Quasi homogeneous associative functions

Internat. J. Math. & Math. Sci. VOL. 21 NO. 2 (1998) 351-358

351

QUASI-HOMOGENEOUS

ASSOCIATIVE

FUNCTIONS

BRUCE R. EBANKS

Department

ofMathematics MarshallUniversity Huntington,

WV

25755,

U. S.

A.

(Received July 15, 1996 and in revied version March 18, 1997)

ABSTRACT. A

triangular norm is aspecialkindof associative function on the closedunitinterval[0,1

].

Triangularnorms(or t-norms)were introduced in thecontextofprobabilisticmetric

space

theory, and they

have

foundapplicationsalso in other areas, such asfuzzyset

theory.

Wedetermine theexplicit formsof all t-normswhichsatisfyageneralized homogeneity propertycalledquasi-homogeneity.

KEY

WORDS

AND PHRASES:

Triangularnorm(t-norm), homogeneousfunction, probabilistic metric

space.

1991

AMS

SUBJECT

CLASSIFICATION

CODES:

39B12, 54E70,39B22. 0.

INTRODUCTION

Let

I

[0, be the closedunitintervalon thereal line

R. A

triangularnorm(or t-norm)is a

map

T: I x

I-->I

satisfyingthefollowingfourhypotheses. First,

T

is associative:

T(x,

Try,

z)) T(T(x, y), z), x,y,zl;

second,

T

isnondecreasingin each variable; third,

T

iscommutative; andfourth, forall x

L

T(x, 1) x.

(0.1)

T-norms

were introducedby

Menger

as ameansof generalizingthetriangle inequalitytostatistical (later, probabilistic)metric

spaces.

Associatedwithafamily

Fpq}

ofprobabilitydistributionfunctionsis a

map

T:

I

x

I-->I

(later,at-norm)such that

Fpr(X

+

y)>_

T[Fpq(X), Fqr(Y)]

(0.2)

for all p,q,rin some

space

Sandallnonnegativereals x andy. Interpreting

Fpq(X)

as theprobability that the distance betweenp andqislessthan x, inequality (0.2) meansthat theprobability that the distance frompto ris less than x

+

y isatleast asgreatas the T-value of theprobabilities that the distance from ptoqis lessthan x and the distancefrom qto risless thany.

For

furtherinformation on thehistory, theoryandapplicationsoft-norms,see Schweizer and Sklar

[2].

35 B.R. EB.NKS

r(t, ty)

n-’{(r)

nit(.,

(0.3)

for some

map

q):I--)Iandsomecontinuousinjection

H:

I--)_{[0, oo).} Such t-norms will becalled quasi-homogeneous. If

H

in

(0.3)

istheidentity

map,

then

T

is calledhomogenous,and theforms of such t-norms areknown already (e.g.

[3]). In

thatcase,eitherT(x, y) Min(x, y)withq)(t) t,orT(x, y) xywithq)(t)

2.

We

shall obtain that resultasacorollaryof the main result of thepresent

paper,

in whichwe determine thegeneralsolutionof

(0.3)

fort-norms.

Beforeproceeding, let us observe some otherproperties oft-norms. Itis easytodeduce the followingfrom the definition oftriangularnorm:

T(x,O) T(O, x)

O,

x I,

T(x, y) <Min(x, y), x,y

I

(i.e.Min isthe "maximal"t-norm),so inparticular

T(x, x) <x, x I.

Note

also that

(0.3)

forces

H

tobe strictly increasing.

For,

settingx y 0there, we have

T(0,

0)

H-l{q)(t)

H[T(0,0)]}.

Since

T(0,0)

0, this means

H(0)

q)(t)

H(0)

for all [0, 1].

Hence H(0)

0, since otherwiseq) 1, which in(0.3)with 0 wouldyield 0 T(x, y) forall x,y

L

in violationof

(0.1).

1.

PRELIMINARIES

In

thesequel,let

/

[0, **] be the extended nonnegative real half-line. The structure of continuoust-normsisknown.

(The

continuityassumptioncanbeweakened somewhat, but that isnot relevanttothecurrentdiscussion.)

THEOREM

1.1. (Schweizerand Sklar

[2:

Sections

5.3-5.5]) Let T: I

x

I

---)

I

be a continuous t-norm.

