 denote the set of disjoint interval systems: f2:= {cole) is a disjoint interval system}. Let f : [0, 11 ~ [0, oo] be a monotone function. Obviously, such an f admits the following representation: There exists a countable set O, and disjoint subintervals (ak, bk] (keO) of the unit interval such that f is constant on each (ak, bk], strictly monotone on [O, 1]\Uk~O(ak, bk] and (ak, bk]S are maximal in some sense. Let us call the set of these intervals the constant support of f and denote it by Suppc(f). More formally, let Suppc(f) = {(ak, bk] I ke O, ak < b~, (ak, bk] c [0, 13, f is constant on each (ak, bk], f is not constant on intervals of type (] containing properly (ak, bk] and f is strictly monotone on [0, 1]\Uk~O(ak, bk]}. The following definition for non-decreasing f is due to Schweizer and Sklar (for more details, see [91). The definition for the non-increasing case is an easy generalization. Let f * : [0, 11 ~ [0, 11 be a function fulfilling the following conditions: (1) f * ( f ( O ) ) = O, f * ( f ( 1 ) ) = 1.
(2) If y is in the range of f, then f * (y) is in f - l ( y ) . (3) If y is not in the range of f, then f*(y) = sup {x If(x) < y} = inf{x [f(x) > y}. If f is non-increasing, then one has to change (3) with the following line: (Y) If y is not in the range of f , then f * (y) = inf{x If(x) < y} = sup {x If(x) > y}. Then f * is called a quasi-inverse of f.
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Generally, such an f * is not unique. But there is a unique left-continuous (right-continuous) f * for a given non-decreasing (non-increasing) f. We need this (unique) quasi-inverse as we will see later. Consider the following definition: If f is non-decreasing then let ft-ll(y) =
1 sup{x:f(x) < y}
if y >~f(1), if ye[0,f(1)].
(3)
If f is non-increasing then let ft-lJ(y) =
0 sup{x: f(x) > y}
if y ~> f(0), if y e [0,f(0)].
(4)
One can see easily that the function defined by (3) (4) is a quasi-inverse of f i.e., it fulfills (1)-(3) ((1)-(3')). Obviously, if f is an additive generator function then f t - q coincides the usual pseudo-inverse f(-l). Throughout this article we call the function defined by (3) or (4) the pseudo-inverse of f The name is motivated by the fact that a function f may have many quasi-inverses but has a unique pseudo-inverse (as in the case of additive generators). Proposition 1. If f is non-decreasing (non-increasing, respectively) then f t - 1 j is the unique left-continuous (right-continuous)function given by (3) (4)for which foft-11[Rtf ) = id~(y) may hold. (2) I f f is continuous and non-decreasing (non-increasing respectively) then its pseudo-inverse has the following form: f t - al(y) = inf{x:f(x) = y},
( f t - lj(y) = sup{x:f(x) = y})
if y~[0, f(0)], and f t - l j is strictly increasing (decreasing) on [f(0),f(1)] (If(l), f(0)]). In this case f o f t - 11]R(:) = idR(f) holds. (3) I f g is a monotonic continuous function and Suppo ( f ) = Suppo( g) then g oft-1] o f = g. Proof. We prove only (3) the proof of the rest can be found in [9]. Suppose that Suppc(f)= Supp~(g). If x¢Suppo(f) then f[-11 o f ( x ) = x, whence g of t-ll o f ( x ) = g(x). If x ~(ak, bk] i~ 0 then ft-11 of(x) = ak, whence g oft-al of(x) = g(ak) = g(x) by right-continuity. []
3. Conditions Theorem 2. Let T be a t-norm, f a non-decreasing function from the unit interval to itself with f(1) = 1 and f (O) = O. Let H c [0, 1] with R( f ) c H and g : H ~ [0, 1] a non-decreasing left-continuous function fulfilling (1) f°gl~(y)= idRty). Further, suppose that f is closed under T, i.e., (2) the value T(f(x), f(y)) should belong to the range of f for any x, y ~ [0, 1]. Define TI(x, y) : [0, 1] ~ [0, 1] as follows: ~g[ T(f(x),f(y))] TI(x' Y) -- (min(x, y) then Ty is a t-norm.
if x, y~[0, 1), if max(x,y) = 1,
(5)
S, Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
0'"
~
71
f(x)
I
I
y
the value of the pseudoinverse at z =f(y) Fig. 1.
Proof. Boundary condition and symmetry are straightforward. Associativity (i.e., Ty(Ty(x, y), z) = Tf(x, Ty(y, z)) holds for x, y, z e [0, 1]) is obvious if any of the variables x, y, z is 1. If x, y, z ~ [0, 1) then
Ty(Ty(x, y), z) = g [ T ( f o g[T(f(x),f(y)),f(z))]]
= (from (1) and (2))
= g[T(T(f(x), f(y)), f(z))] = g[T(f(x),f(y),f(z))]
= (similarly) = Ty(x, Ty(y, z)).
As an alternative proof for the associativity one can check easily that [9, Theorem 5.2.1] proves it in a more general setting. Concerning the monotonicity, let y e [0, 1] and 0 ~< xl < x2 < 1; then Ty(xl, y) <~ Ty(x2, y) holds obviously since f and g are non-decreasing. If 0 ~< xl < x2 --- 1 then in case of y = 1 we obviously have that
Ty(xl, y) <<.Ty(x2, y) = y.
(6)
If y < 1 then (6) holds if and only if
g[T (f(xl),f(y))] <<.y.
(7)
But if we have T(f(xl),f(y)) = f(y) (for instance iff(x~) = 1) then (7) holds if and only if O is left-continuous (see Fig. 1). [] Remark 1. Having a strict t-norm T (e.g. the product t-norm) and a function f(with f(0) = 0, f(1) -- 1) which is strictly increasing with one jump (discontinuity at the point a), define g as one of the pseudo-inverses of f ( a s one of f*'s). One can check easily that T: defined in (5) will not be associative. (One has to choose x, y bigger, z smaller than a in such a way that T(x, y) should be in [ f ( a - ) , f ( a + ) ] \ f ( a ) . ) This example points out the importance of condition (2) in Theorem 2. Remark 2. Let us see what happens if we do not define the values of Ty separately in the boundary. Then we would have the following: Ty(x, 1) --- 9 [ T (f(x),f(1))] = 9[T(f(x), 1)] = 9 of(x) and the last expression is x if and only if xq~Supp~(f). Therefore, we really need to redefine the value of Ty in the boundary if and only if Supp¢(f) ~ {0}, i.e., if and only if there exist at least two different points x, y in [0, 1], where f(x) = f ( y ) .
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Remark 3. If T is strict at the boundary O.e., T(x, y) ~ y, whenever x < 1) or f is strict at 1 (i.e., f(x) > f ( 1 ) if x is close to 1) then (7) holds even if # is not left-continuous. Summarizing what we obtained so far on 9: if T is an arbitrary t-norm then one has to choose # = f [ - u . If we restrict our attention to the t-norms which are strict at the boundary or if we restrict our attention to the f ' s which are strict at 1 then one can choose g as any of the f*'s. We want a transformation which produces a t-norm from every t-norm whence we will choose # = f [ - 1 [ in the sequel.
Corollary 1. Suppose that the conditions of Theorem 2 hold and f is strictly increasing. Then Tz(x, y) = g [ T ( f ( x ) , f ( y ) ) ] is a t-norm for any t-norm T. Proof. Taking into account Remark 2, the statement is obviously true.
