View of Iterative systems, a topological and categorial approach

TIEVIJB:

I}'ANAI,YSÐ

NUBTÉTB-TOUE

ET I}E LA

^THÉORIE

nÏì L'IïPPROXIil{/ïTION,

^Tomc

2,

1973,

pp'

37-zrB

ITERATIVE ^SVSTEMS,

ATOI'OLOGICALANDCATEGORIAI,APPROACH

by

i\{. JA],OI]trANU (c1uj)

Introduction

11.

1. T{clations

I,etAbeasetandletRarrd.Pbetwobinaryrelationsin'A,i'e.R, p C-Ã

^x-

Ã. The following

^notations

will be

used:

RP :

{(x,

Y):12 e ^A((x,

^z)

e ^R

^{and (2,} Y)

e

^P)

R + ^P : _{(x,Y)

:

either

_{(*, Y)}

e ^{R or}

^(x, Y)

e ^P}

-it-1 : {(y, x):

^(x,

y) e

^R}

Ro : J -: {(*, *)'. x

^e,

A}

R'' : llp"-I ^for

^an5t

^{integer n}

^2-

^|

3B M, JALOBEANU 2

R+ : >

_{{Ro :}

i > 1}

(p1us closure

of

^R)

l?* : )

{Rd I

i > 0}

(star closure

of

1l)

1t^ : R + I :

{(x,

y): (x,y)e R or x : !}

R" :R-I:{(x,y):(x,y) €R andxty}

R(*) : _{y:

^(*,

y) e ^R}

^(the

^{fibre of R at}

^x)

R(Y) : _U{n(ø) :xeY} for any Y CA

Dom

R :

_{x,:

R(*) ₊

Ø}

_E:_R*ll(Rr,¡-r

Rv

: Rn (y x Y)

(the

restriction of R to Y CA)

RCP i"fÍ((x,y) cRimplies (*,y)ePfor

^any

x,yaA).

The following

results are well-kuown:

_ l.rgpgsition

^1.1.

^{R" is}

^{the sm,øllest rcflexiue} rcl,at'iotc corøtøining

R . R+ is

the snrallest trans'itiue rel,øt'ion contøining

R

øncl R* is tl'te smøl,l,elt reflexiae and, trans,itiae rel,øtion contø,iníng

R,

Moreoaer :

(R+)":(R^)*-R*, (R")'*:11":(Ã")'r',

(R'F)"

:

R+,

(R*)R: R(R'.) :

R-F,

E 'is øn

eqw,iualence

and the

reløtion ,iniluced

bu

Rr. ^,it,t, the qwotient set

AIE

^,is øn bycler

(i.e. (AIE, R*)

øs

o '

_set).

Ð e f

i niti

^o

n

^{7.1. Tke}

_þøir (A, R)

,itercttiae _s)tstetn,

A

_Y

_C aill _A

be _th,e callecl _set tlte

_t(Y)

set

:

of

Y

^støtes

-

^Dom ønd

Rï R - th

u,íl,l be catled tl,¿e set

of

^io¡t,. exí,t

^f'or

states ^ø-sul¡setof

Y;

^{thg set} i,(Y)

:Y -

^Dom

(R")7t --

^{l,ht set} of enírøttce stcttes

of Y.

Pørti- cul,ørly, the set

t(A)

u,ill, be call,ecl the set

of

terntinø|, stat;es artcl

i.(A) -

^the

set of.

initial

støtes

of

the itercttive systent,

(A,

R).

A relatiorral machine [8] is àn iteral;ive

system (A,

R)

such

that

'í(A)

at(A):Ø. ^{A Pawlak}

^machine

^{is an iterative} syst"m'(,4, R)

such bhat

for

^arLy

tí e A, R(x)

has a'b most one element

(Ã is a

single-r'alued relation).

P" Some topological aspects

ô irER^rIvE sYSTEMS ³⁹

J

re

^sþa'ce.

--

R"

(X)j

^wilL

be

cølled' ^th'e

R 'in

the set A.

.A, c*(X):

U{cn(ø)

"KcX}.

Ë

ctosuie þreseruing. Eaery ^closcd

of ^at'L

^cl,osetl

^s¿ls "I

^(A

, _'o)

^¿s

ñ;tty

^{o¡ all closed sel's tttuy be tlefittes}

1) A

^set

^of

ternr,inøl' støtes 'is cl'osed"

2) IÍ Z

^'is

^ø

cl'osed' set ønd'

R"(y) (Z

^th'en

^Z \)Y ^is

closed"

Proof.

Z C r(A) ^inLplics

^{R" (Z)}

^-- Ø ^and tlieri

_'n!?)

.:

^Z

','

^also

o^rz'¿"i)'->öi ^U'RLZ çY).:3..U ^v U RJp.- z l)Y

^sincc

^{Z is}

^a

"fàã"¿"."i ^and RY'(yí¿à-, rìí"i i. ^R(-yIc

^z.

¿v.

