Weighted Expansions for Canonical Desingularization (Lecture Notes in Mathematics, 910) 3540111956, 9783540111955

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

910

Shreeram S. Abhyankar

Weighted Expansions for Canonical Desingularization With Foreword by U. Orbanz

Springer-Verlag Berlin Heidelberg New York 1982

Author

Shreeram S. Abhyankar Purdue University, Div. Math. Sci. West Lafayette, IN 47907, USA

AMS Subject Classifications (1980): 14E15

ISBN 3-540-11195-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11195-6 Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Table of contents Section

Page

Foreword

v

Preface .

1

Notation.

3

§ 2.

Semigroups

4

§ 3.

Strings .

5

§

1.

§ 4.

Semigroup strings with restrictions

§ 5.

Ordered semigroup strings with restrictions

10

§ 6.

Strings on rings

11

7

§ 7.

Indeterminate strings

14

§ 8.

Indeterminate strings with restrictions

21

§ 9.

Restricted degree and order for indeterminate strings

26

§10.

Indexing strings.

29

§11.

Nets

31

§12.

Semigroup nets with restrictions

33

§13.

Ordered semigroup nets with restrictions

36

§14.

Nets on rings

37

§15.

Indeterminate nets

39

§16.

Indeterminate nets with restrictions

46

§17.

Restricted degree and order for indeterminate nets.

53

§18.

Prechips.

••••.

57

§19.

Isobars for prechips and Premonic polynomials

59

§20.

Substitutions • • • • • .

• •

67

§21.

Substitutions with restrictions

73

§22.

Coordinate nets and Monic polynomials

82

§23.

Graded ring of a ring at an ideal

85

§24.

Graded ring of a ring

88

§25.

Graded rings at strings and nets and the notions of separatedness and regularity for strings and nets.

90

§26.

Inner products and further notions of separatedness and regularity for strings . . . . . • . • • • • .

104

§27.

Inner products and further notions of senaratedness and regularity for nets • • • . • . •

109

§28.

Weighted isobars and weighted initial forms • • • •

113

•

. .

IV

Initial forms for regular strings . • • • •

126

§30.

Initial forms for regular strings and nets

150

§31.

Protochips and parachips • • • • • • • • .

161

§32.

N-support of an indexing string for ?

162

§33.

Prescales • •

§34.

Derived pres cales

165

§35.

Supports of prescales

167

§36.

Protoscales . .

168

§37.

Inner products for protoscales

170

§38.

Scales and isobars .

171

§39.

Properties of derived prescales

176

§40.

Isobars for derived scales .

203

§41.

Isobars and initial forms for scales

205

§42.

Initial forms for scales and regular nets

214

§43.

Isobars for protochips .

220

§44.

Initial forms for protochips and monic polynomials

221

Index of definitions

225

Index of notations .

226

§29.

N

6

163

.

.

.

.

. .

.

FOREWORD At the International Conference on Algebraic Geometry in La Rabida. Spain, January 1981, Prof. Abhyankar lectured on his new canonical proof of resolution of singularities in characteristic zero, giving the general idea of his procedure. Now the notes called "Weighted expansions for canonical desingularization" contain the

part of the algebraic setup to make this

procedure work. In some sense these notes are disjoint from the lectures, namely they do not contain any explanation how the content is used for resolution. This foreword intends to fill this gap,at least partially. First I describe in a very rough manner the method of resolution and some of its features. The three main ingredients of the new proof are (1) a new refined measurement of the singularity, (2) a canonical choice of the center to be blown up, (3) a treatment of (1) and (2) by which the centers of blowing ups are automatically globally defined. (1) and (2) are achieved by a new way of "expanding" an element of a regular local ring. By expansion we mean to find a certain regular system of parameters and to write the given element in terms of these parameters subject to certain (complicated) conditions. The definition of this expansion allows to take into account some regular parameters which are given in advance. In the applications, these parameters will be the ones which occured as exceptional divisors, together with their "history", i. e. the