[] '
4. T-norms generated by pseudo-automorphisms Definition 1. We call q~: [0, 1] ~ [0, 1] a pseudo-automorphism of the unit interval if it is a non-decreasing continuous function with q~(0) = 0 and q~(1) = 1. Let us denote the set of pseudo-automorphisms by Autps [0, 1]. Let us investigate the assumptions of Theorem 2 more carefully. Ty has to be associative. Therefore, a good choice for 9 is any f * by virtue of Proposition 1. Tz has to be monotone at the boundary; therefore, we have to choose g = f [ - i ] which is one of the f*'s. Tz has to be associative so we need condition (2). Notice that the continuity of f yields it as we will prove in Corollary 2. This corollary stands in the focus of our interest in the sequel and then in Section 7 we will drop out the continuity of f a n d will investigate the general case.
Corollary 2. Let ¢p ~ Autps[O, 13. Let us define T~: [0, 1] 2 ~ [0, 1] by ~(pt-'](T(q~(x), q0(y))) /f x, y e [ 0 , 1), Te(x, y) = ( m i n ( x , y) /f max(x, y) = 1.
(8)
Then T~ is a t-norm for any t-norm T (and called the t-norm generated by the pseudo-automorphism tp).
Proof. Since ~0 is continuous, therefore R(q~) = [0, 1] and condition (2) in Theorem 2 holds automatically. Condition (1) in Theorem 2 holds by using Proposition 1.
[]
Obviously, it is meaningless to talk about the range of a t-norm since it is always the closed unit interval. Therefore, we need to define it in a bit different way. Definition 2. Let T be a t-norm. Then the range of T (denoted by R(T)) is defined as follows: R ( T ) : = R(Tlto. 1)2) i.e., it is the range of the restriction of T to the half-open unit square, [0, 1) × [0, 1).
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
I
77
I
1. al
bl
2.
3. al
bl
4. al
F
bl
liftthis part
5. al
b~
6. al
b~
Fig. 3. H o w a fibred t - n o r m arises from its factor t-norm.
exactly at the intervals of the interval system co. Then we have to make as many lifts as again the cardinality of co in order to obtain the proper range of Tpro. This range is [0, 1]\co by using Proposition 2 and finally, the boundary has to be redefined. Now, we have a clear picture about the graph of a t-norm which was generated by a pseudo-automorphism. In the light of Theorem 3 first we have to rescale the axes (with the automorphism) then we have to change the graph in the way which was described in this proceeding. Remark 4. Let (a, b] e co. Then Tvr ~ fulfills the following properties (by the definition of the 'projection generation' and by Proposition2): T ( z l , x ) = T ( z 2 , x) whenever x e [ 0 , 1 ) and zl, z2e(a,b], (a, b ] n R ( T ) = O. Consider the opposite direction: Let T be a t-norm which fulfills the following two properties: (1) There exists (a, b] c [0, 1] such that T (zl, x) = T(z2, x) whenever x e [0, 1) and zl, z2 e(a, b]. (2) (a, b]c~R(T) = O. The question arises now: Is T a fibred t-norm (generated by a projection pro,), where (a, b] e co? The answer is straightforward and confirmative taking into account Example 1. With the help of it one can define the
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
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'inverse vibration' easily. As a consequence, we obtain that the factor t-norm can be determined easily. It is simply the value of the 'inverse fibration', where ~ois determined by the union of the intervals (a, b] having the above mentioned two properties. Example 2. If the intervals of co almost cover the unit interval (i.e., ~(~o) is close to 0) then the generated t-norm is matrix-like. This may lead to the observation that any many-valued 'and'-model which uses an associative 'and'-matrix can be approximated by a fuzzy t-norm Tp,~ (see Fig. 4).
6. Weak additive generator functions Definition 3. Let F: [0, 1] ~ [0, oo] be a non-increasing continuous function with F(1) = 0. Then F is called a weak (additive) generator. Now, we define the following two-place function: G : [0, 1] 2 -o [0, 1], ~min (x, y) if max(x, y) = 1, G(x,y) = [Ft_ll(F(x ) + F(y)) if x, ye[O, 1).
(15)
Proposition 4. The two-place function defined by (15) is a t-norm for any weak generator function F.
at
b, a:
b~ a:~
I~ a~
b4 1
%
3
0
!
2
3
2
0
I
1
2
1
0
0
i
J
0
0
0
0
0
1
2
3
0
tile matrix
/ / I}
F i g . 4. T - n o r m
approximation
of a matrix.
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
73
Proposition 2. I f T is continuous then the range of T~ is [0, 1]\Suppc(q)). Proof. The statement is obvious considering the properties of the function (p and (pt- 11.
5. t-norms generated by projections This section is devoted to the deeper study of t-norms generated by pseudo-automorphisms. The fibred t-norms are defined here. These are t-norms generated by special type of pseudo-automorphisms, the so-called projections. It will be clear that every t-norm which was generated by a pseudo-automorphism is a fibred t-norm. That is, it can be generated by a projection and this fact brings us closer to the understanding what happens with the graph of a t-norm T when we generate T~ from it. This clear picture is due to the simple form of projections. Take an element co from f2. Let us denote ~ ( ~ ) = 1--~,kEo(bk- ak). We define the following functions: 2:[0, 1]--*[0, 1]
2(x)= ~ (bj-aj), j:bj
pro,: [0, 1 ] - , [ 0 , 1 ] , ak--2(ak) pro,(x) =
if x e ( a k , bk] for some k e O ,
¢(co) x-~(x) 3((0)
(9) otherwise.
;[: [o, I] -~ [o, U, 2(x) =
~
(b3 - as),
j:pr~(bj) < x
fibo,: [0, 1] --* [0, 1], fib~,(x) = x¢(og) + Z(x),
and, finally, i: f2 ~ Autps[O, 1]
i(co) =pro.
It is easy to check that pro, is piecewise linear and the steep of the lines are 0 or a fix positive number. An example for the functions defined above can be seen in Fig. 2. One can check easily that thus defined pro is continuous, non-decreasing function with pro,(O) = 0 and pro,(1) = 1, hence, pro, is a pseudo-automorphism of [0, 1]. Such a pseudoautomorphism will be called a projection.
74
S. Jenei / Fuzzy Sets and Systems 103 H999) 67-82 pr(x)
Y
fib(x)
Y\ lha-
/
1
/
/ I
~2~ • a~
br
I
I
al bl a2
t~
b~3
t
t >
ba 1
j
i f
J
J
x
t >
1
x
Fig. 2. Projection and fibration functions•
3. fibo = pr [- 1]
Proposition
Proof. Let y~[0, 1]. Then we have prto-1](y) = inf{x: pro(x) < y} = inf{x: pro,(x) = y} = inf{x: pro(x) = y, x¢(ak, bk](ieO)}, by using that pro is continuous and constant on each (ak, bk]. Since pr,o is strictly increasing on [O, 1]\Oi~o(ak,bk] we have that x=prt-~](y) is the unique solution of the equation x -
,a.(x)
-Y on [0, 1]\Ui~e(ak, bk]. The only thing to be checked now is that x =fibo(y) satisfies this equation. Indeed, one can check it easily since pr~-lj(y) = x, therefore y = pro(x) using Proposition 1 and we have 2(y) = ~,j:pr.(bj)
~fibo[T(pro(x), pro(y))] TP'w(x' Y) = ~min (x, y)
if x, y e [0, 1), if max (x, y) = 1
(10)
is a t-norm. Proof. pro is a pseudo-automorphism and one can use Corollary 2. We call T the factor t-norm of Tp,o and Tp,o is called the fibred t-norm o f T with respect to the interval system co. This name is based on the observation that all the points of [ak, bk] have the same behavior (i.e., T(X1, y) = T ( x 2 , y) if xl, x2 ~ [ak, bk] and y~ [0, 1) due to the properties of pro.