^{As we1l,}

^{it is}

^clear

^th:t

the

conclitions 1)

*äì¡-à"iåri,,i"" in" tã-ittv {R*(y) :Y _{ç_A}.}

poiLts

of

^{(r4, co).}

J.et

unbc the topolop.ical-mod-i-licatiotr of t]rc closutc ^cn'

'i'c' tlre

^clostire

defi'iá ifr ^un@¡:-n ti:

^-X

^C | :

^cn(Y),

^{Y C Aj}

Propo-Àiti ^{orr'2.4. Tlrc}

toþotogical ruorl'ifr'ca|iott' u'^

oI l'ltc

^clostt're oþeralion ^cn

is

^{t,tte cLosure oþerøtion}

gr;;r";;;' iî"t'i,í'iin' ¿oti'i"'o-f R

^(briefly

l,et

cn: exp

,4

--> exp A

aty X (,4

(exp

A:{Y:Y It is clear thai

cr(X)

c"(X UY) :

CR(X)

U ^{r^(v),}

be the map

defined

b1'

cn(X)

: R"(X)

^for

C

^A}).

:XUR(X) :XUR"(X) and cn(Ø):Ø,

Consequently:

ø

I

h

M, J.ALOBEANU 5 lTERÀTIVE ^SYSTEMS 4I

40 4

sei

of

^(tI

,

^cn) containing

X, for

any

X Ç ^A,

^{therefore c¡¡*}

^{is the}

topologicai nrodification

of

cn and rvi11

be

denote<l bS, tt,n.

Proposition 2.5. For

euery su,bset

X of A

^tl,tere

is

^{c¿ swbset}

I

of

tl,te ut'it'iøI sta.tes suc/'t,

thøt X Çu,"Q).

Particwlørl,y,

i(A) is

^{dense encl}

euery s,u,bset

Y of

nonintttal, støtes

is

^{(r. nowlxere ¿lense set}

tn

^(A, wn).

Proof.

I:i(A) n ^{(R*)-l(X) and then}

^/?'F(,I)

) X. Particrlarly,A ç C ^{Ìl'r(iØ)) that}

^is l(.ul)-dense set

in

(,4, wn).

IlY _{C A} -

|,(A) then øn(Y) _¡-¡

I ^i(/1) :Ø

therefore un(A

-

^o4(Y))

:

A.

3"

Sor¡e categorial aspects

Ilelinition 3.I Let (A, R)

^at'r,d (Lì,

P¡

be |,zøo 'ilerøl,iue s3tsl¿11xs:

øri,clf

:A*>B ø

møþþing

of ilte

set

ii,iu.to

tlte

set B. f:(A,Il)->(B,P)

is

^cølled. a morþl,t,isnc of iterøtiue qtstems (briefly is-morphism)

iff

^(*,

^y) e

^1Ì

intþh.es (.t'Q"),

JU))

^€.,

^P for

^{øny K,}

y e

A.

It ^is

easy

to verify that the

composition

of

tr.r'o is-rnorphisms

is

al, is-nrotphism. Aiso

the

iclentical

nappiug e: (A, R) -, ^{(A, Ã) is}

^{an is-mor-}

phisnr. Conseclu-entl¡' :

Propositioir

3.1

AU

iteratiae systents w'ith, rnorþloisms

of

^,iteratiue

systems

forru ø

^{cøtegory denotecl by}

Is.

Proposition 3.2.tÍ Í:(A,R)-->(B,P) is ^øn

is-rnorþh,isno lhen

f

^:

^(4,

^{cn) -+}

^(8, c)

^ønd

_f

: (A, un)

-, ^{(Il, ur)}

^øre cont'ittwoots uøaþþings.

Proof. tr'or

ary

integer

.i> l, (*,y) ç. R'inplies the

existence

of

^a,

sequence ^q.o,

at,

, . .,

a¡ tvittr cto: x, Øi: y

ancl (a,r, cto._r)

g _{1l for}

/¡

: :0,

1,

...,i - ^{1. 'lhen}

^Í(oo)

: _f(*),

_Í(o,o)

- "fb,) and

(J@o),

f(ø,,*,))

^e:

P

f.or h,

-- _0,

_{1, .}

..,i -

^{1, narnel¡'}

(f(x),Í(y)) g'

Pt, llence (x,

y) G

R* iurplies

(Í(x),l(y)) € ^P'i' for

aTry x,

y

€:

A

ancl theref ore _f(ut,n(XD

:/(n*(X)) _C l'*(/(X)) : _uoff$)), for an¡' X çA.

^{Besides, (*,}

^y)

^e.

^{R^}

^is ecluivalerrt

s,itlr (r, y)

= R ar x: y, then (f(*),f(y))

e:

P _{or Í(x)} :f(y),

nanely U(*),

JJ)) ^{æ lrn,}

^îor..

^any

^)í,

^{y Ç 1i.}

Consequently,

ÍGn6)):,f(Ã^(X)) C CIr"(j(X)): rr(f(X)), for any X CA,

Propositlon

3.3

lto

the category

Is

ø morþl,r,isru,

f

^{r,s cL} lmor¿.onloy-

þl'tisut (epimorphisn) i.f øttcl only

if it is

øn injecl,,iue (surjective) nrøþþing;

"f ,t iln

isotnorþlûsrn.

