A characterization of the ordering of continuous t-norms

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 86 (1997) 189-195

A characterization o f the ordering of continuous t-norms 1 E r i c h P e t e r K l e m e n t * , R a d k o M e s i a r 2, E n d r e P a p 3

Fuzzy Loyic Laboratorium Linz-Hayenberg, Institut fiir Mathematik, Johannes Kepler Universitgit, A-4040 Linz, Austria Received December 1993; revised November 1995

Abstract Given two t-norms/'1 and Tz, it is quite often difficult or even impossible to check directly whether TI ~
Keywords." t-norm; Comparison of t-norms; Additive generator; Ordinal sum

1. Introduction Triangular norms (t-norms) and the corresponding t-conorms are used in several branches of mathematics in different manners, e.g., in probabilistic metric spaces, many-valued logic, fuzzy sets, decomposable measures and their applications [3, 7, 9, 11, 17]. A tnorm T is a two-place function from the unit square into the unit interval which is associative, commutative, non-decreasing, and fulfills, for all x E [0, 1], the boundary condition T(1,x) = x. Its dual function S defined via S(x, y ) = 1 - T(1 - x , 1 - y ) is called a t-conorm (see [13]). * Corresponding author. E-mail: [email protected]; www: http://www.flll.uni-linz.ac.at/klement/my.html. l This paper was written during a visit of the second and third author in Linz. 2 Permanent address: Slovak Technical University, SK-81368 Bratislava, Slovakia. 3 Permanent address: Institute of Mathematics, University of Novi Sad, YU-21000 Novi Sad, Yugoslavia.

The following are the most important t-norms (see Fig. 1 ), together with their corresponding t-conorms:

TM(x, y ) = min(x, y), rp(x,y)=x.y,

SM(x, y ) = max(x, y), Sp(x,y)=x+y-x.y,

TL(x,y) = max(x + y -- 1,0), SL(x, y ) = min(x + y, 1 ), Tw(x, y ) =

min(x,y) 0

if m a x ( x , y ) = 1, otherwise,

Sw(x, y ) =

max(x, y ) 1

if rain(x, y ) = 0, otherwise.

The function Tw is an example of a non-continuous t-norm. In what follows, we restrict ourselves to continuous t-norms and t-conorms. We are now interested in the question whether, given two t-norms Ti and T2, T1 is weaker than T2 or, equivalently, T2 is stronger than Tt (in symbols

0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved SSDI 01 65-0 11 4(95)00407-6

190

E.P. Klement et al./Fuz:y Sets and Systems 86 (1997) 189-195

,'."

1

"

0.8

o.e

O.a

0.4

0.4

0.2

0.~

02

04

0

02

04

06

'--0

02

i

04

i

06

08

1

8

!

1

O.e

0.1

0.6

0.6

0.4

OA

0.2

0.2

0

0

~

0

02

04

06

L

•

i

i

Fig. 1. Three-dimensional plots of the four most important t-norms: TM (top left), Tp (top right), TL (bottom left), and Tw (bottom right).

T1 ~
The monotonicity and boundary conditions also imply Tw ~
then Fig. 2 shows that T and Tp are incomparable. In many cases it is difficult or even impossible to decide whether Tl ~
([12, 13], see also Theorem 3.1) simplify this task in the sense that they involve only one-place functions, but still in two arguments (using, e.g., subadditivity). The inequality Tt <~T2 is a necessary but not sufficient condition for T1 being dominated by T2 (in symbols TI << 7"2) in the sense of Tardiff [15]: T1 (T2(a, b), T2(c, d ) ) ~ T2(T1 (a, c), TI (b, d ) ) ,

which arises naturally in the construction of Cartesian products of probabilistic metric spaces.

2. Continuous t-norms and t-conorms

Continuous t-norms (t-conorms) were studied extensively by Ling [8], among others. A continuous t-norm T is called Archimedean if T(x,x) < x for

191

E.P. K l e m e n t et a l . / F u z z y Sets and S y s t e m s 86 (1997) 189- 195

in the following way:

S ( a , b ) = y * ( g ( a ) + 9(b)),

i! 0.8

where for all x E [0, cx~] g * (x) --- 9 - I ( min(x, 9( 1 ))).