VI

order in which they occured. Thus (3) will be achieved by an expansion relative to given global data. The germ for this expansion is the following procedure for plane curves. Given for example the curve defined by f(x,z)=z2+ x3 at the origin, the ordinary initial form will be z2, whereas the weighted initial form, giving weight 3/2 to z and weight 1 to x, will be z2+ x3. Now expansion of f in the sense mentioned above consists in a choice of z and x such that z gives the multiplicity (i.e. mod x the multiplicity is unchanged) and such that among all such choices of z and x the weight that must be given to z is maximal. So this expansion comes with two numbers, the multiplicity n and the weight e, and these have the property that whenever the multiplicity is unchanged by blowing up, then after blowing up the weight will be exactly one less. For more than two variables, the new measurement consists in an iteration of this procedure, where in each step z and x are replaced by either monomials or (weighted) homogeneous polynomials in a certain set of variables. Now for each step a third number, has to be added, which is related to the number of "exceptional" variables used in the present step of the expansion. Then the measure of the singularity with respect to given exceptional variables will be the sequence (n l,e 1"1:n2,e 2"2:"')' and the expansion to be used for resolution is one for which this sequence is maximal (in the lexicographic order). The expansion also gives the variables defining the center of blowing up, and the proof of resolution is obtained by showing that blowing up the prescribed center will improve the measure of the singularity.

VII

We point out two major differences to Hironaka's famous proof. One is the visible change of the singularity in each step. The second is that this new proof does not use any induction on the dimension of the variety. Even if one takes the expansion for granted, the description of resolution given above was not quite correct. The final proof will use another iteration of this procedure. After expanding one equation, one can extract some coefficients. These coefficients have to be expanded again, then the coefficients of the coefficients, etc. This leads to the notion of a web, which is not treated here. After this rather crude description of a very complicated mechanism, we can indicate the content of the paper that follows. It contains the notation which is necessary to deal with the huge amount of information contained in the expansion. Then there is a proof for the existence of weighted initial forms in great generality, maybe more general than is needed for the purpose of resolution. Finally the existence of an expansion as indicated above is proved.

u.

Orbanz

Preface My hearty thanks to Giraud of Paris and Herrmann of Bonn whose encouragement revived my interest in resolution of singularities.

I am also grateful to the Japanese gardner Hironaka

for propagating sympathetic waves.

But then where shall we be

without the blessings of our grand master Indeed, Mathematics knows no national boundaries. Our method may be termed the method of Shreedharacharya, the fifth century Indian mathematician, to whom Bhaskaracharya ascribes the device of solving quadratic equations by completing the square. The said device is given in verse number 116 of Bhaskaracharya's Bijaganita of 1150 A.D. and is thus:

I

';le:tlla\

0f"i1

ef\ S"\

\'\b I

II

4

In my youth I tried to algorithmize local resolution but had to fall back on the college algebra of rings et al for globalization. In middle age my faith in high-school algebra grew and grew to reach globalization. The lesson learnt is that when you make your local algorithms more and more precise, i.e., even more algorithmic, then they automatically globalize. Another viewpoint. curve

f(X,Y)

Understand desingularization of a plane

better and better.

until it engulfs everything. into vectorial variables

X

Let it bloom like a lotus

Let the singletons =

(Xl' ... ,X

m)

and

Let that be the petal at the core of the lotus. lotus blossom.

X

and

Y

Y Now let the

grow

2

Or think of a beehive. Yet another philosphical point is to understand what is a monic polynomial and thereby to enlarge that notion. Krull and Zariski indoctrinated us with valuations and I got addicted to them. this addiction.

Then Hironaka taught us to rid ourselves of But habits don't die.

So now valuations have

entered through the back door in their reincarnation as weights and lexicons. This Introduction consists mostly of definitions. them the reader could get an idea of the proof.

This is written

in a pedantically precise and resultingly boring manner. still experimenting with notation.

But from

I am

For me it is not easy to

transcribe from the mental blackboard onto the paper! My thanks are due to S. B. Mulay and U. Orbanz and A. M. Sathaye for stimulating discussions and help in proof-reading. to Judy Snider for an excellent job of typing.