,if _{ønd only} .if

,it

is a

bijectiue is-morþhisno ^ønd.

(Í(r), fbù) e P

irnþlies (r,

y) e

^R

_for any

x,

y

€.

A,

i.e. ^,iff

f

^øncl

f-'

^nyt

is-nt orþ h,isnr,s.

Proposition 3.4

Two'íterat'iue qtsterus areisom,orþltic

if

ønd only

iJ ^the

ittduced, closwye sþøces

cre

loorceon,torþl,tíc.

Proof. (lt, R)

and

(8, P)

are

isomolpliic iff

there

is

an isonrorpirìsm

Í:

^(A,11)

+ (8, P). Theu f

^and

f-1 ^arc

is-morphísms hence continuous nrappings, i.e.

/

^{is a} horreomorphism. Conversely , let

f

^be

^a

homeornorphisru oI

: cuj6)) tlre

closure space (,4 and therefoLe cn),

/(R(X))

onto

the Cf6)

closure space

U P(/(X))

(,B,

for

cr,), then

any X f(r"(X)) çtI.

^B:uI

-

(x,

y) e Ìt ^iruplies :,c ^{R(x) ancl then} lU) ef(x) U pU@)) tliat

^is

(Í(x),Í(y)) _G P

since

/ ^is bijective.

Consecluently,

/ ^{is an}

is-morp1'risn"

Similarly, ,f-' ^{is an}

is-rnorphism

and

then

f ^{is an}

isomorphism

in

^the'

category Is,

If

Ens is

the

category of sets

let F

:

Is +

_{Ëns be} the forgetful functor (i.e.

the

^furrctor

wich

assigns

to

each

jterative

_system

the

tLnderlying set

òf stut"s,

and

to

each is-morphisrn

the

underlying mapping

of the

undcr-

functor F: Is + Ens is

_føithfot'I

oacljoint

functor "F.

^Moreoaer,

_for'

(A,

A x

^A).

f :

^S

if

ancl only

if ;ì(/) :

¡r(S)

U): A -- B). After that, for

any -8,

the

set

of

is-morphisms from

(8,Ø)

inLo

(A, R)

coincides

with the

set

of

mappings

of B into,4

since f

tøl e ^{IÌ lor}

any f ^:

B

^--+

A,ther'"I'is

a coadjoint of

F' Sitrilarly,/(n)(

'È xE ^for

^atry'

j: _A

-u

B, anll

then the set of mappings of ,4

into B

^is

tlre

same

that the

set

of

is-morphis ns

of

(,4, 1l)

iltoi (8, B x ^B),

^colse-

quently

F"

is an adjoint of

F.

(/ãenotes the napping induced by

f

^:

^A

^{-n B}

fronr tlre

sqtlare

of

¿L

itfto

square

of B, that is f((x, i,)) :

(f(x),

f(y)) ^lor ory of

^{iterøtiae systems}

is

^comþlete

colim'its.

îJ' J";îå'îiÏ3:"lt: "3i'äïåJ' li?

tive

systems. V/e define an iterative system

(A, Il)

with

A : IIA¡

(in Ëns) ^ancl

R : (\ _Þi'(Ro),

^ivhere _{þ0" A} ^{-u Áu.}

1ár 'i

*.i ^{iu th"}

^canonic proiection

(in Ens) and

^-þ¿:

A x ^A _-,

^Ao x,.Ao ^is

,, In other

rvords, _{i|. {xo:}

i e I}

^and

tlre product of the fami111, {(Ao,

R):

is

an is-morphisrn

for

an1, 'i

e I

^t]ne

tlrat

s,

:

_þ¡s

Ior

^{ariy' ¿}

€ /,

^sjrlcc

Îamilly {A,:'i

^e-=

/} ^{(in Jins).}

^Bu

(so(r), s,(y))

e R ^then

^(þos(x),

(s(ø), s(y))

e ^R.

Consecluently, s:

For two

is-morphisns

f, g: ^(A, R)

->

(8, P) u'e define an

^iterative

systenr (I{, Q) with ^K :"{i: x

€:'

A', _l@) : _e@)} and

Q

: Ru. I,et h: (I{,

_Q)

-, ^(A,

^-R)

^{be the}

^inclusio

an is-norphjsm. lf j:

^{(C, S)}

-' ^(/, h : _Si thcn there is a

mappirrg

is thè-differcrcc kerncl

o{

/, ^{g (il}

(j(x),

j(y)) :

(kk(x), hh(5r)) ۓ

R

and

h,

is

an inclusion,

i.e.

^/¿

is

an is-rnorp t'euce

kerncl

for

l, ^{g in}

^Is.

Coproducts

ánð diff"t"nce ^cokerlels may be

construct

in the

^sarne tt.onn"i, Consequently,

the

category

Is

has products, coproducts, difference.

.42 M. JALOBEANU 6 ? IrERATM SYSTEMS ⁴³

Proposition ^4.2.

^{T'lt'ere øre three} functors:

"C:

Cl

-, ^{Is, "U:}

^{ToP -+}

^{Is, "O:}

^{Ord -> Is}

swch thøt C

is

an ad'joint o.f

"C,

^{(J 'is a'n acljoi'nt of}

"U

^ønd

O

'is a'n ad'joint o.f

- "o,

Proof

.