/iII 0.~

Summarizing the results of Mostert and Shields [ 10] and using the notation of Frank [4], we have the following characterization.

0.4

Theorem 2.2. T & a continuous t-norm if and only if there is a unique countable family {]Tk,/~k[}kE~ of pairwise disjoint open subintervals of the interval [0, 1] and a unique family of continuous Archimedean t-norms {Tk}kcx such that T is the ordinal sum of {(Tk, ]3k, Tk)}kEX, i.e.,

0.2

_ ~ ' ~

. ./ 0.2

.

.

.

.

0.4

.

.

.

0.6

.

. 0.8

.

.

. I

T(x, y)

y

Fig. 2. Two incomparable t-norms: plots of the functions x ~-~ T l , ( x , x ) (dashed curve) and x ~

T(x,x).

all x E ]0,1[. A continuous t-norm T is strict if T(x, y) < T(x,z) whenever x E ]0, 1[ and y
Theorem 2.1. T is a continuous Archimedean t-norm if and only if there is a continuous strictly decreasin9 function f : [0, 1] --+ [ 0 , ~ ] such that f ( 1 ) = 0 and

T(a,b) = f * ( f ( a ) + f ( b ) ) , where f * is the pseudoinverse o f f , i.e.,for all x E

[0, ~ ] , f*(x) = f-l(min(x, f(0))). T is strict if and only iff(O) = +e,z, i.e., f is bijective and f * = f - 1 . The function f is called an additive generator of T and it is unique up to a positive multiplicative constant. Note that the dual t-conorm S of an Archimedean tnorm T with generator f is generated by the additive generator 9 given by

g(x) = f ( l - x)

)

if x, y ~ [~k, &], min(x, y)

otherwise.

An example of a simple ordinal sum is shown in Fig. 3.

3. Comparison of continuous Archimedean t-norms Now, let 7"1,T2 be two continuous Archimedean t-norms with additive generators f l and f : , respectively. The full information about Ti is contained in y~ and, as a consequence, it should be possible to decide whether Tj is weaker than/'2 only by means of fl and f z . The first step into this direction was done by Schweizer and Sklar [12, Theorem 7], who proved that if both T~ and T2 are strict, then TI ~< 7"2 if and only if the composite h = f l o f 2 1 is a subadditive function, i.e., if for all s, t ~>0

h(s + t)<~h(s) + h(t). It is known from [8], that for all Archimedean t-norms Tl, T2 we have TI = /'2 if and only if the composite h = f l o f 2 1 is a linear function on [ 0 , f 2 ( 0 ) [ . The following result is due to Schweizer and Sklar [13, L e m m a 5.5.8]:

E.P. Klement et al./Fuzzy Sets and Systems 86 (1997) 189 195

192

~.:~.................................................................... ..,....

0.6 O.II 0.4 0.,I 0.2

0.2

o

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

Fig. 3. Three-dimensional plot (left) and contour plot (right) of the ordinal sum of {(0.1,0,45, Tp), (0.55, 0.9, TL)}.

Theorem 3.1. Let Tj,T2 be two Archimedean tnorms with additive generators f l, f2, respectively. Let h = f l o f 2 1 be the composite function defined on [0,f2(0)]. Then Ti is weaker than 1"2 if and only if h is subadditive. An important class o f subadditive functions is the set o f all concave, continuous and strictly increasing functions h defined on a closed interval [O,M] with M E ]0, oo], fulfilling h(O) = O. In order to see this, it is enough to exploit a

~

a a+b"

w

b (a + b) + a---~---b " 0

and b b : - a- +

Note that the subadditivity o f h on [0,M] does not imply its concavity: the continuous and strictly increasing function h : [0, 4] ~ [0, 3] given by

h(x) =

i f O ~ x ~ < 1, if 1 < x~<3, if3 < x ~ < 4 .