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S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
More formally, define a binary relation R on the unit interval by the following rule: x R y ¢~ x and y belong to the interval [ak, bk] for some k e O. Obviously, it is an equivalence relation and T is the 'factor t-norm' on the quotient set (generated by R). Another interpretation is given as follows: from the semigroup-theoretic point of view a t-norm (if we do not consider the boundary) is a homomorphic image of its fibred t-norm T~, i.e., ~0is a homomorphism from the semigroup ([0, 1), T~) to ([0, 1), T). Indeed, T~o(x, y) = q~t-i]o T(q)(x), ~o(y)) yields q~o T,p(x, y ) = T ( o ( x ) , q)(y)) by using Proposition 1. Theorem 3. F o r a given pseudo-automorphism
q~ there exists a projection pr and an automorphism
~b
such that
Proof. Let pr = i o Suppc(q2) and let O(x) = q)° pr {- ll(x).
(11)
Then ~ is an automorphism of the unit interval. Indeed, ~ ( 0 ) = ~0oprt-ll(0)= ~0(0)=0 and ~,(1) = ~ooprt-il(1)= ~ o ( m i n { ( z : p r ( z ) = 1 } ) = ~0(z*)= q~(1)= 1 since ~ and pr have common constant support. is left-continuous, since it is composition of two left-continuous non-decreasing functions. The right-continuity is straightforward in the continuity points of pr t- 11. If x is not a continuity point of pr E- 11 then pr t- ll(x) = ak for some k and right-continuity follows from the following line:
lim O(t)= lim q2o p r t - ll(t) = I~+X
t-~+X
q)( ~mxPrt-1](t)) = qo(bk) = q)(ak) = O(x ). t
So ~ is continuous. ~b is strictly increasing, since pr t - l l is strictly increasing and ~o is strictly increasing on the range of pr {- 1]. So tp is an automorphism of the unit interval. Now, we have to check that Oopr=¢
and
prt-ilo$-i=¢t-ll.
(12)
Indeed, from (11) we have that ~ko pr(x) = ~o o pr t- 11 o pr(x). The right-hand side is ~o(x) by Proposition 1. On the other hand ~b-1 = po ¢pt-ll since ~ - i o @(x) = pro ~0t- 1] oq~)oprt-1](x ) = x by using Proposition 1. Finally, T~(x, y) = o2t - 11 o T(q)(x), @(y)) = pr t- il o ~b- lo TOp o pr(x), (To)pr(x, y) by using (12). []
(13)
o p r ( y ) ) = T~,(pr(x), p r ( y ) ) =
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
76
Example 1. Let co = { (a, b] [a ~< b} and T be a t-norm. Easy verification shows that Tp,. has the following form:
•T
x
y
if (x, y) ~ [0, a] x [0, a],
if (x, y)m(a,b]x(a,b],
T¢o(X, y) =
T
x
a
if (x, y) m [0, a] x (a, b],
T
a
y
if (x, y) e [a, b] x [0, a],
x y - (__~ b - a)) 4(09) • T ( _~),
if (x, y) ~ [0, a] x [b, 1),
~(¢°)" T(x-(b-a)-~-~-) ' ~o)Y )
if (x, y)e(b,
,(¢o)
• T ( -(~o), a Y -(b -f(~- a))
~(°9)'T(x-(b-a)~) (co). T
if (x, y)~ (a, b] x(b, 1),
' ~(co)a)
if
x-(b-a) y--(b_-a)~ -~] ' ~(o~) ,]
~(~o). Tf X - (b - a)
Y ,
1)x [O,a],
(b
(x,y)~ [b, 1] x(a, b],
if (x,y)e(b, 1)x(b, 1) and, ~(og).T(X-(b-a) -~-(w-) ' y-(b-a)) --~) <<.a,
a)]\ + (b - a) /
if (x,y)e(b,
1)x(b, 1) and,
~(¢o).T(X-(b-a) -~ min(x, y) > a
,
y-(b-a)) -~--(~
> a
if max (x, y) = 1.
(14) It is obvious that in case of a = b, Tp,~ = T. Let us analyze the graph of Tpro. We are interested to k n o w how it arises from the graph of T. First, we have to compress the graph of T with 4(o9) like in the case of a 'summand' at the representation of the continuous t-norms [5]. Hence, we have a t-norm on I-0, ~(o9)] x [0, 4(09)]. Now, we have to cut this graph along the lines {(x, y): x ~ [0, 1], y = a} and {(x, y): y ~ [0, 1], x = a}. Then one has to shift the obtained left and the upper pieces to left and up, respectively with b - a. One has to fill the fibers (i.e., [a, b] x [0, 1] and [0, 1] x [a, b]) as defined above. Then we have to cut our graph with a plane which is parallel to the d o m a i n [0, 1] x I-0, 1], at the high of a and have to lift the higher part of the graph with b - a. Notice that at this step the graph attains its range in accordance with Proposition 2. Finally, the b o u n d a r y has to be redefined. This process can be seen in Fig. 3, when T (the factor t-norm) is the Lukasiewicz t-norm. The lines in the unit square mean levels. In general, when 09 consists of m o r e intervals then the process is almost the same. We have to compress T with ~(¢o). We have to make as m a n y cuts and shifts as the cardinality of co in order to obtain the fibers
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
79
Proof. We prove the associativity, the rest is obvious. G(G(x, y), z) = G(x, G(y, z)) holds obviously if one of the variables is 1. Suppose that x, y, z e [0, 1]. By Proposition 1 we have G(G(x, y), z) = F t- ll(F(x) + F(y) + F(z)) = G(x, G(y, z)).
[]
Definition 4. We say that a t-norm T admits the weak generator function representation if T(x, y) = F t- ll(F(x) + F(y))
(16)
holds for all x, y • [0, 1) and for a weak generator function F. Theorem 4. A t-norm T * admits representation (16) if and only if there exists a continuous Archimedean t-norm T and a pseudo-automorphism of the unit interval q) such that T * = T~. Proof. (i) Suppose that T * ( x , y ) = F t - i l ( F ( x ) + F(y)) holds for x , y • [ 0 , 1) and for a weak generator function F. Let us define pr:= i°Suppc(F)
and
f : = Fopr t-j1
(17)
Then f is a generator function. Indeed, f(1) = 0,f(0) = F(0). f is strictly decreasing since pr t- 11 is strictly increasing and F is strictly decreasing on the range of pr I- 11 f is left-continuous. Indeed, pr 1-11 is left-continuous and non-decreasing, F is continuous and f is the composition of them. f i s right-continuous at the continuity points of pr 1-11, if x is not a continuity point of pr t- 11 then lim f ( t ) = F ( lim prt-ll(t)) = r ( b k ) = r ( a k ) = f ( x ) . t~+x
\t~+x
/
Therefore, f is continuous. Let T(x, y ) : = f ( - l)(f(x) +f(y)).
By using (17) and Proposition 1 we have that f o p r = F o p r t - l l o p r = F.
(18)
pr t- 11oft-a) is left-continuous, and right-inverse of F on the range of F. Indeed, if x • R(F)( = R ( f ) ) then by
using Proposition 1 we have that F o pr t- 11o f t - 1) = f o pro pr t- 11of(- l)(x ) = f o f ( - i)(x ) = x.
Therefore, F t- 11 = pr t- 11oft- 1), by using Proposition 1.
(19)
80
s. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82 Finally, by using (18) and (19) we have Tpr(x, y) = pr t- 11of(- i)(fo pr(x) + fo pr(y)) = F t- il(F(x) + F(y)) = T * (x, y).
(ii) Suppose that T is a continuous Archimedean t-norm, q)eAutps[O, 1] and T * = T,. Then T(x, y) = f t - 1 ) ( f ( x ) + f ( y ) ) holds by Theorem 1. Let us define F : = f o ¢p. Then F is a weak generator function, and F t-ll = q,t-ljof(-1), since it is left-continuous and is the right-inverse of F. From these we have T*(x, y) = T~(x, y) = q~t-ilof(-1)(foq)(x) + f o q~(y)) = Ft-ll(F(x) + F(y)).