Also thanks

Finally, thanks

to the National Science Foundation for financial support under MCS-8002900 at Purdue University.

3

§l.

Notation

In this paper we shall use the following notation:

Q

the set of all nonnegative rational numbers.

Z = the set of all nonnegative integers. Z* = the set of all subsets of

z.

the set of all n-tuples of nonnegative integers. [a,b] = {n E Z: a

n

b}.

4

§2. u E Q

For any

Semigroups

we put

fa

denom(u) and for any

u' c Q

we put

fO

denom(u') We note that an (additive)

n E z: nu E Z}

Q

z:

n E

nu E Z

for all

u E U'}.

is an additive abelian sernigroup and

subsemigroup of

Q.

In fact

Q

Z

is

is a nonnegative

ordered additive abelian semigroup where by a nonnegative ordered additive abelian sernigroup we mean the nonnegative part of an ordered additive abelian group, i.e., the set of all nonnegative elements of an ordered additive abelian group.

Likewise

a nonnegative ordered additive abelian semigroup. is divisible, but

Z

v E G

that

v = nv*.

and

0

Note that if

abelian group then for every unique (nu)w

w* E

=

nw*;

G

we define:

uw

is consistent with regarding

part of uw E G.

G

Moreover

Q

G

w*;

G

there exists

and

u E Q

n E denom(u)

such

there exists a we have

we observe that this notation

as a module over the ring of

we also note that if

then for every

v* E G

is a divisible ordered additive

w E

=

is said to be divisible

G

n E Z

such that for every

rational numbers;

is

is not, in the following sense.

An additive abelian semigroup if for every

Z

w E G

and

G

is the nonnegative

u E Q

we now have

5

§3.

Strings

By a string we mean a system integer

o(x),

x

called the length of

consisting of a nonnegative x, and

an element x(e) E Universe whereby we call such that

x(c)

x(c) E G

or a string on

th

component of

for all

c

and any

0

G

G(o)

so is

G(o);

G x

is a set

a G-string

E Z

we put

G

and any

o.

0

E Z,

we

as an additive abelian semigroup with

we have

c E [1,0];

If

the set of all G-strings whose length is

componentwise addition; x E G(o)

x.

then we may call

Given any additive abelian semigroup may regard

1 s; c s; o(x)

G.

For any set G(o)

the c

for

we note that then for any

nx E G(o)

given by

we also observe that if similarly, if

in an obvious manner

G(o)

G

(nx) (c)

G

=

n E Z nx(c)

and

for all

is actually a group then

is a module over a ring

R

may be regarded as a module over

then R.

Likewise, given any divisible ordered additive abelian group G and any

0

E Z,

by putting

for any (ux) (c)

u E Q

= ux(c)

and

x E G(o)

for all

G, for any

we define supt(i)

{c E

ux E G(o)

e E [1,0].

Given any additive abelian semigroup i

we define

[1,0 (i)]: i (c) 'I O}

G-string

6

and abs(i)

and for any set

i'

abs (i ') For any Q-string

denom(i) and for any set

r o ti'

1:

i (e)

Ls cs o f i )

of G-strings we put {abs (i):

i

n E

i

E I "}.

we define

Z:

ni(e) E Z

for

of Q-strings we put

denom(i') =

n

iE i

denom(i). I

l:s; e:s; o(i)}

7

§4. Let

0

For any

Semigroup strings with restrictions

be a nonnegative integer. r c Z

we put supt(o,r)

and for any string

D

[1,0]

n

r

on any additive abelian semigroup we put

supt(o,D)

supt(o,supt(D»

supt(O,D)

[1,0]

Le.

and for any

c E Z

n supt(D)

we put

supt(o,c)

supt(O, [c,o])

supt(o,c)

[c,o] .

Le.

By a string-subrestriction we mean an object e i t he r { or

t c

t

t

where

Z

is a string on an additive abelian semigroup.

We put subrest(string)

the class of all string-subrestriction.

