Ma h
w
a e&So C
om
pu ing 2(
19
95)39-45
Onsom
econs
uc ionso
new
iangula
n
o m
s
Radk
oMesia
Slo
akT
ec
hnicalUni
e
s
i
yB a
i
sl
a
a
Radlins
k
eho1
1,81368B a is
la
a
Slo
aki
a
Abs
ac
W
ediscuss hep o
pe ieso
w
o
ypeso cons uc iono
anew -no m
omag
i
en -no
mp
opo
sed ecen
lyb
yB.
Deman
,namely hedila
a
iona
nd h eco
n
a
c
ion.Ingene
al,
hed
ila
a iono
a -no misa
no d inalsum -no mand he
con
in
ui y o heou g
oing -no misp ese
ed.On heo
he
hand, he con
ac ionma
y io
la
e hec
on
in
ui
yasw
ellas he
non-co
n
in
ui y o heou g
oing
-no
m.Se
e al examplesa
e
gi
en.
Keyw
o ds:
co
n ac
ion,dila
a ion,o
di
nalsum, iangula
no m.
1In
o duc ion
Am
ong se
e al cons uc ions o he new -no m
s o m
gi
en o ne
s [2,3,4],
we e
ca l
l
wo bas i
c cons uc i
ons a i
sen o m he s e
mi
g oup in
e
p e-
a
ion o a i
a ngula no m
,se
eSc
hweiz
e a nd Skl
a [3].
O dinal sum
:Le
h
]
a
k
;b
k
[;
k
2K
i
be a dis join sys emo open
su bin e als o he uni in e a l [0,1] and le [
T
k
;
k
2K
] b e a sys em
39
40
R.Mesi
a
o gi
e
n -no
m
s.F
o
x;y
2
[0
;
1],p
u
T
(
x;y
)=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
a
k
+(
b
k
0
a
k
)
T
k
((
x
0
a
k
)
=
(
b
k
0
a
k
)
;
(
y
0
a
k
)
=
(
b
k
0
a
k
)) i
x;y
2
[
a
k
;b
k
]
o som
e
k
2K
mi
n(
x
;y
)o he
wi
se
Then
T
isa -no mandi
i
sc
al
ledano dinalsumwi h sum
m
ands
<a
k
;b
k
;T
k
>
,
k
2K
,b iey
T

[
<a
k
;b
k
;T
k
>
;
k
2K
]
.
Semig
oupde o
ma ion
:l
e

:[
0
;
1]
!
[
a;
1],
a
2
[
0
;
1[,bean
i
nc
easi
ngbi
jec io n. Le
T
beagi
en -no m
.F
o
x;y
2
[0
;
1]pu
T

(
x;y
)=

(
0
1)
(
T
(

(
x
)
;
(
y
))
;
whe e

(
0
1
)
:[0
;
1]
!
[
0
;
1]is hepseudo- i
n
e seo

,

(
0
1)
(
x
)=

(
0
1)
(m
ax(
a;x
)).N
o e ha i
a
=0 hen
T

iscal
leda

- ans o m
a i
on
o
T
and
T
and
T

a eisom
o
phi
c(andhence hep ope iessuc
has
con
i
n
ui
y
,s ic
ness,e c.
,a ep e
se
e
d).I
a>
0 hen hede
o m
a-
i
on
T

p ese
e
s hec
on
in
ui
yand
heA
c
him
e
deanp ope
y(and
henil
po ency
)bu hes ic ne
ssm
a
yb
e
iola ed.T
ak
e,e.g., he
p oduc -no m
T
P
andl
e

(
x
)=2
x
0
1
,i
.e.
,
a
=1
=
2and

0
1
(
x
)=
1+log
2
x
.The
n
(
T
P
)

(
x;y
)=1+l
og
2
m
ax
(1
=
2
;
2
x
0
1
1
2
y
0
1
)=m
ax
(0
;x
+
y;
0
1)
:
He
nce
he