[]
Corollary 4. The weak additive generator function is unique up to a positive multiplicative constant. Proof. Straightforward by using Theorems 1 and 4.
[]
Proposition 5. Suppose that O, 1 is not in the closure of Supp~(q)). Then T is Archimedean if and only if T, is Archimedean. Proof. Suppose that T~, is Archimedean and (on the contrary) T is not Archimedean. Then there exists x e (0, 1) such that T (x, x) = x. Let y = ~0t- ll(x). Then y e (0, 1) since 0, 1 is not in the closure of Suppc ((o), and T~o(y, y) = q)t-11T(q)(y), (o(y)) = q)t-11T(x, x) = q)t-11(x ) = y, which is a contradiction. Now, suppose that T is Archimedean and (on the contrary) T~o is not Archimedean. Then there exist xe(0,1) such that T ~ ( x , x ) = x . (Let us define y : = (p(x)). It is equivalent to q)t-lJT(~o(x),~o(x))= x ~ q~r- 11T(y ' y) = x =~ T(y, y) = y. This y belongs to (0, 1), since 0, 1 is not in the closure of Suppc(q)), and this leads to a contradiction to the Archimedean property of T. [] Similar to additive generators we have the following proposition:
Proposition 6. I f F is a weak additive generator function of T then H: [0, 1] ~ I-0, ~ ] defined by H(x) := - log(F(x)) is a weak multiplicative generator of T, i.e., H is a non-decreasing function from the unit interval to itself and it holds true that T(x, y) = G t- ll(G(x). G(y)). On the other hand, if H is a weak multiplicative generator o f T then F: [0, 1] --* [0, oo] defined by F (x) := e -rex) is a weak additive generator of T.
Proof. Straightforward verification.
[]
7. Transformations of the minimum The main idea of this paper is to find new t-norms in the following form: g[T(f(x),f(y))]
if x, y e [0, 1),
(20)
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
I
P
o
¼
o
o
0
0
0
0
0
0
81
12
4
0
I
1
Fig. 5. Discrete Lukasiewicz t-norm.
J O
J
1 4
min(z,u)
0
a
1
bc
a
0
0
'
O I
i
a T { 4)
O I
1
r quadr
Fig. 6. Fibrations of the m i n i m u m t-norm.
where f is a pseudo-automorphism. In order to have the associativity property and monotonicity, it was useful to suppose that g = f t - 1 j which is the (unique) left-continuous right-pseudo-inverse off. In order not to have the value T ( f ( x ) , f ( y ) ) out of the domain of f t - 11 (which is the range of f ) we supposed that f is continuous. This property was essential for having the associativity. But in some cases even the continuity of f can be dropped out. As we mentioned in Theorem 2 we need the following conditions: (1) Let f be 'closed under T ' (i.e., the value of T ( f ( x ) , f ( y ) ) should belong to the range of f for all x, y). Then formula (20) defines a t-norm (of course, T admits the boundary condition). Notice that the pseudoinverse was defined even for the non-continuous case. It puts the question: Which are those f and T such that (1) holds? The investigation and characterization of this problem is open and out of the scope of this paper. Only two family of examples are given. The first can be considered as the discrete case of the Lukasiewicz t-norm. (This t-norm can be found in a forthcoming paper as well [2], but it had been obtained there by using a different approach.) (See Fig. 5.) The second example includes a wide family of t-norms. It is obvious that evaluating T = min, we can use any non-decreasing function f (with f(0) = 0 and f(1) = 1) and (1) holds, whence (20) defines a t-norm. Some
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particular cases are considered here: The t-norms T (") = E(min(nx, ny)) n
where n is an arbitrary natural number, and E(x) is the integer part of the real x, and T
quadr a,£
__ --
f0
if min (x, y) ~< a and max (x, y) < 1, if min (x, y) > a and max (x, y) < 1,
rain (x, y) if max (x, y ) = 1, where a ~ [0, ll, e e [0, a] (introduced by De Baets and Mesiar) is also within our framework (ifa = e): (Notice that T quadr 0.0 = TW the weakest t-norm, see Fig. 6). Remark 5. By duality, all the results have the counterpart concerning triangular conorrns.
References [1] B. Demant, Deformationen von t-Normen, ihre Symmetrien und Symmetriebrechungen, preprint. 12] C. Drossos, M. Navara, Generalized t-conorms and closure operators, Abstracts EUFIT 96 (1996). [3] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141-156. 14] S. Jenei, New family of triangular norms via contrapositive symmetrization of residuated implications, submitted. [5] C.-H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189-212. [6"] R. Mesiar, On some constructions of new triangular norms, Math. Soft Comput. 2 (1995) 39-45. 17"] B. Schweizer, A. Sklar, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961) 169-186. 1.8"] B. Schweizer, A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963) 69-81. 1.91 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983.
t,
FUllY
sets and systems ELSEVIER
Fuzzy Sets and Systems 103 (1999) 67-82
Fibred triangular n o r m s 1 S~mdor Jenei* Department of ComputerScience, E6tv6sLor&ndUniversity,Mtbeum krt. 6-8, 1-1-1088Budapest, Hungary Received May 1996; received in revised form May 1997
Abstract
In this paper, we deal with the formula g [T(f(x),f(y))] where f, g are functions mapping from the unit interval to itself and T is a triangular norm. We are interested in the conditions under which this formula yields a triangular norm. The case, when f is continuous is investigated in detail. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. These new triangular norms are the so-called 'fibred triangular norms' which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. We investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, some examples are given for the case when f is not continuous. © 1999 Elsevier Science B.V. All rights reserved.
Keywords: Oper; Ana; Triangular norm; Homomorphism; Semigroup; Additive generator function
1. I n t r o d u c t i o n
A triangular n o r m (t-norm for short) is a function T from [0, 1] 2 to [0, 1] being commutative, associative, non-decreasing in each place and T (1, x) = x holds for all x e [0, 1]. A t - n o r m T is said to be continuous if it is continuous as a two-place function. A continuous t - n o r m T is called Archimedean if T (x, x) < x is true for all x e (0, 1). Since we will deal with additive generators in this paper, we need the following representation theorem of continuous Archimedean t-norms [5]. Theorem 1. A t-norm T is continuous and Archimedean if and only if there exists a strictly decreasing continuous function f: [0, 1] ~ [0, oo] with f(1) = 0 such that
T(x, y) = f ( - 1)(f(x) + f(y)), * E-mail: [email protected].
1Supported in part by OTKA (National Scientific Research Fund, Hungary) 1/6-14144. 0165-0114/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 1 65-0 1 14(97)00 1 97-8
(1)
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
68
where ft-1) is the pseudo-inverse of f defined by f~_i)(x )
=
~ ( - I(X) (O
/f X ~f(O),
otherwise.
Moreover, representation (1) is unique up to a positive muhiplicative constant. If a t-norm T admits representation (1) then the function f is called an additive generator of T. There are several methods in the literature concerning the construction of new triangular norms from a given t-norm T. We cite here some of them. (1) Suppose that { [ a , bi]}i~r is a countable family of non-overlapping, closed, proper subintervals of [0, 1], denoted by J . Notice that either K = N or there exists ne N such that K = {1, ..., n}. With each [ a , b~] e J associate a t-norm T~. Let T be a function defined on [0, 1] 2 by am
+
I x--am
(bin
a")T~bm~-a,,'b,. - a,./
T(x, y) = min (x, y)
if (x,y)e[am, b,,] 2,
(2)
otherwise.
