(iii) id(x,y) + id(y,z) - 15 id(x,z),. i.e. id is chosen in ... x + y & y z -> xº z ,. (5). Exçwy & yçu ..... 18, 19-30. Gottwald, S.; 1973, 'Uber Einbettungen in Zahlenbereiche.
'62-tt
'tuodwoS Bu1qsl1qn1 pplay .e {q rcA t 'ssauanSorl lo slcadsV,(.spa)sopttl .g puD ,tuuuoJ S ,DID4S.t
TI u r ' r u o p a T q e r r n s e roJ (t")a)edns =({")4xE)pue Jo 41e;a)"Jr-l
g : : T r : " ps ' u o T r " ' i r i ; " ; ü '
-i
>i =.\/0^*,a=) (p( ""ll g : n ) , f : . ( r _ ( b ) *i(öa')-, G 0;' i"" ( ö ) : ( o')(;ar )' )t x; ueruurr= ( b - a ) = ( b t a . ) '!C (b) = ( Q n . >l.( O ) ' , i ö 1 " 1 , " = - ( b u a y '(d>
-t = (a-) :a^eq dnqr a,o er8ol'pn.trb,r-rüeiu'1ä aaeriauei -rno Jo d uorssardxa ue Jo anlp^ qtnrt aqt :oJ ( a) qrlfl 'dns JuT Aq dtanrlcadsa: uanrB .roTlr"iJTtupnb pue uorTezt lerauaS panlen-r{ueur ro3: pasn aJe Jerrualsrxa E puEA (,{TTEuTd ' (gtel salTc uT sE uoTlcun[uo:r p1äq' ,og g pue ) 'uorlectldrulTq pue uorlpJrldurT zcrrnrarseln? JoJ €{ .-os1 al1 ' u n u r r x e u r 6 u l n u r r u r u r; . . dlanrlcadsa-r i rä pauTJap uorrJunf rTp 'uorlcunluoc 6uorleBau Jo sa^r]JauuoJ panlen-Äu€u aql foJ Tsnsn sr se A'v 'r-arTrA sn lar { u>>r}o J (r-u)/r I = tlr rrrr aqt Jo auo tT Jo fasqns atTuTJ aurosro Ir.o] 'sanlen drqs_raqurau Ie^-ratur Tear aroq|'\ aqt 'a'r pazr leiauaä Jo ras aqr sÄen1e_-raprsuoJ rT1l,r an cr301 panTen-Äueu aql Jo sanrE^ qrnfr Jo A ras se puE cr301 panren-dueur 3:o a3en3ue1 Jo salecrpa:d pu' saATlcauuoJ auros 's1as f,zzn1 Jo sanTeA drqsraqurau pazrlerauaB foJ q.apeZ uorlelou rrAou{_TIar\ J o aqt saprsaq .asn TITar alr uorteluasa_rd paTJrun E JoC ' _ r o q fn E luasa-rd aql Burpnl r . a l d o a d -ur BunoÄ Jo dno_r8TIeus e qtTA paliroA pue ZL6L or 996l ruorJ drqs.rossago-rd TrnJ e plarl en'T) araqA ,nzäia1 'dlrs:anrun x J e i , t rT - r e ) a r i r l E Ä l u r e r u p a z : I E a J p u e u a l { l rrio'rJSursrre luaurdoTa^ap auros pue s-raded äroül ur prrrodn, s r l n s a J a q r S u r d a n - r n s l e s u r B - r a d e d s T q r 3a - r o 3 : a - r a r { J. arEp ot qrnu pa8ueqc tou rrq rä-lenlrs sTqr leql siuaas tI rr'Portraca-r a/\Eq ol s-readde lT uBi{l uorluallE eJou qJnu sa^Jasap pue 10afqns aq] Jo luau-l -dora^ap IErTleuaqrBur aqr ur -raded luasa:d aql puodaq -reg .a-raq paldope sao8 IroA sTH leql (ueqt patecrlsrqdos a - r o u rq J n u l n q ) o l J e l r u r s Jauueu E ur crBol z)l-uar.seln? uo pasEq ,d:oaql las panlen-Äueiur e sdola^ap .:. EneI),r t E q l S u r c r l o u ' , , ( 0 1 6 L - S 9 6 L ) e n e l y d q Ä . r a d e dj o , " r r " s palJaI -Bau e,, of, s-raJar . f O L 6 l ) s a I T C U c r B o l p " " , 1 " n ^ _ d u e u tq t T l r d-roaqr f as t{'zzn1 Jo suorrcauuoJ uo dp.ts uAour-ITao' E ur NOIJJNOOUJNI 'I
JNgl^ldOT[AgO ÄTUVI IHJ .{O SI]EdSv :ÄUOEHI JHS LZZNA
il^tos
pleltttoC
per-rgäarg
S.GOTTWALD quantification.