By a string-restriction we mean an object e i t he r { or

t

t E Z

t

where

is a string-subrestriction

8

We put rest (string)

the class of all string-restrictions

and we note that we have defined

supt(o,t)

for every

t E rest (string) , i.e., for every string-restriction Given any string-restriction supt(o,lt) and for any

r c Z

=

we put supt(o,t) n r

D

on any additive abelian semigroup we put

supt(o,t,D) and for any

c E Z

Thus we have defined and

supt(o,t,supt(D))

we put

supt(o,t,c)

t

t, we put

[l,o]\supt(o,t)

supt(o,t,r) and for any string

t.

supt(o,t,{c}) . supt(o,t,z)

for any string-restrictions

z.

Given any additive abelian semigroup restriction

G

and any string-

t, we define

G(o,t)

{i E G(o): supt(i) c supt(o,t)}

and G(o,lt)

{i E G(o): supt(i) c supt(o,lt)}

9

and for any string-restriction G(o,t,k)

k

we define

{i E G(o): supt(i) c supt(o,t,k)}.

10

§5. Let let

0

G

Ordered semigroup strings with restrictions be a nonnegative ordered additive abelian semigroup,

be a nonnegative integer, and let

u E G.

We define G(o

Given any

u)

u}

{iE G(o) : abs(i)

G(o "' u)

{i E G (0): abs (i) "' u}

G(o > u)

{i E G (0): abs (i)

> u}

G(o < u)

{i E G (0): abs(i)

< u}

G(o,; u)

{i E G (0): abs (L) ,; uL

P E {=,",,>,

R[X]Q

and we define

19

and Info [R,X,=uJ: Iso (R,X,2:u)

-+

to be the R-homomorphisms induced by Info[R,X,=uJ Q: * ISO(R,X,2:U)Q

R[XJ

ISO[R,X,=uJ

and we define

Q

ISO(R,X,=U)Q

-+

and Info[R,X,=uJ*: Iso(R,X,2:u)

Iso(R,X,=u)

-+

to be the R-epimorphisms induced by

ISO[R,X,=U J

and we observe

Q

that ker(Info[R,x,=uJ

ker(Info[R,x,=U]Q)

Q)

ISO(R,X'>U)Q and ker(Info[R,X,=uJ)

ker(Info[R,X,=uJ*) IsO(R,X,>u)

Here we have used, and we shall continue to use, the following obvious conventions for sets map

g: S

-+

subset of

S' Sl

where then by

{g (z) : z E So

n S}

that

S' 2

g(S2)

C

mean the map.

S

of

C

Sl and

Sl

If

then by the map

g2: S2

-+

8'

2

S'

Si C

and a

(set-theoretic)

If

Si·

So

is any

we denote the subset

g(SO) S' .

and

S2

C

S

S2

-+

and S' 2

S2

C

Si

induced by

obtained by putting

are such g

we

g2(z) = g(z)

for

20

all

5' 3

If

we denote the subset injective then by any 5 {g

zl E g(5) such that

-1

(zl)}

=

g-l

g(5)

-1

-1

(zl))

({zl})·

=

5

+

9

-1

then by

5'

1

{z E 5: g(z) E 53}

we have that g(g

9

is any subset of

of

5.

If

g-1(5 9

3)

is

we denote the map whereby for (zl)

is the unique element of

zl' i.e., such that

21

§8. Let

R

Indeterminate strings with restrictions

be a ring, let

R, and let

t

X

be an indeterminate string over

be a string-restriction.

We define the of

and

subrings R[X(t)]Q

R[X]Q

by putting R [X( t)] Q

{f E R[X]Q: supt(f)

C

Q(o(X) ,t)}

and {f E R[X]Q: supt(f) c Q(o(X) ,ft)} and we define the sUbrings R[X(t)]

and

R[X(ft)]

of

R[X]

by putting R[X(t)]

=

R[X(t)]Q n R[X]

and moreover for any

R

O

c

R

and with

R[X(ft)] 0 E R

O

=

R[X(ft)]Q n R[X]

we put

and R [X( t)] O

=

R [X( t)] n RO[X]

We observe that then

and

RO[X( ft) ]

R[x(ft)]

n

RO[X].