We have

the

diagram:

kernels ancl difference cokernels and

then Is

has

any limits

and colimits.

Particular-1y, s¡e

can

construct

the

intersection

of a familly of

_.iterative

systems

.trã th"

pullbacks, images and inverse images etc.

in

a similar way'

4.

Some

functorial

^asPects

Let

^{Cl be}

the

categor¡'

of

closure spaces and

Top - ^the

^{categorl' of}

topological spaces

(witñ

côntin11ol1s maþpings).

It is

known

that

C1 ancl

Top

have

limits

and colimits,

The association of a closuïe space (,4, ^co)

to

an iterative system_

(A,

R) delines

a functor c: Is + cl; the

asèociaïion

of the topological

^space (A,

rn) to the iterative

system

(A,,.!ì)

defines

a functor U:Tt _->Top' Ìn" píoporition

3.2 providês

that'C(f) a4 q(fl

^are. morphisrns

in Cl

^rcs- pectii,e

in Top, for ãiry

is-nrorphism

f

^:

^{(A, Â)} +

(.B, P).

The

association

of

a reflexirre and

transitive relation Rt' to a

relatiotl

R

defines a functor O of

the

categoly

Is into the

category

of

ordered ^sets (denoted

by Ord).

^Incleed,

the proþosition

1.1 provid.es

that

(.418,Il'4)

ìs an

ordei"d.

sei for any iterative

sy_stem

(A, R).A1so,

^frcim

the

proof of the proposition ^3.2,

î(R*) C ^P*

^alrd

¡1Rx-r) C P*-t

^Tor any is-mo1plism

f

^:

Ø, nl -'

^Q3,

P).eäòoráingty, / ^is

^{án isotone mapping}

^{of (A,}

^R'R)

^into

(8, P*),

where

A:

AI(R'N

llR*-l

^and

^B : Ble* ll ^P*-'¡.

proposition 4.1

Tke _fømcto.rs

c,

^(J,

o

defined c.tboae øve føithfor'l ,ønd, litniús þreserain'g fwrt'ctors.

cn) -u (Ao,

cn) is a

^continuous

an

is-nrorphism,

for

^anY

i e

^L

lor any i,

^tiherc

^{is a}

^continuous

re

is a napping

^{s :}

lI'

-->,4 ^(since :'

é) f ¿

^:

^{::èi *}

^r

t

å'',".*"0

!;rLrl": 2

(t(X))

(since {/o ^:

i

e:

I}

aLe canonic projections

in lins) for

a,ny set

X _C.,4. 'Ihen

s

is

a continuous rnapping

änc1 therefore (A, ru¡ is

t¡"

_þroduct ^oT'the

laniliy _{(Ar,

^{tur)} in the category}

of the closüre spaces. corrsequcntly,

the functor c

^preserves

the

differeilce kernels and

thé

proclucts a,nd

then C

preserves

the limits'

C1

C

U

I"

Fr

3

fs -__ *'

^{'I'oir Ens}

O

rvhere þ-t,

Fr,

Þ-" are correspondinS; lorgetf-'11

fu¡'ctors

^ancl FrC

-!rU -

^Þ-.

All

this

futr"lotJ

^are

faithful

^{and there ane}

"Irt -

a coadjoint of Fr,

"þ-, -

a coadjoint of

F,

^and

"F" -

^a coadjoint of

Fr.

Since

the

catego.ry^Is,has

linrits änd the functors

C, ^(J,

O

are

limits

preserving functors,

it

Îo11ows

from

theorem

4 of ⁽¹⁾ that c, u,o

^have

^{ad.joilt functors,}

respectively,

,,C, "U,

^"O,

We can construct

the

functors

"C, "(J, "O ^directly. Let

(tl , c)

be

^a closure space. We d.efine a relation

RCA x Aby ^{(x,y) <=Il ii| y}

^a.c(x)

for

any

i, y *.A.It ^is

^clear

^{that R is}

^{a reflexive}

^{relation. R is}

^transitive

iff

the clo;úre ^c is topological,

and

R is an order

tîf Ø,

^{c) is}

^a

7'o-1.opo1og;ica1

space (i.e.

x

æ

c(y) atd y e

^c(x)

ⁱ

a functor C' : Çl _->

Is

and a functor

+ Ord

r,vhere Zo

is the

subcategory

-f : (,4, c)

-,

^(A' ,

c') is a

continuous implies

f(r) C"fþU) Cc'(f(y))

^anð,

lying iteratir¡e

systems. lVloreover, quasidiscrete modification of the ^clos

systemand (8,t) a

^{closure space}

C'(8,

c)

- ^{(8, R,).}

^I"et

/ ^be

^an

is for any x,

y e

^B,

^{y q}

^c(x)

^inp (

^cr,

^(/(c)) and

therefore

f

^:

^(8,

^c)

Conversely,

iÎ g: (8,

^{c) --+} (A, c") is

C ^cn(/(X)) for

an¡,

X C ^B)