In this case T is called the ordinal sum of {([ai, bi], T~)}~r and each T~ is called a summand. For more details see e.g. [5]. (2) If q~is an increasing bijection of the closed unit interval then the following formula defines the so called q)-transform of T:
T~o(x,y) = (o-l(T((o(x), (o(y))), x, y~[0, 1]. Then q~ is an isomorphism between the semigroups ([0, 1], T) and ([0, 1], T¢) from the algebraic point of view. (3) Let (0 be an increasing bijection from [0, 1] to [a, 1] (ae [0, 1)). Let
To(x, y)= qo(-1)(T(q)(x), (o(y))), x, y e [ 0 , 1], where q¢-i)(x) = ~0-i (max(a, x)) for x • [0, 1] and called the pseudo-inverse of ~o. Then we obtain a semi9roup deformation of T. For more details see [7, 8]. (4) and (5) Recently, Demant [ 1] suggested two new constructions with the help of an increasing bijection ¢ from [0, 1] to [0, a] (a ~ (0, 1]). (Further investigation can be found in [6].) The first transformation is called contraction, the second one is dilatation and are defined as follows, (x, y e [0, 1]):
T~)(x, Y)
f,p~-l)(T(,p(x),
~0(y))) if max(x, y) < 1,
%
(rain (x, y)
otherwise,
fq~(T(~0t-1)(x), cpt-1)(y)))
T (~)(x, y) = [ T(x, y)
if T(x, y) < a, otherwise,
where (o(- i)(x) : = ¢p- 1(rain(a, x)). (6) Let n be a strong negation (i.e., an involutive order reversing bijection of the closed unit interval). Define the following operator:
T~,,(x,y)={:(x,y)
ifx>n(Y),otherwise.
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
69
The question is whether Tt,) is a t-norm or not for a given t-norm T. In [3] a sufficient and necessary condition was established for that. An important example the 'nilpotent minimum' was found (T = min, n is an arbitrary strong negation). Another example is given in [4], the nilpotent ordinal sum family. However, the complete characterization of the problem is still unsolved. In this paper a new construction which produces new triangular norm from an arbitrary triangular norm will be defined and investigated. We deal with the formula 9 [T (f(x), f(Y))l, where f, 9 are functions mapping from the unit interval to itself and T is a triangular norm. So, we generalize the o-transform, which was mentioned above in point (2). After this introduction, we give a brief preliminary on pseudo-inverses of monoton functions in Section 2. In Section 3 we are interested in the conditions under which the above-mentioned formula yields a triangular norm. The case, when f is continuous is investigated in detail in Sections 4 and 5. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. Applying this method we obtain a new class of discontinuous t-norms. These new triangular norms are the so called 'fibred triangular norms' which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. In Section 6 we investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, an open problem is formulated and some examples are given for the case when f is not continuous in Section 7.
2. Pseudo-inverses We call co a disjoint interval system if
o9 = {(ak, bk] C [0, 1 ] t k e O ,
(ak, bk]n(at, bt] = 0 if k # l}.
Let 0 denote the set of disjoint interval systems: f2:= {cole) is a disjoint interval system}. Let f : [0, 11 ~ [0, oo] be a monotone function. Obviously, such an f admits the following representation: There exists a countable set O, and disjoint subintervals (ak, bk] (keO) of the unit interval such that f is constant on each (ak, bk], strictly monotone on [O, 1]\Uk~O(ak, bk] and (ak, bk]S are maximal in some sense. Let us call the set of these intervals the constant support of f and denote it by Suppc(f). More formally, let Suppc(f) = {(ak, bk] I ke O, ak < b~, (ak, bk] c [0, 13, f is constant on each (ak, bk], f is not constant on intervals of type (] containing properly (ak, bk] and f is strictly monotone on [0, 1]\Uk~O(ak, bk]}. The following definition for non-decreasing f is due to Schweizer and Sklar (for more details, see [91). The definition for the non-increasing case is an easy generalization. Let f * : [0, 11 ~ [0, 11 be a function fulfilling the following conditions: (1) f * ( f ( O ) ) = O, f * ( f ( 1 ) ) = 1.
(2) If y is in the range of f, then f * (y) is in f - l ( y ) . (3) If y is not in the range of f, then f*(y) = sup {x If(x) < y} = inf{x [f(x) > y}. If f is non-increasing, then one has to change (3) with the following line: (Y) If y is not in the range of f , then f * (y) = inf{x If(x) < y} = sup {x If(x) > y}. Then f * is called a quasi-inverse of f.
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s. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
Generally, such an f * is not unique. But there is a unique left-continuous (right-continuous) f * for a given non-decreasing (non-increasing) f. We need this (unique) quasi-inverse as we will see later. Consider the following definition: If f is non-decreasing then let ft-ll(y) =
1 sup{x:f(x) < y}
if y >~f(1), if ye[0,f(1)].
(3)
If f is non-increasing then let ft-lJ(y) =
0 sup{x: f(x) > y}
if y ~> f(0), if y e [0,f(0)].
(4)
One can see easily that the function defined by (3) (4) is a quasi-inverse of f i.e., it fulfills (1)-(3) ((1)-(3')). Obviously, if f is an additive generator function then f t - q coincides the usual pseudo-inverse f(-l). Throughout this article we call the function defined by (3) or (4) the pseudo-inverse of f The name is motivated by the fact that a function f may have many quasi-inverses but has a unique pseudo-inverse (as in the case of additive generators). Proposition 1. If f is non-decreasing (non-increasing, respectively) then f t - 1 j is the unique left-continuous (right-continuous)function given by (3) (4)for which foft-11[Rtf ) = id~(y) may hold. (2) I f f is continuous and non-decreasing (non-increasing respectively) then its pseudo-inverse has the following form: f t - al(y) = inf{x:f(x) = y},
( f t - lj(y) = sup{x:f(x) = y})
if y~[0, f(0)], and f t - l j is strictly increasing (decreasing) on [f(0),f(1)] (If(l), f(0)]). In this case f o f t - 11]R(:) = idR(f) holds. (3) I f g is a monotonic continuous function and Suppo ( f ) = Suppo( g) then g oft-1] o f = g. Proof. We prove only (3) the proof of the rest can be found in [9]. Suppose that Suppc(f)= Supp~(g). If x¢Suppo(f) then f[-11 o f ( x ) = x, whence g of t-ll o f ( x ) = g(x). If x ~(ak, bk] i~ 0 then ft-11 of(x) = ak, whence g oft-al of(x) = g(ak) = g(x) by right-continuity. []
3. Conditions Theorem 2. Let T be a t-norm, f a non-decreasing function from the unit interval to itself with f(1) = 1 and f (O) = O. Let H c [0, 1] with R( f ) c H and g : H ~ [0, 1] a non-decreasing left-continuous function fulfilling (1) f°gl~(y)= idRty). Further, suppose that f is closed under T, i.e., (2) the value T(f(x), f(y)) should belong to the range of f for any x, y ~ [0, 1]. Define TI(x, y) : [0, 1] ~ [0, 1] as follows: ~g[ T(f(x),f(y))] TI(x' Y) -- (min(x, y) then Ty is a t-norm.
if x, y~[0, 1), if max(x,y) = 1,
(5)
S, Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
0'"
~
71
f(x)
I
I
y
the value of the pseudoinverse at z =f(y) Fig. 1.
Proof. Boundary condition and symmetry are straightforward. Associativity (i.e., Ty(Ty(x, y), z) = Tf(x, Ty(y, z)) holds for x, y, z e [0, 1]) is obvious if any of the variables x, y, z is 1. If x, y, z ~ [0, 1) then
Ty(Ty(x, y), z) = g [ T ( f o g[T(f(x),f(y)),f(z))]]
= (from (1) and (2))
= g[T(T(f(x), f(y)), f(z))] = g[T(f(x),f(y),f(z))]
= (similarly) = Ty(x, Ty(y, z)).