Restricted
quantification
is used sers as resrricrions only and writ.""-ü1",-jnl"ii.n
with crisp inrer_ pre ta t ion by_ , . y r e s p . s u p ; : c M. i . f Addirionallv,'"iä .r""ä rh"'äE;ilership relarion e of classical set theory, the class on of ordinals, the diffe_ rence M\N of crisp sets MrN, the-ioaio.,, d o m f r , g f of domain and range of a mapping f . w"-.rr" the notation (uilier) for tÄ" familv or-ait ui witn index from ser r; and ts indicates rhar rh; f o11o"i"| iorr.,t, of many-valued logic holds rrue, i.e. has (arwalst-;;"rh value 1.
for
e
F X l=x F., t - ^
The m for s (x WITN
many-' -\,v
2.
TI,,üO HIERARCHIES OF GENERALIZED SETS
o X t
X C
2 . 1 .
From
As explained in his works 1g67a) and.in direcr conver_ sation, D. Klaua got the .(rgos, imp.rlsÄ r". nis investigations from conversations with Karl Schröter and Karl Menger. The last named, ir particular, advised him that perhaps he could use for some of his studies , g".,".. rized membership relation a d m it t i n g intermediate v.1.räs between ',true,, and "farse", i.e. between 1 and 0. From t h-el a nv geur ya g eb e g i n n i n g i t \ ' r ' a st h e i d e a t o a l s o u s e the means of of many_ valued 10gic for describing. such a genera Ltzation, and to consider for purpose the g".,".ä1ir"a membership values a s t h e g e n e r a !l li izre d t r u t h values. Hence, the aim of these s t u d i e s r , ü a sv e r y d i f f e r e n t of that of , e . g. Skolem (1g57 , t g 6 0')" i ^ ' J n " C h a n g ( l g O g ) w h e r e the main problem was the cons " . , J i, t".r"f axioms of extens ion_ ality and comprehension in set theory. The starring point for the work oi Klr.ru (lgOS) and the extended version (lgola) is also some set J of urelements over which' inside some basic set an""ry, a whole cumurative hierarchy E (of height,, onry, ;;; ;;;;it"ity) or sers is consideredl rnside ir,ir universe of discourse E, a many_ valued identity = r , , r ,i s d e f i n e d - r a r r a i " , , i a n a so_called ident ity functio.rtia : - l * - r_ , w w i t h p r o p e r t ies : = J id(x,y) (i) itf x = y
tliJ,F!",rJ = id(y,*j
\ lrr/ rd (x,y) *.f 9(y, r) 1< i d ( x , z ) , i. e. id ii-chosen i" s""n a way that by P ( x , y ) = d f l - i a ( x , v ) a merrlc 0 with maxiilat äistance r is defined in J. The recursive definition of = is such ar-,rJ-an" author sers
n
1
) A '
= z t-
fa
_
Y
F
Ä
I
He is ext ens he get '
)
L . L .
