Some notes on T partial order

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R E S E A R C H

Open Access

Some notes on

T

-partial order

Funda Karaçal and Emel A¸sıcı

*

*_{Correspondence:}

[email protected] Department of Mathematics, Karadeniz Technical University, Trabzon, 61080, Turkey

Abstract

In this study, by means of theT-partial order defined in Karaçal and Kesicio ˘glu (Kybernetika 47:300-314, 2011), an equivalence relation on the class oft-norms on ([0, 1],≤, 0, 1) is defined. Then, it is showed that the equivalence class of the weakest t-normTDon [0, 1] contains at-norm which is different fromTD. Finally, defining the

sets of some incomparable elements with anyx∈(0, 1) according toT, these sets

are studied.

MSC: 03E72; 03B52

Keywords: triangular norm;T-partial order

1 Introduction

Triangular norms were originally studied in the framework of probabilistic metric spaces [–] aiming at an extension of the triangle inequality and following some ideas of Menger []. Later on, they turned out to be interpretations of the conjunction in many-valued logics [–], in particular in fuzzy logics, where the unit interval serves as a set of truth values.

In [], a natural order for semigroups was deﬁned. Similarly, in [], a partial order de-ﬁned by means oft-norms on a bounded lattice was introduced. For any elementsx,yof a bounded lattice

xTy: ⇔ T(,y) =x for some,

whereT is at-norm. This orderTis called at-partial order ofT. Moreover, some

con-nections between the natural order and thet-partial orderTwere studied.

In [], it was investigated thatTimplies the natural order but its converse needs not

be true. It was showed that a partially ordered set is not a lattice with respect toT. Some

sets which are lattices with respect toTunder some special conditions were determined.

For more details ont-norms on bounded lattices, we refer to [–].

In the present paper, we introduce an equivalence on the class oft-norms on ([, ],≤, , ) based on the equality of the sets of all incomparable elements with respect toT.

The paper is organized as follows. We shortly recall some basic notions in Section . In Section , we deﬁne an equivalence on the class oft-norms on ([, ],≤, , ), and we de-termine the equivalence class of the weakestt-normTDon [, ]. Thus, we obtain that the

equivalence class of the weakestt-normTD contains at-norm which is diﬀerent from a t-normTD. We obtain that for arbitrarym∈KT, there exists an elementym∈KTsuch that T(m,·) : [, ]→[,m] orT(ym,·) : [, ]→[,ym] is not continuous.

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2 Basic deﬁnitions and properties

Deﬁnition .[] A triangular norm (t-norm for short) is a binary operationT on the unit interval [, ],i.e., a functionT: [, ]×[, ]→[, ], such that for allx,y,z∈[, ] the following four axioms are satisﬁed:

(T) T(x,y) =T(y,x)(commutativity);

(T) T(x,T(y,z)) =T(T(x,y),z)(associativity); (T) T(x,y)≤T(x,z)whenevery≤z(monotonicity); (T) T(x, ) =x(boundary condition).

Example .[] The following are the four basict-normsTM,TP,TL,TD given by,

re-spectively,

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

TP(x,y) =x·y,

TL(x,y) =max(x+y– , ),

TD(x,y) =

⎧ ⎨ ⎩

 if (x,y)∈[, [_,

min(x,y) otherwise.

Also, t-norms on a bounded lattice (L,≤, , ) are deﬁned in a similar way, and then extremalt-normsTWas well asT∧onLare deﬁned asTDandTMon [, ].

Remark .[]

(i) Directly from Deﬁnition ., we can deduce that, for allx∈[, ], eacht-normT satisﬁes the following additional boundary conditions:

T(,x) = ,

T(,x) =x.

Therefore, allt-norms coincide on the boundary of the unit square[, ]×[, ]. (ii) The monotonicity of at-normTin its second component described by (T) is,

together with the commutativity (T), equivalent to the monotonicity in both components,i.e., to

T(x,y)≤T(x,y) whenever x≤xandy≤y. (.)

Deﬁnition .[] A functionF: [, ]×[, ]→[, ] is called continuous if for all con-vergent sequences (xn)n∈N, (yn)n∈N∈[, ]N,

F

lim

n→∞xn,nlim→∞yn

= lim

n→∞F(xn,yn).