^ttren

^y

U n(/(r)) and then

(l@),

f(y)) e

-- (A, R) is an

is-morphism. This rnorplrisrns o1

C'(8,

^c)

iuto (tt, ^ll)

^and,

44 M' JALoBEANU I

l)rove

Lltat

C' is a

coarljojlrt

functor of C (tliat is

equivalerrt rvith

: C

^is ärr

adioiut ^for thc ìr;.i;;

^C')_

arrcl.ru"'"át'

^replace

C' by "C' T¡e

^proo{

äiË';';; ^iiì"; ;""ì;ìäi'c, ^b;u"t " ?î,,Tir!r,,:äfit

conseqttc'trv:

c(x\,Yx,^/aA ^s

⁴

slrntts 'intô

lhc c

^o-[

y; :.'ß: ffi f ill 1';:î"",1J ùoi;"ttt!ãt,

^{is a}

1o -

^toPolo-

is

arr

ordel' In this

wa)¡, ^we

ji'?'jllïh1'#'ïoli

^"

_:i'

T,,'l"i å:"ìi:

i:

4.1

^{alnð' 4.2.}

Ccmbining

the

above results

we

^obtain:

Proposition4'4'Tlceveat'eten.flt'ncl'ors'øcljoitr'tsinþøirs'so

thcú

ih;iå'U"*t"g

cliøgrøttt'

[o

be cotnnt'utøtiue

g lrIRÄrlvE SYSTEMS 45

øt'iott,

of

atç iterøtiae sltsteut

(A,

R) : ø

e /)),

^zaltere

R: )

{11¿ :

i' e I}'

P

r

o p o

siti

^o

n

^{S.S Ttce ttoøþþing (A,}

_{R,: i a I}) -' ^(A, ¡l

{Rt ^:

øe

ç.

l\\

^ctefincs a fai;;lntl futnctor Z : Cis --> Is.

'Þroo¡. Let i:

^(A,

{R,:

^{i' ez,I})}

-, ^(8, {P,,.i.eJ}) ^be

^a

Then, foi any i e I,'(r", y)

e=

R, irnplies (f(x)' f(y)) e _4t

(x,y\ *n {n,:i e1} i-i-,tics (I@)'f(y)) e O {P,:i e ^tl.

V(ji'-1,Ø, ñ ^{R,.t:ie l)) -n(8, O {Po:l e 1}) is a}

^u,or

category Is.

P

r op

^o

siti

^o

u 5.4 The _{Y: \A,} ^R) -Ì g,

_{R,:^'i'

eI}).,

uhere ^R¿

|

_R

_¡o,

_{ør'ty 'í Q.}

I, def'ír

^hfwl' _-fomctot'

^{of Is}

¡nto Cis' Tltis

frnr,ctoris an

^adjoitt't

of S

and

a 0f

V.

the set oI is-morphisns of

S(.B,

{P

bi ) ^are

natura'ilY ecluivalent'

nd

tl

en the category

Is

can be considered'

a

. Proposition 5'4 proved

that

rhe category

Is _'e

subcategory

of the

category Cis'

proposi tion 5.5The

cctegory ^of .cotnþ|,ex itera.tiae ^systenr,s is cotnþl,ete _- and, cocomþlete.

Proof. In the saÍre ^lnanner as in

_proposition

3.6.,

^r,ve

have to pfove titat the category Cis has ploclucts unq difference

kernels.

Let

{(A¡, {R¡¡:

i e /)

^:

¡ a J} ^be a familly of

complex

iterative

^systems.

For any

'i

ç

^{.1, we define R,}

: _U

_{

is

the

canonic projection (in E ra1ly irducedl..y _þ¡,

j _{e J.}

The

with the canonic

projectiors

{(A

j, _{Rjt: i

t,:

I}): j c _J}.

^Properiy

þ¡

is

a cis-morphisu,

lor

any

j

e, J.

morphisms

of

a cornple

{(A¡,{R¡ ieI}):je

s¡

:

_þ¡s

for

any

I ^e,J

But for

aî5r 'i G,

I,

^x,

ls

CI

o

t,o ^Ord.

'lop

C LU

U

^{,,7: T}

l

tl

I'rool. Tlrc.[u[ctor "J:To|.-+

^c]

"is

tlrc

iirclttsion (a topological ^si'acc

js a

cLostrr.c orrc,¡

arrj h"n'^r,-aäjoirrt fuilctor' / -

^tlLe topological ^{llì'odl [l-}

cation

o1

thc

closure Sl,ace.

The cornrnutatiíity

_{.of .} ^{the -diaPram follorvs}

fronr the .orrstructiãrr'of tn" run"torä]"Þätiiätá¡y, lC : t¡ :

TO is

just the

ProPositiorr 2'4'

5.