)
c U
Perhap start ents i hierar these I charac T o part ia stages 1/ ! _- n \ v,
ordina set of exist r Here we whole r partial B a se notatic
qcns . pacnpo-r1ur sr v3x uorf Elou e juorlcn;JSuoJ sTql uo paseg 'sasJanTun 1er1.red aqf ITe Jo uorun sr slas paz:1e_raua8 f o ^ as-ra^run aloqA "yt 'lO uoprx al.r a-rag l ,,lf,oddns,, aql roJ lCl I (")DlC "YJ ' . .= r { c n s m < - d 6 : g u o r l D u n J p u e r o > / 1 e u t f _ r o" " r dt u e t s r x a r e q l lgl lVl t o u s a o p a - r a q t q r T q A - r o 3 :n < - r a ^ : y , s u o r t c u n J a s o q f T I e J o t a s aqf r{lTlr to^ Jo uorun aql sr ,- oA f as aql , | + ,osIEuTpJo -rossaJJns JoJ'y.€r TIe Jo uorun eqf sr 16 ,o =y T n sleurpro tTurTI leg 'i = oA . uorlrnrtsuor sTqr Jo sa3e1s" a A T S S A C J n Sa q l u 6 : r t o A d q a f o u a p p u p s a s J a ^ r u n 1e-r1-red . ITEc sn tal sTqf ureldxa oJ T T B fa p _ a ; o r u u r u o r l c n J t s u o r '^\ uT SanIe^ qlTA uOrfJUnJ JTfSrJalJe.reqJ paz:1e:auail e se paurJap sr lnou sJas pezlle-raua8 asaql sr slas pazlle-raua3 Jo Äqc:e_ra-rq Jo auo qceg 'parrnJlsuoJ anrlplnunc_alTuTlsue-rl ITnJ e lT -ra^o pue .paurnsse sr slua -iualarn Jo I tas B 6ure8y.laleT auTt troqs e egg6L) tlefs puocas aqr asneD uorsuaqarduroc qlT^1 uralqo-rd sTqr sdeq-ra4 leqf drqs-raqruau pazrle-raua8
.
paz:1ar
z. z
.uors-ran palrurrT e dluo sla8 aq u o r s u a q a - r d u r o cf o J t n q . Ä t T I e r a u a 8 ITnJ ur ÄlTTeuorsuaJXa Jo uorsJa^ panlen-dueur alqelrns e ano_rdol alqe sr aH '
(L)
1f,zn + x
, ) ^-
(e)
,
fc\
(r) 'L96l
'ene1y;
r " ) r ' 1 A - * - ' { i l 5 xj
Xz p Ä lx= z ? x e - 2 " .Mf ,) L g ' K 3 z < - - K t x g X 'X' ' a= xz =a
Pa -di ST aATlEIl s lui aqt I
-uoT SU€ aql a-r€ Jo s anI e^ o t p
{ t T 3 puE drqsr aq sd {uoJJ su -JAAUOJ
( L 6 z' d
sanord uaql -roqlne aql suorlrurJap asaql uo-rc
' 1 n /t4^ = n ; n{ f . * A . J P = d A 5 x ( (^'''-*i"oE JP = Ktx Kq paur;ap a.re A5 uorlElaJ uorsnlJur pue 3_uorlETa-r drqs:aquraur .panlen_Äueur 'a'r 'pazlle_raua8 aqt ^''= Ä1t1uapr pazlleraua8 sTqr gtTlt '
( (nA=n) r," E ^dA u 1nA=n) ^dE.r*A ) = (d x) an-rJ s p T o q a . r a q l g a p r s u r I e ^ a I a u r e s a q t u r Ä . x s t a s r o J teqt den e qJns ur spaaco-rd uorsJnJal sTr{r f,oJ dals ureur aqJ .Z ''t= X else rhan zadeh's ,,"n(x), and for fuzzy_sets. ArB: AEB in the sense of Zad,eh (gtS) holds rrue it{(AsIüe)= r s " t , 1, . T h e r e f o r e , i n t h e following rde use the name ,,fir)| for all the generaLLzed sets in V, independent of the finiteness or infiniteness of the w of generalized, membership, i.e. ".t truth values.