-de o m
a iono hes i
c p oduc
-no m
T
P
i
s he
nil
po en
Luk
as
iewicz - no m
T
L
,(
T
P
)

=
T
L
.
Rec
en
l
y,Dem
an
[1] h as sug ges ed
wo new
ype
s o -no mc
on-
s uc i
on
s. L e

:[0
;
1]
!
[0
;a
],
a
2
]0
;
1], b e a g i
eninc
eas i
ng bi-
je
c ion. F
o
x
2
[0
;
1] w
e dene he pseudo -i
n e se o

by

(
0
1)
(
x
)=

0
1
(m
in(
a; x
)). Le
T
be a gi
en -no m
.F
o
x; y
2
[0
;
1], w
ede
ne:
Con
ac ion
:
T
(

)
(
x;y
)=
(

(
0
1)
(
T
(

(
x
)
;
(
y
))) i max (
x;y
)
<
1
T
(
x; y
)o he wise
;
Onsom
ec
ons uc i
on
so
n
ew iangul
a no ms
41
Dila
a ion
:
T
(

)
(
x
;y
)=
(


T
(

(
0
1
)
(
x
)
;
(
0
1
)
(
y
))

i
T
(
x; y
)
<a
T
(
x;y
)o he wise
:
Bo h hecon
ac io na
nd hedil
a
a i
ono a -n o m
T
a eagai
n
-no
ms ,s
e
e[
1]
.No e ha in
hecase
a
=1bo h hedil
a a i
onand
hecon
ac i
ona e he us ualsem
ig oup ans o m
a
iono
Sc
h
w
e
ize
andSkl
a [
3]
,
T
(

)
=
T

and
T
(

)
=
T

0
1
.F
u he no e ha he
onl
y -no ms p e
se
e
db
ya bi
a yde o m
a ion, ans o m
a ion,and
con
ac i
ona e hel
im
i -no
m
s
T
M
an
d
T
W
.H
o
w
e
e
, heonl
y -
no mp e
se
ed b
yana bi
a ydila a ion is
T
M
,whil
e(
T
W
)
(

)
6
=
T
W
whene
e

(1)
6
=1.
2C
on
ac ionso -no m
s
F
o agi
enb
i
jec
ion

:[
0
;
1]
!
[
0
;a
]wi h
a<
1andagi
e
n -no m
T
,
he
al
ueso heco n
ac
ion
T
(

)
on hehal
-opensq
ua e[
0
;
1[
2
depend
on he
alueso
T
on he ha
l
-ope
nsqua
e[
0
;a
[
2
onl
y( he em
ainde
o hedomain isc
on
ai
nedin hebo de s o heuni
sq
ua ewhe eall
-no m
scoi
ncide).H
ence henon-con
in
ui
y (a nd heabse
nc
eo he
A c
him
edeanp ope y
)o
T
ne
edno be
ue o i
s con ac i
on
T
(

)
.
On
heo he hand, hecon
in
ui y o
T
m
a
ybe i
ol
a e
db
y
T
(

)
, o o,
while
he A c
him
e
deanp ope
y ema insp
ese
ed.
Examp le 1
i
)L
e

(
x
)=
a
1
x
,
a
2
]0
;
1[
,andl
e
T
=
T
P
.Then
he c
on
ac ion
T
(

)
is de ne
dby
T
(

)
(
x;y
)=
(
a
1
x
1
y
i max
(
x; y
)
<
1
x
1
y
o h
e
wise
:
No e ha
T
(

)
is n o c
on in uous al houg h
T
is c
on inuous. F
u -
he , he s ic ness
T
(

)
(
x; y
)
<T
(

)
(
x; z
)
o each
x>
0
,
y<z
,
holds ue.
42
R.Mesi
a
ii)Le

(
x
) =
a
1
x
,
a
2
]
0
;
1[
,andle
T

h
<
0
;a; T
P
>;
<a
;
1
;T
W
>
i
be ano
d
inalsum -no
m. Then
T
isnon-c
on-
inuo
us(and non-
A
chim
e
de
an)bu
T
(