Cornltlex int'cral'ivc ^sysloms

D c

ti

ⁿ

iti ^{olr 5.1' Lel} I

^be

n

^sel'

^ølltl

^lel'

{R:i e./}

^bc

^a

^fant'ily

^ol

*.r"(.'i, in, _' ^I e ¹⁾⁾

^'wiLt

be'cnlletl

^ft'

,:

^{'1 G}

/)) ^be

^tr'vo

cornlle" i1"t"li)::

1ed

a

^mórphisni

of

cortlllex-.iteratlv'c

.(x'Y) e R,

^irnPlics

^(l(*)'

J\Y))

e

^t'¿

Propositions.l'At't'conoþlex,iterø{;iaeqtstetns.aithtnorþlrisnt.s.

"¡ ,inipirí1t"ioi¡r'i -iltrloì,'

iorno a-ca'eg,ry denoteil Dy cis'

M. JALOBEANU 1ß ¹¹ ITERATIVE SYSTEMS 4V

Proof, First

part is easy. F'or the second, 7et x,

j

be

two

classes modulo

E, and iet be

(*,3r)

e ^Ri.

^Then,

^{for any n} e.t

^and

y e,y,

^(x,

^y) e

^R,

iff E is a left-permitted and riglrt-permitted

relation.

P

r

o p o s

i t i

o

n

6.4 (I-st isomorphism theorern).

Letf

: (A,

{Ilt: iel})

-u

(8,

_{Pn:

i e I})

be

a

cis-n+orþhistm, ønd

let K(f)

be the h,ernel

"f Í If (All{ff), {Ri i ^a /}) ls

tl'r,e qwotietr,t comþLex iterøliue systenr, øncf ff@),

{R^n,: í e I}) is

tlte f-incage comþlex 'iterøtiue systenr, then

ue

ltaue tke contrnuíat'iue d,iagr am, :

" (A,{R¡:ie.I\) r - (8,{P,:ieI}) ,J _1'

(,41K(f),

{Ri

^:

ie /}) 4' (f(tI),

_{Itnnn, :

i

^e=

l})

u,lrere /z

is the

canonical cis-morphism,

g is the

inclusion

of the

_f-image

- ^{that is}

^{a strong} cis-mononorphism, al.: d

f

^is

^the

^{bijection indt',ced natu-}

ra11y

by Í.

^IT

Í ^{is a}

^strong cis-morphism then

/ ^is

^an isonorphism

in

^Ciss,

ancl /ø

is

also

in

Ciss. ConsequentI5,,

if f

^is

ⁱⁿ

^category Ciss thelL

all

above diagram

is in

Ciss.

ProoJ. We have

the

decomposition

f :

Sfh

in

Ens.

If / ^{is a}

^cis-mor-

phisnr tlren g,

f

^{and /z} are cis-morphisms.

the

propositions 6.1, 6.2 and 6.3

conplete the

proof.

Proposition

^6.5

^(II-ed

isomorphisn theorem). Let

hbe

øtt' inclu- sion

of (8, U',:

^i.

e.I\)

'ínto (A, jl?o: i.

e: Ij)

^an'cl

^{lel E}

^{be øn'} equiuøl,ence

i,n

A. .tf AIE

i,s considered. øs

a

swbset

of A, let (B', {Pi:

^i,

e I})

^{be the}

contþlex 'itera.t'iue systenø deþnecl

by B' : B ) ^{(AlE), Pl'} : P¡s,, i eI.

Tltere ?s ø

cis-bijection

f

^{: (BIE,}

{Pl:i ^{a I}) -- (B',} {Pi:i €/}). If

^E

'is

a

ccngru,ence thett

f ^{is ø}

^strong cis-'isornorþloisn't,.

Proof. liunclamentaTly

B' : _{{x:7; e tIlE,} ß,) _{B =LØ}}

^and.

x G

^BlEs,

iff t €

^,B'.

^'Ihis bijection (in

Iìns)

is

a cis-morphism. As a

matter of

fact,

we cal-l apply the proposition 6.4 to the cis-morphisrn tlt,

^rvhere

t:(A,{Rn:i.eI}) -r(AlE,{R:¡:i e /}) ^{is the}

canonical cis-epimorphism.

Therr there is a cis-bijectiot f :

^{tlo: (B}

lK(th,), {Pi

^:

i

Ç.

I})

^--> (tlt(B),.

{Rnr,tot: i,

e I})

^{and. K(t/r,)}

-

^{EB, th,(B)}

:

B'.

P

r op

^o

siti

^o

ir 6.6 (III-rd isomotphisn

theorerr).

Let E

^ctttd,

E'

be Luo equiaøl,ence relat'ions

in A

ønd. let

(A, lRo:i e ^Ij)

^{be ø cornþl'ex iterø-}

tiae sustent,.

IÍ _{E' Ç} E tlten ue

lt,aae il'te

folloain'g

corntnottatiue diagrøno ::

(¡,

_{1lr:1

€

^1})

ElA" _{Ri: i e

^{1)) h}

>((AlE)lK(g),

_{{R f':}

i e

¹⁾⁾

*g

46

anvi e. /,thcr

('l>¡s(x\,y'is(r))

*.R,,,namelv(s(r),s(y)) €R,'

Colsequently

t,iÉ, _{{r i:}

^{t. G}

/ji ^-- _1it.i,,

1

lt'

:

i e ^{1}) is a}

cis-molphism.

