An. S¸ . Uni . O idius Cons an ¸a Vol. 10(1), 2002,87–94
POINTWISE AND GLOBAL SUMS AND
NEGATIVES OF BINARY RELATIONS
Tam´as Gla osi s and ´
A p´ad Sz´az
Abs ac
Fo any wo ela ions Fand Gon one g oupoid X o ano he Y,
we de ine ( F+G)(x) = F(x) + G(x) o all x∈Xand
F⊕G=

(x+z, y +w) : (x, y)∈F , (z, w)∈G

.
Mo eo e , i in pa icula Xand Ya e g oups, hen we may also
na u ally de ine ( −F)(x) = −F(x) o all x∈Xand
ªF=

(−x , −y) : (x, y)∈F

.
By using hese de ini ions, we p o e some basic heo ems abou he
images o subse s o Xunde he ela ions −F,ªF,F+Gand
F⊕G. In pa icula , we show ha
(F⊕G)(x) =
[
x=u+

F(u) + G( )

o all x∈X. The e o e, in con as o he in e sec ion con olu ion [1] ,
he union con olu ion o ela ions need no be in oduced. Mo eo e , i
is also wo h men ioning ha he esul s ob ained can, o ins ance, be
applied o ansla ion and addi i e ela ions [2] .
1 A ew basic ac s on ela ions and g oupoids
A subse Fo a p oduc se X×Yis called a ela ion on X o Y. In
pa icula , he ela ions ∆X={(x, x ) : x∈X}and X2=X×Xa e
called he iden i y and uni e sal ela ions on X, espec i ely.
Namely, i in pa icula F⊂X2, hen we may simply say ha Fis a
ela ion on X. No e ha i Fis a ela ion on X o Y, hen Fis also
87
88 T. Gla osi s and ´
A. Sz´
az
a ela ion on X∪Y. The e o e, i is equen ly no a se e e es ic ion o
assume ha X=Y.
I Fis a ela ion on X o Y, hen o any x∈Xand A⊂X he se s
F(x) = {y∈Y: ( x, y )∈F}and F[A] = Sx∈AF(x) a e called he
images o xand Aunde F, espec i ely. Whene e A∈Xseems unlikely,
we may w i e F(A) in place o F[A] .
I Fis a ela ion on X o Y, hen he alues F(x) , whe e x∈X,
uniquely de e mine Fsince F=Sx∈X{x}×F(x) . The e o e, he in e se
F−1o Fcan, o ins ance, be de ined such ha F−1(y) =©x∈X:y∈
F(x)ª o all y∈Y.
I Fis a ela ion on X o Y, hen he se s DF=F−1(X) and RF=
F(X) a e called he domain and ange o F, espec i ely. I in pa icula
X=DF(and Y=RF) , hen we say ha Fis a ela ion o Xin o (on o)
Y.
A ela ion Fon X o Yis called a unc ion i o each x∈DF he e
exis s y∈Ysuch ha F(x) = {y}. In his case, by iden i ying single ons
wi h hei elemen s, we usually w i e F(x) = yin place o F(x) = {y}.
I Xis non oid se and + is a unc ion o X2in o X, hen he o de ed
pai X(+) = ( X, + ) is called a g oupoid. In his case, we may also na u ally
w i e x+y= + ( x, y ) o all x, y ∈X.
Mo eo e , i Xis a g oupoid, hen we may also na u ally w i e A+B=
©x+y:x∈A , y ∈Bª o all A , B ⊂X. Thus, he amily P(X) o all
subse s o Xis also a g oupoid.
No e ha i Xis, in pa icula , a g oup, hen P(X) is, in gene al, only
a semig oup wi h ze o elemen {0}. Howe e , we can s ill na u ally use he
no a ions −A={ − x:x∈Aªand A−B=A+ ( −B) .
2 Poin wise and global sums and nega i es o ela ions
De ini ion 2.1 I Fand Ga e ela ions on a se X o g oupoid Yand
F+Gis he ela ion on X o Ysuch ha
(F+G)(x) = F(x) + G(x)
o all x∈X, hen F+Gis called he poin wise sum o Fand G.
While, i Fand Ga e ela ions on one g oupoid X o ano he Yand
F⊕G=©(x+z, y +w) : (x, y)∈F , (z, w)∈Gª,
hen he ela ion F⊕Gis called he global sum o Fand G.
Poin wise and global sums and nega i es o bina y ela ions 89
Rema k 2.2 Thus, we ha e
DF+G=DF∩DGand DF⊕G=DF+DG.
The global sum F⊕Gis, in gene al, qui e di e en om he poin wise
one F+Ge en i DF+G=DF⊕G.
Example 2.3 I Xis a g oupoid, hen
(1) ∆X+ ∆X= ∆Xi and only i x=x+x o all x∈X;
(2) ∆X⊕∆X= ∆Xi and only i o each x∈X he e exis u , ∈X
such ha x=u+ .
The e o e, i in pa icula Xis a g oup, hen ∆X⊕∆X= ∆X, bu
∆X+ ∆X= ∆Xi and only i X={0}.