is subadditive but non-concave. Using Theorem 3.1, we obtain the following sufficient condition for 7'] ~< Tz: Corollary 3.1. Let Tl, Te be two continuous Archimedean t-norms with additive generators f l and f 2, respectively. I f h = f l o f - 21 is a concave function on [0, f2(0)], then we have Ti <~T2.

a

b . ( a + b ) + - f f - ~ .0

for all a,b E [0,M] with 0 < a + b<~M < +oo. In this case we get a

h(a)>~ ~

. h(a + b),

If the composite function h = f l o f 2 1 is differentiable on ]0, f2(0)[, then its concavity is a consequence o f the non-increasingness of its first derivative. Note that in such a case we get for each x E ]0, f 2 ( 0 ) [

hi(x)h(b) >>-~

x (1 + x ) / 2 x- 1

b

• h(a + b),

and, consequently, h(a + b)<~ h ( a ) + h(b). If a = b = 0, then the subadditivity is trivial. The same is true in the case a + b = M = oo, since then either a or b are infinite and, consequently,

h(a + b) = h(oo)<~h(oo) + h(min(a,b)) = h(a) + h(b).

f ' l ( f 21(x)) - f'l(U) f~(f zl(X))

f12(u)'

where u = f 2 j (x) E ]0, 1[. Then the non-increasingness o f f 2 1 reverses the monotonicity, hence h ~is nonincreasing (inx) if and only i f f ~ / f ~ is non-decreasing (in u). We therefore have shown the following result: Corollary 3.2. Let Ti, 1"2 be two continuous Archimedean t-norms with differentiable additive generI ! ators f l and f 2 , respectively. I f g = f l / f 2 is a

193

E.P. Klement et al./Fuzzy Sets and Systems 86 (1997) 189-195 on ]0, 1[, then we have

this reasoning is shorter than both the direct proof and the proof of Theorem 2 in Sherwood [14].

The duality between t-norms and t-conorms yields the following result for the comparison of two Archimedean t-conorms:

Example 3.2. Many applications deal with the Frank [4] family of t-norms {Tff}sE[0,~], where

non-decreasing f u n c t i o n Tt <~T2.

T~(x, y ) Corollary 3.3. L e t S1,$2 be two continuous Archim e d e a n t-conorms with differentiable additive generators gl a n d 92, respectively. I f g = g~/gt2 is a non-increasing f u n c t i o n on ]0, 1[, then we have

l"p(x, y) =

/

(sX - )-1)(s~ s -- 1 )

otherwise.

\

R e m a r k 3.1. Corollaries 3.2 and 3.3 can be proved also using some results of Hardy et al. [6]: as a consequence of (148) in [6] the monotonicity of gl/g2~ implies the monotonicity of gl/g2, and (105) (which uses (103) and (102)) in [6] implies the comparability of $1 and $2 and, subsequently, of Tl and T2. R e m a r k 3.2. From (103) in [6] we get another sufficient criterion for the comparability of t-norms: if the function x H [ ( f l o f ~ l )(x)]/x is decreasing, then T~ ~
Frank showed that this family is continuous with respect to the parameter s. Note that trivially T0F = TM >~ Ty for all s E ]0, cx~]. For each s E ]0, oo[, TF is a strict t-norm whose generator is given by

fs(x) =

TM(x,y) TSS~ x , s t ,Y) =

Te(x, y ) Tw(x,y)

[max(x s + yS _ 1,0)]Us

i f s = --e~z, i f s = 0, i f s = +e~, otherwise,

having, for - e ~ < s < +cx~, the additive generators -log x (1 - x S ) / s

i f s = 0, i f s E ]-cx~,0[ U ]0,+cx~[.

ss ~< TSS ~< T ~ss. It is easy to see that Obviously, T_o¢ for - e ~ < s < t < + e ~ we always have

-logx

if s =

1,

log ssx- ~- 1_ l

ifs¢

1.