'qreo.rdde srqapez aqf of acuataJJrp a8.re1 e a{TI sTool sTqr 'f sf,Tl tV .d1uo fas anTen qln.rf aqf roJ saser alTuTl aql ot uorfefTurTT a q f : p a u r e l d x a f c T f , f s ag ot spq teql .raded sTitl lcadse .raqfoue sr J o l s a .ratur Jo alou qDnlnl 'alaq Trstap ur aqr.rcsap fou 'pano-rd a J , e suorfe.rado asaqf, f,oJ S^IET dueur TTIA a|I qcrqA 'asec drana u1 .sJas Kzznl J o s a T T T u r E Jf o ; a u o d . r e l T u T J u T uB of uorfe.rado d:eurq e uo.rJ paz\1e.raua3 sr sfas fzznl Jo lcnpo-rd uersatJec aql pup uorf Jas.raluT aql .uorun aqf 3ur4e1 Jo suorfe:ado aqr Jo auo qJea pue 6paur;ap rrr" tlzznl Jo lCnpord uersaJJec aql 'pacnpo-r1ur sr suorJeJado uorl . sf as fzzn1 f,oJ e.rqa8ie -efueuaTduroc i{au Jo dTTureJ aToqn v f a s s d o l a n a p . r a q f . r n J , ( q g g Ot ) e n e T X . . r a d e d l a n b a s a q J '(qg6t) qapez ur uorJeluaualduroc aqf Jo uor jezrle"raua8 e 'alqelrene osTe sr uorfeluaualduroa pazrferaua8 e
'a'T J O ar{l Jo asu ueql . L
puTI e -dueur I I
aql
'aJu
las dSol oururJ aq aql
Sutaq ure8e
d a n L _ v x 3 nl P = ( d _ x ) 3 n q8no.rql paurJap sJas ßzznl Jo aruaraJJrp e qtTfl 'sauo puB solaz aTqBurJap ÄTqrl-nr {frn saJrlleT a^rfnqrrtsTp afe o^fit sfas aql TTp leql sano-rd aq puv '(qg0t) qapez se derr aures aql ur v'A sanrlcauuoJ panlen-dueur 'sanlen drqs-raquraur aql Jo asn dq 'a'r Surpuodsa-r_rocaql Jo urnururlu 'dsa-r u n u r r x e u a q l d q s f a s K z z n l s T q Jo uorlJasJalur pup uorun aql Surur;ap dpea-r1e selr EneTX (eggO t ) I.roA sTq uI SCNIddVI^IONV SNOIJVTSU I\ZZfrT 'SIUS
I\ZZNI. CO VUggClV 'E
J o a u A leqt d:or 3ur ppa apEul 'auo lt
s Tq suoT S-Ia aiuo s ) paJnpc auos
'sJas p a z l l e _ rauaB 'o'au asaqf f o J Ä r o a q r e l a s l u a u r d o l a ^ a p aql 1 e r a u a 8 J o uo .{11oqrr patertuaruoc s.rear( aqf u r s . r a A o T T o J l s f s r q pue T J ensTx seaJaqA 6acuanllur 1ea:3 srr{ .roJ suoseal aqt Jo auo seA srqf pue s p T a T J lua-raJlTp uT stas f,zzn1 Jo suorterTldde ur palsa-rafur Jsour seA 'SuruurSaq K-ran aqf rrro-rJ ,qapez tEr{] pazlseqdura aQ o1 seq f r SurpueJSJapunsrur pro^p or '(0g01 'urlef) -rarT.rea s_read auos srsaqt .ralseu ,egg6L) paqsTlqndun ue ur dpea-r1e pazlTpa-r u a a q a n e q ( q g q O l eneT) Jo saurT aql 3uo1e (qqOf) LIapeZ Jo uorfezlleraua8 pue rTleurals,{s aql , (ZtOt) pTeAff,oO pup (ZtOt .0/6t) luaulearl eneT) ur dluo a.re ( S q O f L IapeZ ot sacuaraJar palur-rd ls"rrJ ) '(8961) qapez aqt q8noqtl! Jo luapuedapur dTTnl selt (egg;t) eneT) Jo raded aql .relncrlred u1 .gg6L ur seA slrlJ .:acled Ll.apez a q l S u r u r a c u o c u o r l e u i r o J u r b u r n r a r a J d l a l e TpaururT r a l J e uaas dno.r8 sreneT) ur s e n q a p e z ( E g O t ) p u e (egge f) eneT) Jo saqceo-rdde olrt aqt uaaAtaq uorlcauuoJ sTrtJ
'(g) '(E
(ot; (6) ur pa 'cr3o1
Pu p
anIEl
0
(8)
AXO1HJ JgS AZZNC
l8
S. GOTTWALD