Proposition .[] A function F : [, ]×[, ]→[, ] which is non-decreasing,i.e.,

which satisﬁes(.),is continuous if and only if it is continuous in each component,i.e.,

if for all x,y∈[, ],both the vertical section F(x,·) : [, ]→[, ]and the horizontal

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Proposition .[] A non-decreasing function F: [, ]×[, ]→[, ]is lower semi-continuous if and only if it is left-semi-continuous in each component,i.e.,if for all x,y∈[, ]

and for all sequences(xn)n∈N, (yn)n∈N∈[, ]N,we have

supF(xn,y)|n∈N

=F sup{xn|n∈N},y

,

supF(x,yn)|n∈N

=F x,sup{yn|n∈N}

.

By the same token, the upper semicontinuity of a non-decreasing functionF: [, ]× [, ]→[, ] is equivalent to its right-continuity in each component.

Proposition .[] For a non-decreasing function F:In_→_R₍_I_{an interval}_),_{the following} conditions are equivalent:

(i) Fis continuous;

(ii) Fis continuous in each variable,i.e.,for anyx∈In_{and any}_i_{∈ {}_{, , . . . ,}_n_}_,_the unary function

u→F(x, . . . ,xi–,u,xi+, . . . ,xn)

is continuous;

(iii) Fhas the intermediate value property:For anyx,y∈In,withx≤y,and any c∈[F(x),F(y)],there existsz∈In,withx≤z≤y,such thatF(z) =c.

Deﬁnition .[] LetLbe a bounded lattice, letTbe at-norm onL. The order deﬁned as follows is called at-order (triangular order) for at-normT.

xTy: ⇔ T(,y) =x for some∈L.

Example .[] LetL={,a,b,c, }and consider the order≤onLas in Figure . We choose thet-normTW. Thena≤b, butaTW b. We suppose thataTW b. Then there exists an element∈Lsuch thatTW(,b) =a. If= , then it is obtained thata= ,

a contradiction. If=a,borc, then we have thatTW(,b) =  =a, a contradiction. If= ,

then it is obtained thatTW(,b) =b=a, which is not possible. So, there does not exist any

element∈LsatisfyingTW(,b) =a. Thus,aTW b. Then the orderTW onLis as in Figure .

Proposition .[] Let L be a bounded lattice,let T be a t-norm on L.Then the binary relationTis a partial order on L.

Deﬁnition .[] This partial orderTis called aT-partial order onL.

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[image:4.595.118.479.80.152.2]

Figure 2 The order_TWonL.

Deﬁnition . LetTbe at-norm on [, ] and letKTbe deﬁned by

KT=

x∈[, ]| for somey∈[, ], [x≤yimpliesxTy] or [y≤ximpliesyTx]

.

We will use the notationKTto denote the set of all incomparable elements with respect

toT.

3 The equivalence of any twot-norms

LetLbe a lattice and letTbe anyt-norm onL. In [], a partial order for at-normTonL

was deﬁned. In this section, we deﬁne an equivalence relation with the help of the sets of all incomparable elements with respect toT. The above introducedT-partial order allows

us to introduce the next equivalence relation on the class of allt-norms on ([, ],≤, , ).

Deﬁnition . Let ([, ],≤, , ) be the unit interval. Deﬁne a relation∼on the class of allt-norms on ([, ],≤, , ) byT∼Tif and only if the set of all incomparable elements with respect to theT-partial order is equal to the set of all incomparable elements with respect to theT-partial order, that is,

T∼T: ⇔ KT=KT.

Proposition . The relation∼given in Deﬁnition.is an equivalence relation.

Proof LetT,TandTbet-norms on ([, ],≤, , ). SinceKT=KT, it is obtained that

T∼T. Thus, the reﬂexivity is satisﬁed.

Let T ∼T. Then we have that KT =KT, and sinceKT =KT, it is obtained that

T∼T. Thus, the symmetry is satisﬁed. LetT∼TandT∼T. Then we haveKT=KT

andKT =KT. SinceKT=KT, it is obtained thatT∼T. This means that the relation

∼satisﬁes the transitivity. So, we have that∼is an equivalence relation.