For

tr,vo cis-morphisrrrs

/, ^g: (tI,{R,:i e /}) + (B,tPr:

ø

e

^1}) _-we

define

a

complex

iterätive

system

(I(,

_{Qn:

i _{e I}) Ylh} /{.:

K--erD".

(/,3}

anð.

Q,: ^{R,ir, for arry}

^'i

^e I. I.et

^h':

^(k,,lQn: i e ^Ij) l^V.{llo: I F ^/})

be

thä'

inclusion napping, i.e. a

cis-morphism'

Let l:

_{(C, {Sr}

"

i e I\) -'

such

that Ít : gt.

Then Lhere

is

a

definition

of

1(), ancl,

for

any

i Q I,

' Í,i

?J"5"itlnï"'?'; ^ß!:h jl?il.n

ifference kernel f.or

f, ^{g in the}

^cate-

gory

Cis.

The

existence

of colimits follows from theorem 5 of

(1) since th.e category

Is is

cocomplete and

the functor

^{T/: Cis}

-' ^Is

^has

an adjoint

Y"

Nattîaliy,

\'/e can cdnstruct

directly

coproducts

and

clillerence cokernels' 6" Isornorphisrn theore¡ns

Definition 6.1 A cis-morphisrn f:(A,{R,:i'el}) -t

-n

(8, {Pr:

^i,

e I,\)

wi.tl be called

a

^stron'g factorizøtion' ^(1,1).

or

ø. strong^cis'

ntoìpki,èn'if, for''an'y ieI

øncl

for ^øn'y

^%,)t

GA, (Í(*)'Í(y))

QPuء

,imþlies (x,

' pròpïsiti'ou y) e

^R,.

_G.l rltt

contþlex'iteratiue systen,ts zø.itlr str.ot'tg__cis- morþh,osrní form ø category d,enoted öy Ciss. Ciss

ls

ø sr,ùcategory of Cts. lVIove-

orr'ì,

Cirs

is a

bøløncld category,'i.e. euery bi.jectioto'is

an

isom,orfisno.

Def initio n

^6.2

^A

^rel,ation

Kç

^be calleclleft-þertn'itted

in

tlte conr.þlex itera.t'iae systettr' (A, _{Rn:

I

^anil

i e' l,

^KRn

_Ç

_4".

Arr, eqati,uøiettce reløtion

*¡u

^{uà cøit'eit}

a in

_{(A, {Rn:} ^i'

a I}) iÍ it is ø

left-þerruitted' ønd.

ø

right-þerru"itte

It

is"clôar

that

the identical

ielation I

^is a congruence

in

er¡ery complex

iterative

system

ivith the

same

sei of

states.

Definition

^6.3

I,et .f,(A,{Rn:i c/}) -'(B'{P,:ieI})

^bç..q

cis-utorþhisut,. The bitoøry relation defined ^irø

A

by

K(Í):{(x,5r): Í(*):fj)l uill

bc cctl,l,ed tlte hetnel

of

the cis-nr,orþlr"isnt

f.

Proposition

^6.2

^For

^øny cis-ncorþhism

f, ^{K(f) is}

^øn eclu'íualence

reløtion.

K(f) is

a. congluence

iff f ^{is a}

^strong cis-morþh'isn't"

The proof is

easy.

I,et

(A,

{Ro i e ^{1}) be} a

complex

iterative

system

anrl let E be

^an equivalenòe

ii'A. If ^{Ri is the}

rest?iction

of

R¿

to the ^quotient

^sel.

^AlE,

fo'r

any i e,I, ^{then thå}

^complex

^iterative

^systäm

ØlE,-{Ri: i e /}) ^will

be calied the quotient

complex

iterative

system

relating to

E-

Proposition 6.3

The canon'ical ruøþþing

f :A-nAlE ^is,ø.cis-

eþ'imorþhisrn

in

the qwotient comþIex iterøtiue systery,

f9r ^&ny

equcaøle'nce

E; Í ii ø ^strong

cis-eþiruorþkiswiJ

and'

onl,y

if E t's a'

congruence ^1"t1'

(A,{Rn:i=I}).

(AlE,tR::ee1))

where

f

^,

f'

^{and /t. are canc nical} cis-epimorphisms

ard

^h,

is

an isorrorphism"

If E is

a congruence then

the

above diagram is

in

Ciss,

i.e. all

morphisms are strong.

$

Áo M JALOBEANU 12 +(t

"å,""':;::iî"å'nJåffi

"fi nl.fl iÏ

^fn"¿

morphisms since

/{(g) is

a congruence

ir.

^A _lE' ^.

kem,ark.

The final

results can be transposed

lor the

simple ite,ratir¡e

ryst"Àt.--Ái.o, th"

most

of

above results are

indepe'dent of the kirtd

of ,álotlorrn, and

they

^m.ay

^be

formulated

for n-ary

relations.

REVUE I}'ANALYSD

^NUMÉRIQUD

ET DB I,A

THÉORIE

DE

L'APPROXIR'IATION,

Tomc 2, ^lÐ73, _{Pp. 49-53}

RÐIIÐRDNCES

[1 ] ll ^{e rr a b o} u, J., ^{c.riièrcs d.e} reþrésentøbili,té ^d,es foncteurs, c.R. acad. sci. ^{Paris, Ð6{1,}

752 -755 ^(1etis).