of course, from the formal point of view it is a spectaLization. And rea1ly, t h i s \ , ü a y\ , / a s c h o s e n t o a v o i d s o m e t e c h n i c a l difficurties connected with rhe case o, =[0;ii-ur,ä"Irirry caused by the existence of subsets or-[0, rJ which do not contain their infimum resp. supremum. However, it was arways - which never was the conviction disproved in accepting this restriction that the remaining tur"" would be sufficiently representative also for i = [0, I] The paper Jahn (1969) indeed strongly supported this view; and Klaua (1970) sketched the development of the algebra of fuzzy sets in the way of his (rg66b) paper for the "rr" claimed , al-l the resulrs of (1970) äre'a, in Klaua ( I 966b) or a bit more general. on the other hand, zadeh and his followers also often restrict the considerations to fuzzy sets with finite support and therefore with de facto finitely many generalized membership values also. Furthermore, ii one is accepting the point of view that an order preserving transformation of the set of general ized. membership valuäs does not affect the general laws of fuzzy set theory, all considerations concerning finitely many fuzzy ,"t, with finite support can be done via a s,-,itaf te order preserving transformation of Io, r] already "isomorphi";ii;"'i;"ror" system of fuzzy sets with a suitable wn as set of general tzed, membership values. one more restriction of this paper - unnecessary from a theoretical point of view, but .r"ä simprifying ,oäu of the developments - should be discussed here. rt concerns the consistent use of many-valued logic begun with definitions (9)' (10) of fuzzy inclusion relation and fuzzy equality as rea1ly many-valued relations capable of assuming irrrth values different from 0 and r. rn fact, this ,"Ihod of analysis is not continued. rnstead, the author considers the relations for ftzzy sets de facto as binary ones, e.g. he distinguis.hes for fuzzy inclusion (9) only( xS,^,y)'= I
l . n dn e wn o r i o n s , e . B . ' i r , , r . - ; i ; " i ; t :r u" z1z t }
used by Zadeh (1965 very useful for cha sets. And it depend which one of these vexity of a fuzzy s be characterized b-v t + 0, and equipoll (ll) of uniqueness tzed by the usual e t-components At, Bt I,^/ith the set alg up to rhis point, i theoretic construct equipollent sets in to do so also for t of cardinals and fo this way, two notio arose, based on def Let us refer to the kind. Each generali is a crisp set of f From the main po the essential use o result is that the structure of all ca to the operations o relation, by an app morphic to the alge1 kind. In this sense distinction of ftzz. first kind. For the ftzzy sel first level in the r both kinds of cardir Cardinals of firr for them a represenl cardinals of crisp { card,,A of f irst kin< ( a t l 0 f t e w ) o f r t A. And, as proved ir representation can i the sum, the product of the first kind ol For given fuzzy s
by (rtl 0 f rew) a
Äq pa3uasarda.r_g lp-rec = q- pue dq 1Ä=r I O Ital patuasarda-r y lp-rec = e q l T n g , v s J a s fzzn1 .rärria ,og . sfas LzznJ lo puTT tsfTJ aqf Jo sTBurpJpr JoJ uorfeTa] Sutrap.ro aqf pue lcnpo-rd aqr (ulns agl JO SUOTfeZT.ralcBreqc anrS 0f pasn aq osrp uBc uorlefuasa:da-r .