As an alternative proof for the associativity one can check easily that [9, Theorem 5.2.1] proves it in a more general setting. Concerning the monotonicity, let y e [0, 1] and 0 ~< xl < x2 < 1; then Ty(xl, y) <~ Ty(x2, y) holds obviously since f and g are non-decreasing. If 0 ~< xl < x2 --- 1 then in case of y = 1 we obviously have that
Ty(xl, y) <<.Ty(x2, y) = y.
(6)
If y < 1 then (6) holds if and only if
g[T (f(xl),f(y))] <<.y.
(7)
But if we have T(f(xl),f(y)) = f(y) (for instance iff(x~) = 1) then (7) holds if and only if O is left-continuous (see Fig. 1). [] Remark 1. Having a strict t-norm T (e.g. the product t-norm) and a function f(with f(0) = 0, f(1) -- 1) which is strictly increasing with one jump (discontinuity at the point a), define g as one of the pseudo-inverses of f ( a s one of f*'s). One can check easily that T: defined in (5) will not be associative. (One has to choose x, y bigger, z smaller than a in such a way that T(x, y) should be in [ f ( a - ) , f ( a + ) ] \ f ( a ) . ) This example points out the importance of condition (2) in Theorem 2. Remark 2. Let us see what happens if we do not define the values of Ty separately in the boundary. Then we would have the following: Ty(x, 1) --- 9 [ T (f(x),f(1))] = 9[T(f(x), 1)] = 9 of(x) and the last expression is x if and only if xq~Supp~(f). Therefore, we really need to redefine the value of Ty in the boundary if and only if Supp¢(f) ~ {0}, i.e., if and only if there exist at least two different points x, y in [0, 1], where f(x) = f ( y ) .
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
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Remark 3. If T is strict at the boundary O.e., T(x, y) ~ y, whenever x < 1) or f is strict at 1 (i.e., f(x) > f ( 1 ) if x is close to 1) then (7) holds even if # is not left-continuous. Summarizing what we obtained so far on 9: if T is an arbitrary t-norm then one has to choose # = f [ - u . If we restrict our attention to the t-norms which are strict at the boundary or if we restrict our attention to the f ' s which are strict at 1 then one can choose g as any of the f*'s. We want a transformation which produces a t-norm from every t-norm whence we will choose # = f [ - 1 [ in the sequel.
Corollary 1. Suppose that the conditions of Theorem 2 hold and f is strictly increasing. Then Tz(x, y) = g [ T ( f ( x ) , f ( y ) ) ] is a t-norm for any t-norm T. Proof. Taking into account Remark 2, the statement is obviously true.
[] '
4. T-norms generated by pseudo-automorphisms Definition 1. We call q~: [0, 1] ~ [0, 1] a pseudo-automorphism of the unit interval if it is a non-decreasing continuous function with q~(0) = 0 and q~(1) = 1. Let us denote the set of pseudo-automorphisms by Autps [0, 1]. Let us investigate the assumptions of Theorem 2 more carefully. Ty has to be associative. Therefore, a good choice for 9 is any f * by virtue of Proposition 1. Tz has to be monotone at the boundary; therefore, we have to choose g = f [ - i ] which is one of the f*'s. Tz has to be associative so we need condition (2). Notice that the continuity of f yields it as we will prove in Corollary 2. This corollary stands in the focus of our interest in the sequel and then in Section 7 we will drop out the continuity of f a n d will investigate the general case.
Corollary 2. Let ¢p ~ Autps[O, 13. Let us define T~: [0, 1] 2 ~ [0, 1] by ~(pt-'](T(q~(x), q0(y))) /f x, y e [ 0 , 1), Te(x, y) = ( m i n ( x , y) /f max(x, y) = 1.
(8)
Then T~ is a t-norm for any t-norm T (and called the t-norm generated by the pseudo-automorphism tp).
Proof. Since ~0 is continuous, therefore R(q~) = [0, 1] and condition (2) in Theorem 2 holds automatically. Condition (1) in Theorem 2 holds by using Proposition 1.
[]
Obviously, it is meaningless to talk about the range of a t-norm since it is always the closed unit interval. Therefore, we need to define it in a bit different way. Definition 2. Let T be a t-norm. Then the range of T (denoted by R(T)) is defined as follows: R ( T ) : = R(Tlto. 1)2) i.e., it is the range of the restriction of T to the half-open unit square, [0, 1) × [0, 1).
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
I
77
I
1. al
bl
2.
3. al
bl
4. al
F
bl
liftthis part
5. al
b~
6. al
b~
Fig. 3. H o w a fibred t - n o r m arises from its factor t-norm.
exactly at the intervals of the interval system co. Then we have to make as many lifts as again the cardinality of co in order to obtain the proper range of Tpro. This range is [0, 1]\co by using Proposition 2 and finally, the boundary has to be redefined. Now, we have a clear picture about the graph of a t-norm which was generated by a pseudo-automorphism. In the light of Theorem 3 first we have to rescale the axes (with the automorphism) then we have to change the graph in the way which was described in this proceeding. Remark 4. Let (a, b] e co. Then Tvr ~ fulfills the following properties (by the definition of the 'projection generation' and by Proposition2): T ( z l , x ) = T ( z 2 , x) whenever x e [ 0 , 1 ) and zl, z2e(a,b], (a, b ] n R ( T ) = O. Consider the opposite direction: Let T be a t-norm which fulfills the following two properties: (1) There exists (a, b] c [0, 1] such that T (zl, x) = T(z2, x) whenever x e [0, 1) and zl, z2 e(a, b]. (2) (a, b]c~R(T) = O. The question arises now: Is T a fibred t-norm (generated by a projection pro,), where (a, b] e co? The answer is straightforward and confirmative taking into account Example 1. With the help of it one can define the
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
78
'inverse vibration' easily. As a consequence, we obtain that the factor t-norm can be determined easily. It is simply the value of the 'inverse fibration', where ~ois determined by the union of the intervals (a, b] having the above mentioned two properties. Example 2. If the intervals of co almost cover the unit interval (i.e., ~(~o) is close to 0) then the generated t-norm is matrix-like. This may lead to the observation that any many-valued 'and'-model which uses an associative 'and'-matrix can be approximated by a fuzzy t-norm Tp,~ (see Fig. 4).
6. Weak additive generator functions Definition 3. Let F: [0, 1] ~ [0, oo] be a non-increasing continuous function with F(1) = 0. Then F is called a weak (additive) generator. Now, we define the following two-place function: G : [0, 1] 2 -o [0, 1], ~min (x, y) if max(x, y) = 1, G(x,y) = [Ft_ll(F(x ) + F(y)) if x, ye[O, 1).
(15)
Proposition 4. The two-place function defined by (15) is a t-norm for any weak generator function F.
at
b, a:
b~ a:~
I~ a~
b4 1
%
3
0
!
2
3
2
0
I
1
2
1
0
0
i
J
0
0
0
0
0
1
2
3
0
tile matrix
/ / I}
F i g . 4. T - n o r m
approximation
of a matrix.
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
73
Proposition 2. I f T is continuous then the range of T~ is [0, 1]\Suppc(q)). Proof. The statement is obvious considering the properties of the function (p and (pt- 11.
5. t-norms generated by projections This section is devoted to the deeper study of t-norms generated by pseudo-automorphisms. The fibred t-norms are defined here. These are t-norms generated by special type of pseudo-automorphisms, the so-called projections. It will be clear that every t-norm which was generated by a pseudo-automorphism is a fibred t-norm. That is, it can be generated by a projection and this fact brings us closer to the understanding what happens with the graph of a t-norm T when we generate T~ from it. This clear picture is due to the simple form of projections. Take an element co from f2. Let us denote ~ ( ~ ) = 1--~,kEo(bk- ak). We define the following functions: 2:[0, 1]--*[0, 1]
2(x)= ~ (bj-aj), j:bj
pro,: [0, 1 ] - , [ 0 , 1 ] , ak--2(ak) pro,(x) =
if x e ( a k , bk] for some k e O ,
¢(co) x-~(x) 3((0)
(9) otherwise.