T ~ is a nilpotent t-norm, its generator is given by f o~(x) = 1 - x. Then f~(u) f~(u)

_

-1 - s u logs

-

1

--(1

-

s-U),

logs

s~- 1 i.e., for each s E ]0, oo[ \{ 1 }, f ~ / f ~ is non-decreasing on ]0, 1[. The same is true for f ~ / f ~ because of f~(u)

_

-1 --

f~(u)

=

TL(x, y )

logs ( 1 +

$1 ~ $ 2 .

fax)

i f s = 0, if s = 1, i f s = cx~,

TM(x, y )

U,

-1/u

hence implying TV~ ~< T~ for all s E ]0, cx~[. Now let us prove that Ty ~< T~ whenever 1 < s < t < cx~ (the case 0 < s < t < 1 is completely analogous). Define g :]0, 1[ --~ [0, e~[ by g(u) -- f ~ ( u ) _ (s u - 1 ) t U l o g t = C . 1 - a__~__~ f~s(U) (t u - 1 )s ~ log s 1 - b u' where C = ( l o g t ) / ( l o g s ) > O and 0 < b = 1 / t < 1Is = a < 1. Then g is non-decreasing on ]0, 1[ if and only if ( 1 - b u )( - a u log a) ~>( 1 - a u )( - b u log b),

f ; ( x ) _ xt_S, f~(x)

i.e., if and only if

i.e., f ~ / f t 2 is a non-decreasing function on ]0, 1[. Hence, Corollary 3.2 implies TSS~< Tss. Note that

a u log a 1 au b"log-----~ >~ ~1 -" b

(3.1)

194

E.P. Klement

et al. IFuzzy

Sets and Systems

Put f(u) = 1 - au and h(u) = 1 - h”. Then, by the Cauchy Mean Value Theorem, for each u E IO, 1[ there exists an Y E IO, u[ such that _1 - au = f(u) -f(O) 1 - 6” h(u) - h(0)

f’(r) = -h’(r)

a’loga = F&b

86 (1997)

189-195

Proof. Since we have T ,< TM for each t-norm T, the inequality TI < T2 is equivalent to

for all k2 E KI and all x, y E IX:‘, pg)[. Now fix k2 E

This proves inequality (3.1) and, consequently, f :/f i is non-decreasing, i.e., TF
K2, x,y ~]c$‘,fig)[ such that

and choose the unique kl E KI

Then TI (x, y) d T2(x, y) is equivalent

to

4. Comparison of continuous t-norms Let T, and T2 be two continuous t-norms which, according to Theorem 2.2, can be represented in a uni ue way as ordinal sums of Archimedean t-norms, ? and {(c(f), Pp’, Tk(2))}k~~2, {@,‘)J$‘), T;%EK, respectively. If the t-norm T, has an idempotent element x which is contained in some interval ]a?), pf’[, then T, cannot be weaker than T2: since TL2’ is Archimedean, we get in this case the contradiction T2(x,x) < x = Tl(x,x). Theorem 4.1. Let TI and T2 be two continuous t-norms being represented as ordinal sums oj {(“:‘),Pf), T~‘))}AEK, and {(@),Pr), Tk(2))}k~~,, respectively, such that for each k2 E K2 there is a (unique) kl E Kl with

(4.1) Putting x a =

(2)

a:’ (2)

and

b =

Y-Q 8:) - NE”

Pk> - ‘k>

and taking into account that a, b E [0, 11, inequality (4.1) is equivalent to fk,,k?(Tk(,“(fki,lkz(a),fg,~z(b)))~ Then either the left-hand

TLj2’(a,b).

side is non-positive,

i.e.,

hkz(u)+ or it is positive, in which case we have

and let fy’

be an additive generator

of the Archi-

medean t-norm TL”. Then Tl d T2 tf and only if jar each k2 E K2 and the corresponding kl E KI,

hkz=

(1) fk, o f b.'kzo (f x’,-’

is subadditive which

on [0, f:‘(O)]

where

u = f:)(a)

[0, ,fg’(O)]. up to points u and v in

and v =

This latter inequality,

alent to the subadditivity the points fulfilling

u + v)), f:‘(b),

i.e., U,V E

however, is equiv-

of hk2 on [0, ,~~‘(O)] up to

h>(U)+

h>(U)+ where the affme defined by fk,,kz(u)

&J(U) + &(V) 3 hk>(min(f$O),

=

transformation

fk,,&:

[w

-+

[w

iS

Then the subadditivity of hk2 on [0, f:‘(O)] (for all k2 E Kl ) is sufficient to ensure Tl < T2.‘ Analogous reasoning as in Section 3 yields the following sufficient condition for the comparison of continuous t-norms:

E.P. Klement et al./ Fuzz)' Sets and Systems 86 (1997) 189-195

Corollary 4.1. Let T1 and T2 be two continuous tnorms as in Theorem 4.1. Keeping the notations of Theorem 4.1, suppose that for each k2 E K2 the function gk: defined by (1) t u gk2(U ) = ( f k , ) ( f k , , k 2 ( )) (2) t

(fk2) (u)

is non-decreasing on ]0, 1[, then we have Tl <~T2. R e m a r k 4.1. In his recent paper [5], Gottwald deals with left continuous t-norms stronger than TL (see, [5, Theorem 5]). Restricting ourselves to continuous tnorms w e can apply Theorem 4.1. Recall that f ( x ) =1 - x is an additive generator of TL. Then TL ~< T,

where T is the ordinal sum of {(~k,/~k, Tk)}k~X, if and only if for all u, v C [0, fk(0)] and for all k E K (where fk is an additive generator of Tk) we have 1 - ~k

f ~ l ( u ) + f ~ l ( v ) - f ~ l ( u + v)<~ flk - ~ " The latter condition is obviously fulfilled if the tconorm Tk is stronger than TL or i f / ~ ~< (1 + ~k )/2. Example 4.1. It is easy to see that each t-norm T with an additive generator f , where f ( 1 - x ) is subadditive, is weaker than TL, i.e., T~< TL. B y Theorem 4.1, the converse assertion is true for continuous t-norms T, i.e., T ~< TL if and only i f x ~-~ f ( l - x ) is subadditive. Note that the concavity o f f is sufficient to ensure the subadditivity of x H f ( 1 - x ) . Furthermore, each continuous t-norm weaker than TL is nilpotent. This means that neither a strict t-norm nor a t-norm with a non-trivial idempotent can be weaker than TL.

195

References [1] J. Aczrl, Lectures on Functional Equations and Their Applications (Academic Press, New York, 1969). [2] D. Butnariu and E.P. Klement, Triangular norm-based measures and their Markov kernel representation, J. Math. Anal. Appl. 162 (1991) 111 143. [3] D. Butnariu and E.P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions (Kluwer, Dordrecht, 1993). [4] M.J. Frank, On the simultaneous associativity of F(x, y) and x + y - F(x, y), Aequationes Math. 19 (1979) 194-226. [5] S. Gottwald, Approximate solutions of fuzzy relational equations and a characterization of t-norms that define metrics for fuzzy sets, Fuzzy Sets and Systems 75 (1995) 189 201. [6] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities (Cambridge University Press, Cambridge, 2nd ed., 1952). [7] E.P. Klement, Construction of fuzzy a-algebras using triangular norms, J. Math. Anal. Appl. 85 (1982) 543 565. [8] C.H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189-212. [9] R. Mesiar, Fundamental triangular norm based tribes and measures, J. Math. Anal. Appl. 177 (1993) 633 640. [10] P.S. Mostert and A.U Shields, On the structure of semigroups on a compact manifold with boundary, Ann. Math. 65 (1957) 117 143. [11] E. Pap, On non-additive set functions, Atti. Sem. Mat. Fis. Univ. Modena 39 (1991) 345-360. [12] B. Schweizer and A. Sklar, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961) 169 186. [13] B. Schweizer and A. Sklar, Probabilistic Metric" Spaces (Noah-Holland, New York, 1983). [14] H. Sherwood, Characterizing dominates on a family of triangular norms, Aequationes Math. 27 (1984) 255-273. [15] R.M. Tardiff, On a generalized Minkowski inequality and its relation to dominates for t-norms, Aequationes Math. 27 (1984) 308-316. [16] R.R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems 4 (1980) 235-242. [17] H.J. Zimmermann, Fuzzy Set Theory and Its Applications (Kluwer, Dordrecht, 1991).