(696t) pTer,rtoC sTql '(etl6t) .puv .V u T p a n o : d s E re sTeurp-rer ael aql J"{ J o ( ^ > r J o o.l täl "1::tt:d*oc-f ^TrueJ aqf uorteJuasa.rda_r se seq purT lsrTJ I Jo yl'p-rea 'V Tsurp-reJ slT rAS f.zzny e uanrS :sf as dsrrc Jo sTeurp.reJ Jo dlrure; e se alqTssod sr uorfefuasa.rda.r p uaqt f,oJ asnpcaq fea.rt ot Jarsea qcnur a-re puTT lsf,TJ Jo sreurp-re3 'aprcuroc s t a s [ z z n l Jo sTeurprpr Jo spuT{ q]oq 'slas tlzznl as-ra^Tun a^TrpTnunJ a T o q ^ J o a q f u r f , sf,TJ T a A a T Jo slas [zzng aqf foJ .a.r ,t1apez slas f,zzn1 a q ] r o d J o .puT{ lsfTf Jo sleurp-rer qfTA se aTqTssod sr sfas Kzznl Jo uorlcuTlsTp raurJ p puT{ puocas Jo sTeurpler qtTA 6asuas sTqt uI .puT{ fsJTJ Jo sTeurpJer TTe Jo aJnJcnJfs crerqaSle aqr or crqd.rou -osT sr u o r l p T a r a c u a n . r 8 u o c a f e t r d o r d d e u e d q . u o r l e la-r 8ur-rap-ro aqr ol pue pue u n s l c n p o r d s u o r l e r a d o a qr ol J o lcadsa: qlTA 'puTI puoras aqt sTeurpleD a r n l cnrfs J o T T e J o cte-rqa31e gqr fo aJntJnJts luarfonb aqt teql sr lTnsaJ aqr '(zt) roJ ^= dJrlenba p a n l e n d ü r r g o a s n Terluassa aql dlauteu , (Zt) . ( ur arua-raJJrp t t ) ureu urorg J o od . s l a s A- t, uz T z n g J o l a s " qdfs r - r c e s r JTastT fas f,zzn1 E Jo .raqunu Teurp.rer paz\1e_raua8 qref, .puTT puoJas .dsa-r tsrTJ Jo sTeurp_rec se iuaqt of JaJar sn ta.I 'ssauanbrun J o ( Z t ) . d s a - r ( f f ) u o r t r u r J a p u o p a s e q 6a s o J e Jas [zznl Jo Jaqunu T'urp-rec aqf Jo suor]ou orlrf .Äen sTqr ursTeurp-rpc aqf 3ur-rap-ro-TTaA a q r f o J J o p u e s Teurprer Jo lcnpo"rd pue uns suorlrurJap aql roJ J o o s Te os op ol T e n s n pue ' slas Kzzn4 0l ,{en p-ren:o;1q3re.r1s e uT sf as luaT l0drnba JO
SASSETC
ATq'fTNS
SE
STEUTP.IBJ
JO
UOTfC'.'']SUOJ
JT]A.IOAqf
.turod sTqf ol las IEnsn ael azrle:aua8 ol aTqTssod s e 6 f T dn passnJsrp stas Azzng roJ suortou crerqa8le las aqt L{tTfl '(qggot 6ene1y) sluauoduror-l 0 I f, roJ.g'-,v Surpuodsarror s-rred a c u ä 1 1 o d r n b J J o T T e a q l J o Ä q pazr lrirn -f af JB-rBqJ aq dTrsea ueJ ,tlSutpuodsa:-roc ssauanbrun Jo ( I I) uorJrurJap u o p a s e q g 6 y s l a s f z z n 1 fo acuaTlodrnba pue .0 + ] JoJ slnr-l slT dlrxanuoc a q l T T p t l q p a z r-ralre.reqJ aq J o T e n s n Ä1rsea uer (EgOt) r{apeZ asuas ur f z z n l v go,t11xan J o l a s V _uoJ : a u o a T q E l T n s a r o u a q l s r s u o r l o u a s a q l lo auo qcTqA 'fapTSuor ot K1-rado.rd a q l s p u a d ap tT puv .stas l e r c a d s J o t\zzn1 sarl-rado-rd uorrezrJalceJeqJ J o f,oJ TnJesn Ä-ran J o are saurJainos suortou asaql Jo qtog .(EgOt) qapez Äq pasn IZ
osTB sr sTqr sP fas f,zzn1 e B
due :o; paurJap las Kzznl e Jo lui d:an aq of pano:d .Surddeur as g Surddew f,zz lua lenrnba a-re g . V -puodsa_r"roc aql qtT ßzzn1 Jo aJuaIIo( paseq a-rp pue (el, sJaqrunu TPUrp-reJ
'osTe (,L6l) ^ortI -cnJJsuoJ leraua8 ßzznl-1 aql . asJnoJ pale8auun auo JSor ulal uo-rJ d, _uoJ e sT XTJIa; JO eTnuJoJ e sr suorlerado d:elr lecre-rqaBl e osTe aJnlJn-rl s d-rana leql saAo1er1_red pazI -e.rado drelJuT _r . TesJA^TUn ,{ue a-rnl t.. '(Zte t) zl-rer\urq qlnrl Jo a_: enel)I ur suor: sTrlt ur (I16t, aBen8uel aqt _Jo:-. a J . o t üs u o l : : saleu puE luai.:: -. -dueur 3:o ai?:: -_,
ÄUOEHJ LAS AZZNC
22
S. GOTTWALD
( b r il u^
I
a- < b and the
t e w)
iff
it
holds
for
gr-(bt
representations
ffiffi,r
true
all
of
o /
by (ar * !-tl o f t€
ä o !
by
(atbr
+ a.B(t)
t € w
sum and product
a + b
+
* * k are
given
for
w ) ,
. ü
b t a ( r )| 0 + r e w )
with
A(r)
= t " r , t