Howe e , in some e y pa icula cases, he global sum o ela ions may
coincide wi h he poin wise one.
Example 2.4 Le Xbe a non oid se , and o all x, y ∈Xde ine x+y=
x. Then Xis a semig oup such ha , o any wo ela ions Fand Gon X,
we ha e F⊕G=Fwhene e G6=∅, and F+G=Fwhene e G(x)6=∅
o all x∈DF.
Analogously o De ini ion 2.1, we may also na u ally in oduce he ollow-
ing
De ini ion 2.5 I Fis a ela ion on a se X o g oup Yand −Fis he
ela ion on X o Ysuch ha
(−F)(x) = −F(x)
o all x∈X, hen −Fis called he poin wise nega i e o F.
While, i Fis a ela ion on one g oup X o ano he Yand
ªF=©(−x , −y) : (x, y)∈Fª,
hen he ela ion ªFis called he global nega i e o F.
Rema k 2.6 Thus, we ha e
D−F=DFand DªF=−DF.
The global nega i e ªFis, in gene al, also qui e di e en om he poin -
wise one −Fe en i DªF=D−F.
90 T. Gla osi s and ´
A. Sz´
az
Example 2.7 I Xis a g oup, hen ª∆X= ∆X, bu −∆X= ∆Xi and
only i −x=x o all x∈X.
Howe e , in some e y pa icula cases, he global nega i e o a ela ion
may coincide wi h he poin wise one.
Example 2.8 I Xis a g oup such ha −x=x o all x∈X, hen
−F=Fand ªF=F o any ela ion Fon X.
Conce ning he images o se s unde he ela ions −F,ªF,F+Gand
F⊕G, we can easily p o e he ollowing heo ems.
Theo em 2.9 I Fis a ela ion on a se X o a g oup Y, hen
(−F)(A) = −F(A)
o all A⊂X.
Theo em 2.10 I Fis a ela ion on one g oup X o ano he Y, hen
(ªF)(A) = −F(−A)
o all A⊂X.
P oo . I y∈(ªF)(A) , hen he e exis s x∈Asuch ha y∈
(ªF)(x) , and hus ( x, y )∈ ªF. Hence, i ollows ha ( −x, −y)∈F,
and hus
−y∈F(−x) . Thus, since F(−x)⊂F(−A) , we also ha e y∈ −F(−A) .
The e o e, ( ªF)(A)⊂ − F(−A) .
Now, by w i ing ªFin place o Fand −Ain place A, we can also see
ha
F(−A) = ¡ª(ªF)¢(−A)⊂ − (ªF)¡−(−A)¢=−(ªF) (A),
and hus −F(−A)⊂(ªF)(A) is also ue. ¤
Co olla y 2.11 I Fis a ela ion on one g oup X o ano he Y, hen
(1) ªF=Fi and only i F(−x) = −F(x) o all x∈X;
(2) ªF=−Fi and only i F(−x) = F(x) o all x∈X.
Theo em 2.12 I Fand Ga e ela ions on a se X o g oupoid Y, hen
(F+G)(A)⊂F(A) + G(A)
o all A⊂X.
Poin wise and global sums and nega i es o bina y ela ions 91
Theo em 2.13 I Fand Ga e ela ions on one g oupoid X o ano he Y,
hen
F(A) + G(B)⊂(F⊕G)( A+B)
o all A , B ⊂X.
P oo I w∈F(A) + G(B) , hen he e exis y∈F(A) and z∈G(B)
such ha w=y+z. Mo eo e , he e exis a∈Aand b∈Bsuch
ha y∈F(a) and z∈G(b) , and hus ( a, y )∈Fand ( b, z )∈G.
Hence, i ollows ha ( a+b , w ) = ( a+b , y +z)∈F⊕G, and hus
w∈(F⊕G)( a+b) . Thus, since ( F⊕G)( a+b)⊂(F⊕G)( A+B) , we
also ha e w∈(F⊕G)( A+B) . ¤
Co olla y 2.14 I Fand Ga e ela ions on one g oupoid X o ano he
Y, and Ais a subg oupoid o X, hen
F(A) + G(A)⊂(F⊕G)(A).
3 Some u he esul s on he global sums o ela ions
Theo em 3.1 I Fand Ga e ela ions on one g oupoid X o ano he Y,
hen
(F⊕G)(A) = [
u+ ∈A
¡F(u) + G( )¢
o all A⊂X.
P oo I y∈(F⊕G)(A) , hen he e exis s x∈Asuch ha y∈
(F⊕G)(x) , and hence ( x , y )∈F⊕G. The e o e, he e exis ( u , z )∈F
and ( , w )∈Gsuch ha ( x , y ) = ( u+ , z +w) . Hence, i ollows
ha z∈F(u) and w∈G( ) , and mo eo e x=u+ and y=z+w.
The e o e, y∈F(u) + G( ) , and hence y∈Sx=u+ ¡F(u) + G( )¢⊂
Su+ ∈A¡F(u) + G( )¢.
While, i y∈Su+ ∈A¡F(u) + G( )¢, hen he e exis u , ∈X, wi h
x=u+ ∈A, such ha y∈F(u)+G( ) . The e o e, he e exis z∈F(u)
and w∈G( ) such ha y=z+w. Hence, i is clea ha ( u , z )∈Fand
( , w )∈Gsuch ha ( x , y ) = ( u+ , z+w) . The e o e, ( x , y )∈F⊕G,
and hence y∈(F⊕G)(x)⊂(F⊕G)(A) . ¤
Rema k 3.2 The A={x}pa icula case o he abo e heo em shows ha ,
in con as o he in e sec ion con olu ion
(F∗G)(x) =
x=u+
¡F(u) + G( )¢,