;[: [o, I] -~ [o, U, 2(x) =
~
(b3 - as),
j:pr~(bj) < x
fibo,: [0, 1] --* [0, 1], fib~,(x) = x¢(og) + Z(x),
and, finally, i: f2 ~ Autps[O, 1]
i(co) =pro.
It is easy to check that pro, is piecewise linear and the steep of the lines are 0 or a fix positive number. An example for the functions defined above can be seen in Fig. 2. One can check easily that thus defined pro is continuous, non-decreasing function with pro,(O) = 0 and pro,(1) = 1, hence, pro, is a pseudo-automorphism of [0, 1]. Such a pseudoautomorphism will be called a projection.
74
S. Jenei / Fuzzy Sets and Systems 103 H999) 67-82 pr(x)
Y
fib(x)
Y\ lha-
/
1
/
/ I
~2~ • a~
br
I
I
al bl a2
t~
b~3
t
t >
ba 1
j
i f
J
J
x
t >
1
x
Fig. 2. Projection and fibration functions•
3. fibo = pr [- 1]
Proposition
Proof. Let y~[0, 1]. Then we have prto-1](y) = inf{x: pro(x) < y} = inf{x: pro,(x) = y} = inf{x: pro(x) = y, x¢(ak, bk](ieO)}, by using that pro is continuous and constant on each (ak, bk]. Since pr,o is strictly increasing on [O, 1]\Oi~o(ak,bk] we have that x=prt-~](y) is the unique solution of the equation x -
,a.(x)
-Y on [0, 1]\Ui~e(ak, bk]. The only thing to be checked now is that x =fibo(y) satisfies this equation. Indeed, one can check it easily since pr~-lj(y) = x, therefore y = pro(x) using Proposition 1 and we have 2(y) = ~,j:pr.(bj)
~fibo[T(pro(x), pro(y))] TP'w(x' Y) = ~min (x, y)
if x, y e [0, 1), if max (x, y) = 1
(10)
is a t-norm. Proof. pro is a pseudo-automorphism and one can use Corollary 2. We call T the factor t-norm of Tp,o and Tp,o is called the fibred t-norm o f T with respect to the interval system co. This name is based on the observation that all the points of [ak, bk] have the same behavior (i.e., T(X1, y) = T ( x 2 , y) if xl, x2 ~ [ak, bk] and y~ [0, 1) due to the properties of pro.
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S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
More formally, define a binary relation R on the unit interval by the following rule: x R y ¢~ x and y belong to the interval [ak, bk] for some k e O. Obviously, it is an equivalence relation and T is the 'factor t-norm' on the quotient set (generated by R). Another interpretation is given as follows: from the semigroup-theoretic point of view a t-norm (if we do not consider the boundary) is a homomorphic image of its fibred t-norm T~, i.e., ~0is a homomorphism from the semigroup ([0, 1), T~) to ([0, 1), T). Indeed, T~o(x, y) = q~t-i]o T(q)(x), ~o(y)) yields q~o T,p(x, y ) = T ( o ( x ) , q)(y)) by using Proposition 1. Theorem 3. F o r a given pseudo-automorphism
q~ there exists a projection pr and an automorphism
~b
such that
Proof. Let pr = i o Suppc(q2) and let O(x) = q)° pr {- ll(x).
(11)
Then ~ is an automorphism of the unit interval. Indeed, ~ ( 0 ) = ~0oprt-ll(0)= ~0(0)=0 and ~,(1) = ~ooprt-il(1)= ~ o ( m i n { ( z : p r ( z ) = 1 } ) = ~0(z*)= q~(1)= 1 since ~ and pr have common constant support. is left-continuous, since it is composition of two left-continuous non-decreasing functions. The right-continuity is straightforward in the continuity points of pr t- 11. If x is not a continuity point of pr E- 11 then pr t- ll(x) = ak for some k and right-continuity follows from the following line:
lim O(t)= lim q2o p r t - ll(t) = I~+X
t-~+X
q)( ~mxPrt-1](t)) = qo(bk) = q)(ak) = O(x ). t
So ~ is continuous. ~b is strictly increasing, since pr t - l l is strictly increasing and ~o is strictly increasing on the range of pr {- 1]. So tp is an automorphism of the unit interval. Now, we have to check that Oopr=¢
and
prt-ilo$-i=¢t-ll.
(12)
Indeed, from (11) we have that ~ko pr(x) = ~o o pr t- 11 o pr(x). The right-hand side is ~o(x) by Proposition 1. On the other hand ~b-1 = po ¢pt-ll since ~ - i o @(x) = pro ~0t- 1] oq~)oprt-1](x ) = x by using Proposition 1. Finally, T~(x, y) = o2t - 11 o T(q)(x), @(y)) = pr t- il o ~b- lo TOp o pr(x), (To)pr(x, y) by using (12). []
(13)
o p r ( y ) ) = T~,(pr(x), p r ( y ) ) =
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
76
Example 1. Let co = { (a, b] [a ~< b} and T be a t-norm. Easy verification shows that Tp,. has the following form:
•T
x
y
if (x, y) ~ [0, a] x [0, a],
if (x, y)m(a,b]x(a,b],
T¢o(X, y) =
T
x
a
if (x, y) m [0, a] x (a, b],
T
a
y
if (x, y) e [a, b] x [0, a],
x y - (__~ b - a)) 4(09) • T ( _~),
if (x, y) ~ [0, a] x [b, 1),
~(¢°)" T(x-(b-a)-~-~-) ' ~o)Y )
if (x, y)e(b,
,(¢o)
• T ( -(~o), a Y -(b -f(~- a))
~(°9)'T(x-(b-a)~) (co). T
if (x, y)~ (a, b] x(b, 1),
' ~(co)a)
if
x-(b-a) y--(b_-a)~ -~] ' ~(o~) ,]
~(~o). Tf X - (b - a)
Y ,
1)x [O,a],
(b
(x,y)~ [b, 1] x(a, b],
if (x,y)e(b, 1)x(b, 1) and, ~(og).T(X-(b-a) -~-(w-) ' y-(b-a)) --~) <<.a,
a)]\ + (b - a) /
if (x,y)e(b,
1)x(b, 1) and,
~(¢o).T(X-(b-a) -~ min(x, y) > a
,
y-(b-a)) -~--(~
> a
if max (x, y) = 1.
(14) It is obvious that in case of a = b, Tp,~ = T. Let us analyze the graph of Tpro. We are interested to k n o w how it arises from the graph of T. First, we have to compress the graph of T with 4(o9) like in the case of a 'summand' at the representation of the continuous t-norms [5]. Hence, we have a t-norm on I-0, ~(o9)] x [0, 4(09)]. Now, we have to cut this graph along the lines {(x, y): x ~ [0, 1], y = a} and {(x, y): y ~ [0, 1], x = a}. Then one has to shift the obtained left and the upper pieces to left and up, respectively with b - a. One has to fill the fibers (i.e., [a, b] x [0, 1] and [0, 1] x [a, b]) as defined above. Then we have to cut our graph with a plane which is parallel to the d o m a i n [0, 1] x I-0, 1], at the high of a and have to lift the higher part of the graph with b - a. Notice that at this step the graph attains its range in accordance with Proposition 2. Finally, the b o u n d a r y has to be redefined. This process can be seen in Fig. 3, when T (the factor t-norm) is the Lukasiewicz t-norm. The lines in the unit square mean levels. In general, when 09 consists of m o r e intervals then the process is almost the same. We have to compress T with ~(¢o). We have to make as m a n y cuts and shifts as the cardinality of co in order to obtain the fibers
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
79
Proof. We prove the associativity, the rest is obvious. G(G(x, y), z) = G(x, G(y, z)) holds obviously if one of the variables is 1. Suppose that x, y, z e [0, 1]. By Proposition 1 we have G(G(x, y), z) = F t- ll(F(x) + F(y) + F(z)) = G(x, G(y, z)).