(i)

T

satisfiesT(x,x)

<

xfor all x (0, 1)if andonlyif

T

admitstherepresentation

T(x,

y)=Y[e,(x) +

g(y)],

(.)

where

g:

I--)/

is continuous,strictly decreasingwith

g(1)

0,where f:

/--)I

isonto,continuous, strictly decreasing on

[0,

g(0)] with

f(u)

0foru>

g(0),

and where

f

g istheidentity

map

on

I

(i.e. g is a"quasi-inverse"

off).

(ii) If T(x, x) xforall xeI,then

T

Min on

I

I.

(iii) Otherwise,thesemigroup (/,

T)

isan ordinal sumof semigroups (Sk,

T/)}.

Here

each

S

k is a

proper

closed (nontrivial) subintervalof

I,

each

T:

admits arepresentation oftheform

(1.1)

on

S/

x

St,

withg/on

S

t

andf/

onto

S

t,and

T=

Min on(Ix/)

k(Sk

xSk)

[2].

REMARK. Any

t-normof the formgivenbycase (i) of Theorem 1.1 is called Archimedean. If inadditiong(0) ,,*,then

f=

g-1

and Tiscalledstrict.

We

shall use Theorem 1.1 tofind theformsofquasi-homogeneoust-norms.

We

shall also need thefollowingknown result.

THEOREM

1.2.

(See

e.g. Aczel and Dhombres [4: Chapter 15, Theorem

1].

The general solution of

g(tx) a(Og(x)

+

tgt), (1.2)

QUASI-HOMOGENEOUS ASSOCIATIVE FUNCTIONS 353

g(x)

l/x)

+c,a(x) 1,b(x)

t(x);

g(x) c, aarbitrary,b(x)

c[

a(x)]; (1.4)

or

g(x) cm(x)

+

d, a(x) mfx),b(x) d[ re(x)];

for all x

L

wherecand d arcarbitraryconstants,

I---

+

isanarbitrarysolution of thelogarithmic functionalequation

/(xy)=/(x)

+/lv),

x, y

I,

andm:

I--

+

is anarbitrarysolutionofthemultiplicativefunctionalequation

m(xy) re(x)re(y), x, y I.

,2.

DETERMINATION OF

QUASI-HOMOGENEOUS

T-NORMS

We prove

firstthatquasi-homogeneity impliesthat

T

mustbe continuous and thatonlycertain formsarepossiblefor

p.

LEMMA

2.1. Ifat-norm

T

is quasi-homogeneousin the sense of(0.3),then

T

is continuous and there exists a constantct

>

0 such that

,(t)

.

(2.,)

PROOF:

Settingx y in(0.3),wefindthat

H[T(t, t)]

p(t) If(1), (2.2)

sinceT(1, 1) 1. Since

H(1)

0,thisyields

9(t)

H(1)

-

H[T(t,

t)], (2.3)

and showsimmediatelythat

p

is monotonic, with

9(1)

1.

It

showsalsothattpismultiplicative,sinceby (0.3)wehave

q,(xy)

n(1)-

H[T(xy,

xy)]

H(1)-

9(x) H[T(y,

y)]

(x)

x,yl.

Moreover, (2.3)showsthattp(t) < for

<

1, for otherwise, since

rI

isstrictlyincreasingwewould have T(t,t)

>

forsome

<

1,contradicting

T <

Min. Thus tp must beofthe form

(2.1)

for sometx

>

0.

Now

(0.3)takes theform

T(tx, ty)

H

-I

n[T(x, y)]

}.

(2.4)

354 B.R. EBANKS

HoT(x,y)=Ho

y.-, y.l

Ho

,I

II

Y

whileif x>y, then

In

eithercase,puttingtn Min(x, y)and

M

Max

(x, y),weobtain

T(x, y)= I’[

-I.

Since

H

is continuous, this shows that

T

is continuous andcompletesthe

proof

ofthe lemma.

We

observe inpassingthatthe full force of the definition oft-normwasnotused in

Lemma

2.1.

In

fact,Lemma 2.1 is valid also whenever

T: I

x

I

---)

I

is quasi-homogeneous, satisfies T(1,x) =T(x,1)

x, and thediagonal mapt-->T(t,t)isnon-decreasingand satisfiesT(t, t)

<

for all

<

1.

Nowwemayassume that

T

is continuous and use Theorem 1.1 toobtain additional information about the structure ofT. Thenextstepis to deal with the Archimedean case.

THEOREM

2.2.

T: I

x

I

-

I

is aquasi-homogeneoust-normsatisfying T(x, x)

<

xforall x (0, 1)if andonlyif either

T(x, y) xy, (2.5)

forall x,y I,with_{9 given by}

(2.1)

and with

H(t)

II(1)

a//2

for somect

>

0, or

T(x, y)=

+

-1 /x,

ye

otherwise

(2.6)

ri(0)

o.