)
=
T
P
isc
on inuousa
nd
A
chime
de
an.
4
F
o acom
posi
ion la
wo
w
oc
on
ac
ionsw
eha
e
he ol
lo
wing
esul
.
P oposi
ion 1
L
e

:[
0
;
1]
!
[0
;a
]
and
:[
0
;
1]
!
[0
;b
]
b
e wo
bije
c ionswi h
a

1
and
b

1
.L
e
T
b
eagi en -no
m.Then
h
T
(

)
i
(
)
=
T
(


)
;
i.e., he
-c
o
n
ac iono a

-c
on
ac iono
T
is h
e


-con
ac ion
o
T
.
4
Thep
oble
mo -no m
si
n
a i
an
u
nde gi
en

-c
on
ac
ionwillbe
pa iall
ys
ol
edin henex
sec
ion.
3Dila a ionso - n
o m
s
N
on- i ial di
la a i
ons(i
.e.
,when
a<
1) a
ea
l
w
a
yso di
nalsum
swi h
w
osum
m
ands.
P oposi
ion2
L
e

:[
0
;
1]
!
[0
;a
]
b
eagi eni
nc
e
asingb
ijec ion
whe
e
a
2
]
0
;
1[
andle
T
b
eagi en - no m.Then he

-dila a iono
T
isano
dinalsumwi
h wos
ummands,
T
(

)

h
<
0
;a;T
=a
>; < a
;
1
;T
a
>
i
;
whe
e
T
=a
is he
ans o
ma ion
T
o wi h
esp
ec o he mapping
=a
:
[0
;
1]
!
[0
;
1]
, whil e
T
a
is he de o ma ion o
T
wi h
esp
ec o heline
a
ans o ma ion

a
:[0
;
1]
!
[
a;
1]
,

a
(
x
)=
a
+(
1
0
a
)
1
x
,dep
endin g
onl y on
T
and a (indep
en den o

up o
he al ue
a
=

(1)
),
T
a
=
T

a
,
T
a
(
x; y
)=

max
h
0
;T
(
a
+(1
0
a
)
1
x; a
+(1
0
a
)
1
y
)
0
a
i
=
(1
0
a
)
:
4
Onsom
ec
ons uc i
on
so
n
ew iangul
a no ms
43
Rema k1
F
o A c
him
edeanco n in
uous -no m
sw
eha
e
he
ollo
wing
esul :le
be an ad di
i
egene a o o agi
en -no m
T
[5]andle
hel
e de i a i
eo
in hepoin
1b
en
on - i
ial,
0
0
(1)
2
]
01
;
0[.
Thenl
im
a
!
1
0
T
a
=
T
L
,wh e
e
T
L
i
s
heLuk
asi
ewicz -no m
.T
he p oo
ollo
ws om he ac ha i
T
has anaddi
i
egene
a o
hen
T
a
has
anaddi i
egene a o
(
a
+(1
0
a
)
1
x
).
4
I i
se
asy os
ee ha o a bi
a ydi
la a
ion
he -no m
T
M
e
m
ains
s abl
e. F
u he
, o eac
h
a
2
]0
;
1[i
is [
T
W
]
a
=
T
W
andhence o
a bi
a y

i
is
h
T
W
i
(

)

h
<
0
;a;T
W
>;<a;
1
;T
W
>
i
.Apply
ing
he

-dila a ion
o
T
W
inni el
ym
an
y
im
es w
ege
anew -no m
T
3
a
dependingonlyon
a
and in
a i
an
unde

-dil
a
a
io n,
T
3
a

h
<a
n
;a
n
0
1
;T
W
>
;
n
2
N
i
:
Ana u alque
s i
ona ises: o agi en ans o m
a ion

,a e he e
som
eo he

-dil
a a ioni
n
a ian
-no ms up o
T
M
(acon
i
n
uous -
no m
)and
T
3
a
(adi
sc
on
in
uous -no m
)?I i
sob
i
ous ha e
ach