[21 ]l 1i k 1 e, 4., Iteiatiue' syste¡xs; an. alg,ebrøic ctþþroaclt, B'11. ¡\cacl. Polou. Sci., Sér. Sci' Math. Astronorn. Phys , ^P0, 51-55 ^(1972)'

L3l Il 1i k 1 e, 4., Co,tþIer it¡tuîatiue syslems, Brill. Àcad. Polon. Sci., Séi:. Sci. IvIath' .Asttonom. PhYs., ^P{}, 57-02 (19:

[4] ^{c c c} h, [', Toþologicat Z. ì¡ioliir ¿nd ì\I. I(atetov) Prague, 1966.

iÉl G; ^{ã r'g} ". ^c u, ^{G. "} a.ird. ^{Stu' la} cató.gorie des systèmes logiques, F.ev.

Rãutn. Nlatir. Pure , ^{489-495 (1969)'}

[6] II ^a t c h c t, \á. s., syslcril.es forntels et catégoyics, c.l{. acad. sci. Paris 460, 3525-3529

( 1e65).

l7) ^J al o l¡ e a ^r.r u, I\[., ^Son,te toþological ancl caíegot'ial ^a_sþccts oJ c.omþtr'ter theory, \I-athe- rnatical lrorindations 'o{ _ðornpuLet Sciencc, Warsarv, Àugust, 2l-27 ^(1972)' [B] Ks.aso*.iec, $/., Ilclationøt Maihines, Bull. Acac1. Polo., Sci., Sér. Sci. Math'

Âstr-onorn. Ph)'s., 18, 545-549 ^(1970)'

[g] ^{iU i} t c h e l l, ll., Thío,ry of' Categori.es, Àcacì.emic Prcss, Nerv lorl< ancl l,orrrloir 1965.

¡iOj f ^{as. 1a1<,} L., ùIaszlnty Tirogra,tlooaatoe (iir^polish), Algorytilry,_ 1t|, 5-19 ^(1969).

ilfi S"hecht"t, ittot:lé-ïlicheliite, Cot¡tyibr.t!,ii ìa ^studittt algebricoJogic rtl s¡:rttc'

tttr'ilorrclalionale,Doxtc¡r¡'lTliesis,UniversitateaBucuregti'1972' Rcccil'ed 1 XII 1972.

SUPPORTING SPHERES trOR FAMII_.IES

OF SETS IN PRODUCT SPACBS

l)y

HORSl'KRAX'IER

(c1nj)

In.sLitttt ttl cle I zotttþ'i Stab'ili, Clt'tj

I.et

fuI

be a

given set

in the

n-dimensional Ëuclidean space R". We

slrall

^clenote

by

conv IVI tlne convex

hull of Il[,

^i.e,

^the

intersection

of

aii corl\/ex sets

in R"

which colltain

the

set

M. Itt the

following we lleecl the

llotion

of convexly conllected sets introduced

by

o. r{aNNErt ancl rr. RÀD-

srnÖu iu tsl

Def iiritio n l. A

^set

^M in R"'is

callecl clnüexl,y connectecl,'íf there 'is no hyþerþLøne

If

such tlxat

H À M : _Ø

and'

M

contøitt,s þoints

it't

botlt' tl,r,e oþen lcøl'fsþøces deterntinetl by H.

Retnørh. The notion of convexly connected sets is used

in

_[B] also

u'ith all other

meaninS.

It

is easy

to verify that

a collllected set is also convexly collnectecl a1ld

that the union of

convexll¡ connectecl sets

having a point in

commotl ís

con\/exly connected.

Def initio n.2. Let M

^{be ø set}

in R". A

mãxinl,;LlconueriycotLtLec' ted, sttbset

of

fuI

uill'

be cølled ø clnuexly connected col'lxþltxent of Ìl'L

O. Ifanner

and

H.

^Radströrr harre

showt'chat

^there

is a

1lllicll1e de- composition

of a

set

M itfto

convexly collnected cc1xpone11ts.

îhe1'

[1¿1rs

also provecl

the

following

resúlt ([5], corollary

¹⁾

| I1 M (

1i"

is

compact ancl haS

almost ø convexiy

connected cornponents,

then for

each point

1ô

in

conrr AtI there exist

ø

(or fer,r'er) points a.1, ct2,

...,

^{ct,, of r4zl} such',-hat þ belongs

to

conv _{or.,

or, ...

^ø,,).

I-I. KRAr{rìR a1ld

A

B. NrtlrETH

have given in _[6]

bhe

follorving

two clefinitions :

Def inition 3.

^Tl't'ere

uilL

be saicl tl'tat ^tl'te føtnil5t

F of sets'in

a tnett'ic sþøce

E

l.tas

a

srl'þþorti'ng sþh,ere,

i'f

^th,ere

is a

sþltere

S in L

^ltøttilog

4 - Relue clanalyse numórìc1uc et de la théorie de l'apProximatior, tome 2, 197:ì.