PROOF" Suppose T

is aquasi-homogeneous t-normwithT(x, x)<xon(0, l).

By

Lemma

2.1,

T

is continuousand(phas theform

(2.1)

for some

u >

0. ThusbyTheoreml.l(i),

T

has the form

(l.I)wheregis aquasi-inverse

off.

Insertingtherepresentation (I.I)into(2.4),weget

:[[g(tx)

+ g(ty)]

n

-

ttt

H

f[g(x) + g(y)]

}.

(2.7)

First, weestablish thatg(0) (andhence

f=

g-1

and

T

is strict).

Suppose,

tothecontrary, thatg(0) <*,,.Sinceg is continuous, we can choosepositive socloseto0thatg(t) >

-g(0).

Recallingthat

g(1)

O, f(O)

and

f(u)

0 foru> g(0),wededuce from(2.7) by puttingx y that 0

=3’[2g(t)]

n-l{f

(:t

H(1)}

for all sufficientlycloseto0. Butthisimplies H(0) 0 H(1), contradictingtheinjectivityof

H.

Thus

g(0)

and

f=

g-l.

QUASI HOMOGENEOUS ASSOCIATIVE FUNCTIONS 355

g(tx)

+

g(ty) g

H-l{

ta

FI

g-

[g(x) +

g(y)]

},

(2.8)

wheregis astrictly decreasing homeomorphismof

I

onto

+.

Fixing temporarily, let u g(x),v g(y)and define

gt(x):=g(tx),k (x):= g

lI-l[ta

I’l(x)], forx I. Then

(2.8)

becomes

gt

g’l(u) +

gt

g-l(v)

k

g-l(u

+

v),

Sincegtandk arestrictlymonotonicbydefinition, the solutionofthisPexiderequationis gt

g-l(u)

atu

+

bt,

k

g-l(u)

atu +

2b

forsome

"constants"

a andb (dependingont). Freeing

I

andrecallingthe definitions ofgtand kt,wehave now

g(tx) a(t)g(x)

+

b(t), (2.9)

g

l’I-l[tct

II(x)]

a(O g(x) + 2b(t),

(2.10)

validfor allt,x I.

The generalsolution ofequation

(2.9)

isgivenin Theorem1.2.

We

eliminate solution

(1.4)

here because gisstrictlymonotonic.

We

consider solutions

(1.3)

and

(1.5)

separately.

Case

1.

Suppose

the solutionof

(2.9)

isof the form

(1.3).

Sincegisstrictly decreasing,so is thelogarithmicfunction/ That is, there exists a constant b<0 for which

g(x) b logx

+

c,a(t) 1,andb(t) logt. (2.1 I)

Substitutingtheseinto(2.10),wefind that

:

I’I(x)

II(xt2),

t, x I.

With x 1, this yieldsH(q)=

q

H(1)

forallqe 1. Furthermore, inserting (2.11)into(1.1),

withf

g-l,

we obtain

(2.5)

forT.

Case

2.

Suppose

the solution of

(2.9)

isoftheform

(1.5).

Sinceg is strictly monotonic, the sameis true of the multiplicativcfunction m, sore(x)

x[

j.

Moreover,

sinceg(0) andg(1) 0, we must have

g(x)

c(x

1),a(O

t,

andb(t)

(t

1)

(2.12)

355 B.R. EBANKS

rl(q)

rI(1)

[1/2(q

+

1)

Finally, (2.12)combinedwith(1.1) (and

f=

g-l)

yields

whichsimplifiesto

(2.6).

Theconverse iseasilyverified,and thatcompletesthe

proof

ofTheorem2.2.

Now

we arereadytoestablish the main result.

THEOREM

2.3.

T: I x

1

--

I

is aquasi-homogeneous t-normif andonlyif

T

isgiven by (2.5), by

(2.6)

forsome

I

< 0,orby T(x,y) Min(x,y).

In

the last case,{pisgiven by

(2.1)

forsome >0, and

H

hasthe form

H(x)

H(1)

x

t.

REMARK.

Ifwedef’me

TI

tobe thet-normgiven by (2.6),then

lim

Tf(x,y)=

xy,and

fm

Tl(x,y)=

Min(x,y).

-o

Defining

T

O tobe theproductonI

I

and

T.**

tobe Min, we canrestatethe conclusion of Theorem 2.3 asfollows. Theonly quasi-homogeneoust-normsare the members of thefamily

TI

for <

I

<0.