-
dil
a a ionin
a i
an
-n
o m
T
3
die e
n
om
T
M
shouldbeano di
nal
sumo
y
pe
T
3

h
<a
n
;a
n
0
1
;T>
;
n
2
N
i
;
whe e
T
isa -no msuc
h ha
T
=
T
a
=
T
=
a
.Requi
ing hecon
in
ui
y
o
T
,we ha
e he ollo
wi
ng esul .
P opo
si i
on3
L
e
T
b
eac
on inu
ous
-no ma
ndle
a
2
]
0
;
1[
.Then
T
a
equal s
T
i and on ly i
T
is he memb
e o he ex en de
dY
ag e
's
amily
h
T
y
p
;p
2
]0
;
1
]
i
[6], i.e.,
T
y
1
=
T
M
an d o
p
2
]
0
;
1
[
, he -
no
m
T
y
p
is ge ne
a e
dbyana
ddi i e ge ne
a o
p
,
p
(
x
)= (1
0
x
)
p
,
x
2
[0
;
1]
.
4
No e ha he p o o i
sbasedonam
o died Ca uc
hy un
c i
o nal equa-
ion. Fu he , le
b e an addi i e gene a o o a gi en -no m
T
. Th en
T
=a
has a n a ddi i e gene a o

(
= a
), see[5],and hus
T
equa ls

44
R.Mesi
a
T

=a
i an
donlyi
di
e
s om

(
=a
)onlyb
yam
ul i
pli
ca i e
cons an
.F
o n
il
po
en
-no m
T
wi h heno
medg
ene a o
( his
i
s hecase o heY
age
's -no m
s) hi
sm
ea ns ha
=a
is hei
den
i
y,
i
.e.
,

(
x
)=
a
1
x
.W
eha
ejus shown hene
x esul .
P oposi
ion4
L
e

:[0
;
1]
!
[
0
;a
]
,whe
e
a
2
]0
;
1[
,be an inc
e
a
sing
bije
c ion.I

isno line
a hen heo
nl yc
on i
nuous -no
min a ian
unde

-dila
a ionis he s onges -no m
T
M
.I

isline
a
, hen he
onlyc
on inuous -
no msin a ian unde

-dila
a iona
e hememb
e
s
o he
amily
h
T
3
a
;p
;
p
2
]
0
;
1
]
i
,whe e
T
3
a;p

h
<a
n
;a
n
0
1
;T
y
p
>
;
n
2
N
i
.
4
No e ha
T
3
1
=
T
M
.F
u he
,i isusual
op
u
T
y
0
=
T
W
( hele
l
im
i m
em
be o heY
age am
il
y).T
hene
ac
hm
em
be o
he
am
il
y
h
T
3
a;p
;
p
2
[0
;
1
]
i
,whe e
T
3
a;
0
=
T
3
a
,isi
n
a i
an
unde
he

-dil
a a ion
o

(
x
)=
a
1
x
.
Re
m
a k2
No e ha
he

-c
on
ac i
onac sasanin
e
seo he

-
dil
a a i
on, heopposi e be
ingno ue,i
.e
., o a bi
a y -no m
T
i is
h
T
(

)
i
(

)
=
T
.No
w,i
isob
ious ha i
agi
en -no m
T
i
si
n
a ian
wi h e
spec
oagi
en

-dil
a a ioni ha
s
obein
a ian
alsowi h
espec o heco
esponding

-c
on
ac i
on.
Re e e
nces
[1]D
em
an
, B
.,De
o m
a io ne
n
on -No m
e
n,i
h eSy
m
m
e ien und
Sym
me i
eb ec
hungen,
p
ep in
.
[2] F
odo , J.C.
,A
em
a k on cons
u c
ing - no m
s,
Fuzzy Se s and
Sys ems
41
(19 91), 19 5-1 99 .
[3] Sc
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