92 T. Gla osi s and ´
A. Sz´
az
he union con olu ion o ela ions no be in oduced since i coincides wi h he
global sum.
Now, as a use ul consequence o Theo em 3.1, we can also p o e
Co olla y 3.3 I Fand Ga e ela ions on one g oup X o a g oupoid Y,
hen
(F⊕G)(A) = [
∈X
¡F(A− ) + G( )¢
o all A⊂X.
P oo I y∈(F⊕G)(A) , hen by Theo em 3.1 y∈Su+ ∈A¡F(u) +
G( )¢. The e o e, he e exis u , ∈X, wi h x=u+ ∈A, such
ha y∈F(u) + G( ) . Hence, i ollows ha y∈F(x− ) + G( )⊂
F(A− ) + G( ) , and hus y∈S ∈X¡F(A− ) + G( )¢.
While, i y∈S ∈X¡F(A− ) + G( )¢, hen he e exis s ∈X
such ha y∈F(A− ) + G( ) . The e o e, he e exis s x∈Asuch ha
y∈F(x− ) + G( ) . Hence, by de ining u=x− , we can see ha
u∈Xsuch ha x=u+ and y∈F(u) + G( )¢. The e o e, y∈
Sx=u+ ¡F(u) + G( )¢⊂Su+ ∈A¡F(u) + G( )¢, and hence by Theo em
3.1 y∈(F⊕G)(A) . ¤
Mo eo e , as a simple e o mula ion o he abo e co olla y we can also
s a e
Co olla y 3.4 I Fand Ga e ela ions on one g oup X o a g oupoid Y,
hen
(F⊕G) (A) = [
u∈X
¡F(u) + G(−u+A)¢
o all A⊂X.
P oo I y∈(F⊕G)(A) , hen by Co olla y 3.3 y∈S ∈X¡F(A−
)+G( )¢. The e o e, he e exis s ∈Xsuch ha y∈F(A− )+G( ) .
Thus, he e exis s x∈Asuch ha y∈F(x− ) + G( ) . Now, by de ining
u=x− , we can see ha y∈F(u) + G(−u+x)⊂F(u) + G(−u+A) .
The e o e, y∈Su∈X¡F(u) + G(−u+A)¢.
While, i y∈Su∈X¡F(u) + G(−u+A)¢, hen he e exis s u∈X
such ha y∈F(u) + G(−u+A) . The e o e, he e exis s x∈Asuch ha
y∈F(u) + G(−u+x) . Now, by de ining =−u+x, we can see ha
y∈F(x− ) + G( )⊂F(A− ) + G( ) . The e o e, y∈S ∈X¡F(A−
) + G( )¢, and hence by Co olla y 3.3 y∈(F⊕G)(A) . ¤
Poin wise and global sums and nega i es o bina y ela ions 93
Rema k 3.5 Now, by using he p eceding esul s, one can also easily es ablish
some p ope ies o he images o se s unde he ela ions
F−G=F+ ( −G)and FªG=F⊕(ªG).
Re e ences
[1] ´
A. Sz´az, The in e sec ion con olu ion o ela ions and he Hahn–Banach ype heo ems,
Ann. Polon. Ma h. 69 (1998), 235–249
[2] ´
A. Sz´az, T ansla ion ela ions, he building bloks o compa ible ela o s, Ma h. Mon is-
nig i, o appea
Ins i u e o Ma hema ics and In o ma ics,
Uni e si y o Deb ecen,
H-4010 Deb ecen, P . 12,
Hunga y
94 T. Gla osi s and ´
A. Sz´
az