Deﬁnition . For a givent-normTon ([, ],≤, , ), we denote byTthe∼equivalence class linked toT,i.e.,

T=T|Tis at-norm on [, ] andT∼T.

Proposition . shows that the equivalence class of thet-normTD contains at-norm

which is diﬀerent fromTD.

Proposition . Let the t-norm TDbe on[, ].Then TD={TD}.

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Example . Consider thet-normT: [, ]×[, ]→[, ] deﬁned by

T(x,y) = ⎧ ⎨ ⎩

xy

 if (x,y)∈[, ) _,

min(x,y) otherwise.

ThenKT=KTD. Firstly, let us show thatKT= (, ). Letx∈(, ) andy= x

. Theny<x, but

yTx. Suppose thatyTx. Then, for some,T(,x) =_x. Sincex=y, it is not possible

= . Then_x=T(,x) =x

, whence it is obtained that= 

, a contradiction. Since for any

x∈(, ) there exists an elementy=_x such that _x<xbut_x Tx,x∈KT. Conversely,

for anyt-normT, it is clear thatKT⊆(, ). So, it is obtained thatKT= (, ).

Now, we will show thatKTD= (, ). Letx∈(, ). For anyy∈(, ) withx<y, it is obvious thatxTDy. Otherwise, it would beTD(,y) =xfor some. Sincex=y,= . Thus, it must bex=  for,y∈(, ), a contradiction. Since there is an element y withx<ysuch that

xTDy,x∈KTD. This shows thatKTD= (, ). So, it is obtained thatKT=KTD.

Deﬁnition . LetT be at-norm on [, ] and letKx↓,Kx↑be deﬁned by

Kx↓=

y∈[, ]|y≤xandyTx

,

Kx↑=

y∈[, ]|x≤yandxTy

for anyx∈(, ).

Lemma . Let T be a t-norm on[, ]and x∈KTbe arbitrarily chosen.If T is continuous at(x,y)for all y∈[, ],then Kx↓=∅.

Proof Let T be at-norm on [, ] and let x∈KT be arbitrarily chosen. Suppose that Kx↓=∅. Then there exists an elementy∈[, ] such thaty≤x, butyTx. Since the t-normT(x,·) : [, ]→[,x] is continuous, there exists an elementz∈[, ] such that

T(x,z) =yfory∈[,x] by Proposition .. So, it is obtained thatyTx, a

contradic-tion. Therefore we have thatKx↓=∅.

Lemma . Let T be a t-norm on[, ]and the function T(x,·) : [, ]→[,x]be

con-tinuous.Then,for all y∈[, ]with y≤x,we have that yTx.

Proof LetTbe at-norm on [, ] and the functionT(x,·) : [, ]→[,x] be continuous. Suppose that there exists an element y∈[, ] such that y≤x andyT x. Since

T(x,·) : [, ]→[,x] is continuous, there exists an elementt∈[, ] such thatT(x,t) =

yfory∈[,x] by Proposition .. Thus, it is obtained thatyTx, a contradiction.

Therefore, for ally∈[, ] withy≤x, we have thatyTx.

Theorem . Let T be a t-norm on[, ]and KT=∅.Then,for arbitrary m∈KT,there exists an element ym∈KT such that T(m,·) : [, ]→[,m]or T(ym,·) : [, ]→[,ym]is not continuous.

Proof LetTbe at-norm on [, ] andKT=∅. Suppose thatT(x,·) : [, ]→[,x] is

con-tinuous forx∈KT. Choosem∈KTarbitrarily. Then there exists an elementym∈KTsuch

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[, ]→[,ym] is continuous, then it is obtained thatmTymby Lemma ., a

contradic-tion. Letym<mbutymTm. SinceT(m,·) : [, ]→[,m] is continuous, then it is

ob-tained thatymTmby Lemma ., a contradiction. Therefore, for arbitrarym∈KT, there

exists an elementym∈KTsuch thatT(m,·) : [, ]→[,m] orT(ym,·) : [, ]→[,ym] is

not continuous.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The present study was proposed by FK. All authors read and approved the ﬁnal manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Received: 4 December 2012 Accepted: 18 April 2013 Published: 30 April 2013 References

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doi:10.1186/1029-242X-2013-219