[]
Definition 4. We say that a t-norm T admits the weak generator function representation if T(x, y) = F t- ll(F(x) + F(y))
(16)
holds for all x, y • [0, 1) and for a weak generator function F. Theorem 4. A t-norm T * admits representation (16) if and only if there exists a continuous Archimedean t-norm T and a pseudo-automorphism of the unit interval q) such that T * = T~. Proof. (i) Suppose that T * ( x , y ) = F t - i l ( F ( x ) + F(y)) holds for x , y • [ 0 , 1) and for a weak generator function F. Let us define pr:= i°Suppc(F)
and
f : = Fopr t-j1
(17)
Then f is a generator function. Indeed, f(1) = 0,f(0) = F(0). f is strictly decreasing since pr t- 11 is strictly increasing and F is strictly decreasing on the range of pr I- 11 f is left-continuous. Indeed, pr 1-11 is left-continuous and non-decreasing, F is continuous and f is the composition of them. f i s right-continuous at the continuity points of pr 1-11, if x is not a continuity point of pr t- 11 then lim f ( t ) = F ( lim prt-ll(t)) = r ( b k ) = r ( a k ) = f ( x ) . t~+x
\t~+x
/
Therefore, f is continuous. Let T(x, y ) : = f ( - l)(f(x) +f(y)).
By using (17) and Proposition 1 we have that f o p r = F o p r t - l l o p r = F.
(18)
pr t- 11oft-a) is left-continuous, and right-inverse of F on the range of F. Indeed, if x • R(F)( = R ( f ) ) then by
using Proposition 1 we have that F o pr t- 11o f t - 1) = f o pro pr t- 11of(- l)(x ) = f o f ( - i)(x ) = x.
Therefore, F t- 11 = pr t- 11oft- 1), by using Proposition 1.
(19)
80
s. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82 Finally, by using (18) and (19) we have Tpr(x, y) = pr t- 11of(- i)(fo pr(x) + fo pr(y)) = F t- il(F(x) + F(y)) = T * (x, y).
(ii) Suppose that T is a continuous Archimedean t-norm, q)eAutps[O, 1] and T * = T,. Then T(x, y) = f t - 1 ) ( f ( x ) + f ( y ) ) holds by Theorem 1. Let us define F : = f o ¢p. Then F is a weak generator function, and F t-ll = q,t-ljof(-1), since it is left-continuous and is the right-inverse of F. From these we have T*(x, y) = T~(x, y) = q~t-ilof(-1)(foq)(x) + f o q~(y)) = Ft-ll(F(x) + F(y)).
[]
Corollary 4. The weak additive generator function is unique up to a positive multiplicative constant. Proof. Straightforward by using Theorems 1 and 4.
[]
Proposition 5. Suppose that O, 1 is not in the closure of Supp~(q)). Then T is Archimedean if and only if T, is Archimedean. Proof. Suppose that T~, is Archimedean and (on the contrary) T is not Archimedean. Then there exists x e (0, 1) such that T (x, x) = x. Let y = ~0t- ll(x). Then y e (0, 1) since 0, 1 is not in the closure of Suppc ((o), and T~o(y, y) = q)t-11T(q)(y), (o(y)) = q)t-11T(x, x) = q)t-11(x ) = y, which is a contradiction. Now, suppose that T is Archimedean and (on the contrary) T~o is not Archimedean. Then there exist xe(0,1) such that T ~ ( x , x ) = x . (Let us define y : = (p(x)). It is equivalent to q)t-lJT(~o(x),~o(x))= x ~ q~r- 11T(y ' y) = x =~ T(y, y) = y. This y belongs to (0, 1), since 0, 1 is not in the closure of Suppc(q)), and this leads to a contradiction to the Archimedean property of T. [] Similar to additive generators we have the following proposition:
Proposition 6. I f F is a weak additive generator function of T then H: [0, 1] ~ I-0, ~ ] defined by H(x) := - log(F(x)) is a weak multiplicative generator of T, i.e., H is a non-decreasing function from the unit interval to itself and it holds true that T(x, y) = G t- ll(G(x). G(y)). On the other hand, if H is a weak multiplicative generator o f T then F: [0, 1] --* [0, oo] defined by F (x) := e -rex) is a weak additive generator of T.
Proof. Straightforward verification.
[]
7. Transformations of the minimum The main idea of this paper is to find new t-norms in the following form: g[T(f(x),f(y))]
if x, y e [0, 1),
(20)
S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
I
P
o
¼
o
o
0
0
0
0
0
0
81
12
4
0
I
1
Fig. 5. Discrete Lukasiewicz t-norm.
J O
J
1 4
min(z,u)
0
a
1
bc
a
0
0
'
O I
i
a T { 4)
O I
1
r quadr
Fig. 6. Fibrations of the m i n i m u m t-norm.
where f is a pseudo-automorphism. In order to have the associativity property and monotonicity, it was useful to suppose that g = f t - 1 j which is the (unique) left-continuous right-pseudo-inverse off. In order not to have the value T ( f ( x ) , f ( y ) ) out of the domain of f t - 11 (which is the range of f ) we supposed that f is continuous. This property was essential for having the associativity. But in some cases even the continuity of f can be dropped out. As we mentioned in Theorem 2 we need the following conditions: (1) Let f be 'closed under T ' (i.e., the value of T ( f ( x ) , f ( y ) ) should belong to the range of f for all x, y). Then formula (20) defines a t-norm (of course, T admits the boundary condition). Notice that the pseudoinverse was defined even for the non-continuous case. It puts the question: Which are those f and T such that (1) holds? The investigation and characterization of this problem is open and out of the scope of this paper. Only two family of examples are given. The first can be considered as the discrete case of the Lukasiewicz t-norm. (This t-norm can be found in a forthcoming paper as well [2], but it had been obtained there by using a different approach.) (See Fig. 5.) The second example includes a wide family of t-norms. It is obvious that evaluating T = min, we can use any non-decreasing function f (with f(0) = 0 and f(1) = 1) and (1) holds, whence (20) defines a t-norm. Some
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S. Jenei / Fuzzy Sets and Systems 103 (1999) 67-82
particular cases are considered here: The t-norms T (") = E(min(nx, ny)) n
where n is an arbitrary natural number, and E(x) is the integer part of the real x, and T
quadr a,£
__ --
f0
if min (x, y) ~< a and max (x, y) < 1, if min (x, y) > a and max (x, y) < 1,
rain (x, y) if max (x, y ) = 1, where a ~ [0, ll, e e [0, a] (introduced by De Baets and Mesiar) is also within our framework (ifa = e): (Notice that T quadr 0.0 = TW the weakest t-norm, see Fig. 6). Remark 5. By duality, all the results have the counterpart concerning triangular conorrns.
References [1] B. Demant, Deformationen von t-Normen, ihre Symmetrien und Symmetriebrechungen, preprint. 12] C. Drossos, M. Navara, Generalized t-conorms and closure operators, Abstracts EUFIT 96 (1996). [3] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141-156. 14] S. Jenei, New family of triangular norms via contrapositive symmetrization of residuated implications, submitted. [5] C.-H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189-212. [6"] R. Mesiar, On some constructions of new triangular norms, Math. Soft Comput. 2 (1995) 39-45. 17"] B. Schweizer, A. Sklar, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961) 169-186. 1.8"] B. Schweizer, A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963) 69-81. 1.91 B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, 1983.