WORDS WV

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 the}probability 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 q^{to ris}less thany.

For

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

[2].

In

light ofthe interpretation above,we examine now those t-norms satisfying an additional property. Iftheprobabilities

Fpq(X)

^and

Fqr(Y)

^{on thefightsideof}

^(0.2)

^{areshrunkbyafactor a I,}

then itmaybereasonableto

suppose

that theprobability

Fpr(X ⁺

^{y)isalsoshrunkbya factordepending}

ont,possibly aftersomechangeof scale.

More

precisely,^weshallstudythoset-norms

T

whichsatisfy

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 becalledquasi- homogeneous. If

H

in

(0.3)

istheidentity

map,

then

T

is calledhomogenous,and theforms of sucht- 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 easy^todeduce 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 a}representation 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)

among maps

g,a, b:^I--)

/,

isgiven bythefollowing:

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--- +

^{isanarbitrary}solution of thelogarithmic functionalequation

/(xy)=/(x)

+/lv),

x, y

I,

andm:

I-- +

^{is anarbitrarysolutionofthe}multiplicativefunctionalequation

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)

For any

given x,y

I,

if x< ywe have

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 a}quasi-homogeneoust-normsatisfying T(x, x)

<

xforall x (0, 1)^{if and}onlyif either

T(x, y) xy, (2.5)

forall x,y I,with9 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), contradicting^theinjectivityof

H.

Thus

g(0)

and

f= ^g-l.

Nowwe can write(2.7)as

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 k_t,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) log^t. (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)

forsomeconstants

I

^{< 0and c}

^>

0. Insertingthese into(2.10) and simplifying,we arriveat

355 B.R. EBANKS

rl(q) rI(1) [1/2(q ⁺

¹⁾

Finally, (2.12)combinedwith(1.1) (and

f= ^g-l)

^yields

whichsimplifies^to

(2.6).

Theconverse iseasilyverified,and thatcompletesthe

proof

ofTheorem2.2.

Now

we arereadytoestablish the main result.

THEOREM

2.3.

T: I x

1

-- ^I

^{is a}quasi-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-norm}given by (2.6),then lim

Tf(x,y)=

^xy,and

fm ^Tl(x,y)=

^Min(x,y).

-o

Defining

T

_{O to}be theproductonI

I

and

T.**

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

TI

^{for <}

^I

^<0.

PROOF OF THEOREM

2.3:

Let T

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

T

iscontinuousand0^{has 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)

^aH(1), I.

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

1)

^suchthatT(x0,x

0)

x_0,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 x0

Sk Sk

^{sothat x}0< y<_ x<_ 1. Thenwe choose --, so that(tx, ty) belongs to

D

_1.

Now (0.3)

Y

and

(2.1)

yield

H[T(x,y)] H(x

O)

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 noS_kin 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,

Cambridge,

New

York,Melbourne, 1989.