PROOF OF THEOREM

2.3:

Let

T

beaquasi-homogeneoust-norm. Then by Lemma2.1,

T

iscontinuousand0has theform(2.1)for some0t> 0.

Now

weapplyTheorem1.1. Ifcase(i)of Theorem 1.1holds,then wefind that

(2.5)

or

(2.6)

holds byTheorem 2.2.

In

case(ii) ofTheorem 1.1,wehave

T=

Min, andequations

(0.3)

and

(2.1)

yield Min(tx, ty)

rI-l{/t

1I Min(x,y)}.

With x y 1, we obtain

H(t)

a_{H(1), I.}

Finally, we consider case (iii) ofTheorem 1.1. Choose anyx0 (0,

1)

suchthatT(x0,x

0)

x0,and let

Sk

be

any

subintervalof

Ix

0, suchthat

T

kadmits

representation

(1.1)inthe

square

S_k

x

Sk.

Then

T(x,y) Min(x,y), (x,y)

.

D

D

2,

where

D

and

D

2are therectangles

O

l={(x,y) lO<y<xO<x<l}, 02={(x,y)

lO<x<xo<Y<_l}

in

I

I.

We prove

that in fact this casecannotoccur.

Let

us confine our attentiontothelowertriangle (x,y) 0

<

y<x<

}.

Choose(x,y)in x

0

Sk

Sk

sothat x0< y<_ x<_ 1. Thenwe choose --, so that(tx, ty) belongs to

D

1.

Now (0.3)

Y and

(2.1)

yield

QUASI-HOMOGENEOUS ASSOCIATIVE FUNCTIONS 357

whichimpliesthatT(x, y)isindependentofxfor x>y >x_0. Specifically,

T(x, y)=

n

-I

17(%)

(2.13)

when x>_ y >x0, whichis inconsistent withrepresentation (1.1) for the restriction

T/

of

T

to

SkS

k.

Indeed,as in theproofof Theorem 2.2,

T

_kmust be strict:

Tk(X,Y)

gk-l[

gk(x)

+ gk(Y)],

x,

yeSk.

Thisrepresentation, with strictly monotonicgk, shows that

Tk(x,

y)isinjective in x for eachy

.

S

_k.

But

therighthand side of

(2.13)

isindependent ofx. This contradictionshows that there can be noSkin the interval

[x

0,

].

A

similarargumentshows that there can be no

Sk

in the interval[0,

x0].

Thus there can be no

proper

ordinalsum. That is, case(iii)of Theorem 1.1 isincompatiblewithquasi-homogeneity. (Inother words, if aquasi-homogeneous

T

satisfies

T(xo,xO)

x0for some x e (0, 1),then it satisfiesT(x,x)

r

for all x e (0, 1).)

The converse is easily verified, and thiscompletesthe

proof

of the theorem.

From

Theorem 2.3, thestructuretheorem forhomogeneoust-normsiseasilyextracted.

COROLLARY

2.4. T:

I

x

I

--)

I

is ahomogeneous t-normifand onlyif either T(x,y) xy,with

to(t)

2,

orT(x,y) Min(x,y),with

t0(t)

t.

PROOF: Suppose T

is ahomogeneoust-norm.

We

applyTheorem2.3 in thespecialcase in which

rI

in

(0.3)

istheidentity

map.

When

T

isof theform(2.5),wehave(of.Theorem2.2) H(x)

rI(l)x

’.

This will give

rI(x)

xonlyift 2; in

(2.1)

this yields

to(t)

2.

When

T

is given by (2.6)for some

1$

<0,wenotethat

H(x)=l-I(1)[21-(xl

+1)]/.

Such

H

can never be theidentitymap,so thiscasecannotarise. Finally,when T(x,y) Min(x,y), the accompanying

FI

is (cf. Theorem

2.3)

H(x)

H(I)

x

a.

This

H

istheidentityonlyift 1, in which case(2.1)becomes

to(t)

t.

Thesimpleconversecompletesthe

proof.

[1] MENGER, K.

Statistical metrics,

Proc.

Nat.Acad. Sci.

USA

28 (1942), 535-537.

[2]

SCHWEIZER,

B.

and

SKLAR,

A. Probabilistic Metric

Spaces,

North-Holland,

New

York, Amsterdam, Oxford, 1983.

[3] SCHWEIZER,

B. personalcommunication.

[4] ACZEL, J.

and

DHOMBRES,

J.

FunctionalEquationsin Several Variables,Cambridge

U. Press,