22

R[X]Q

R [X( 1) ] Q

We also observe that for any

and

R[X] .

R[X(l)]

C E [l,o(X)], in an obvious manner

we have R[X({C})]Q

R[X(C)]Q

where as usual

and

R[X({C})]

R[X(c)]

stands for an isomorphism.

Given any

u E Q

and

P E

R-submodules X(t)(RP)Q

and

we define the

u

X( t) (RP)

of

R[X(t)]Q and R[X(t)]

respectively by putting

X(t)(RP)Q

and for any

=

R

O

C

R

with

u

and

iEQ (0 (X)Pu,t)

0 E R

O

X(RP)Q

n R[X(t)]

we put

and Iso(RO,X(t) ,Pu) and we note that ISo(R,X(t) ,pu)Q

ISO(R,X,PU)Q n R[X(t)]Q

and IsO(R,X(t) ,Pu)

ISO(R,X,PU)Q n R[X(t)].

23

Given any

U E Q

and

P E

we define

Iso[R,X(t) ,Pu]Q: R[X(t)]Q

R[X]Q

and ISO[R,X(t) ,Pu]: R[X(t)] to be the R-homomorphisms induced by

ISO[R,X(t),PU]Q: R[X(t)]Q

-s-

R[X]

ISO[R,X,PU]Q

and we define

Iso(R,X(t),pu)Q

and ISO[R,X(t) ,Pu]*: R[X(t)]

Iso(R,X(t),Pu)

to be the R-epimorphisms induced by Given any

u E Q

ISO[R,X,PU]Q"

we define

Info[R,X(t) ,=u]Q: Iso(R,X(t)

R[X]Q

and Info[R,X(t),=U]: to be the R-homomorphisms induced by

R[X] ISO[R,X,=U]Q

Info[R,X(t) ,=u]5: Iso(R,X(t) and

and we define

ISO(R,X(t),=u)Q

24

to be the R-epirnorphisrns induced by

ISO[R,X'=U]Q

and we note

that ker(Info[R,X(t) ,=u]Q)

ker(Info[R,X(t) ,=u]8) ISO(R,X(t) ,>u)Q

and ker(Info[R,X(t),=U] )

ker (Info [R,X( t) ,=u] *) IsO(R,X(t) ,>u).

Given any

u E Q

and any string-restriction

k

we define

the

R-subrnodules

and

of

R[X(t)]Q

and R[X(t)]

respectively by putting

iE Q(o(x)=u,t,k) and

n

R[X(t)]

and we define the

ideals

and

in

R[X(t)]Q

and

R[X(t)]

25

respectively by putting u

X(t,k) (R2}Q and X(t,k)(R2}Q and for any

P E

n R[X(t)]

as an alternative notation, we put

ISO(R,X(t,k),PU}Q

and ISO(R,X(t,k),Pu}

26

§9.

Restricted degree and order for indeterminate strings

Let over

R

be a ring and let

X

be an indeterminate string

R. Given any string-subrestriction

t, we observe that

is naturally isomorphic to

R[X(ft)]Q[X(t)]Q

the following definitions.

For any

by

F(t) [i]

F E R[X]Q

we denote the unique element in

and this leads to and

i E Q(o(X),t),

R[X(ft)]Q

that F

For any

F E R[X]Q

iEQ

1:

F(:') [i]X

i

(0 (X) ,t)

we define

supt(F(t»)

{ i E Q(o(X) ,t): F(t) [i]

f

O}

and we put Ord [R,X] (F( t»)

min abs(supt(F(t»))

Deg [R,X] (F( t) )

max abs(supt(F(t»))

and

and we note that: F

For any

o .. F ' c

Ord[R,X] (F(t»)

00

..

Deg[R,X] (F(t»)

we put Ord[R,X] (F'(t»)

R[X]Q

{Ord[R,X] (F(t»): FE F '}

such

27

and Deg[R,X] (F'(t») For any

F' c R[X]

{Deg[R,X] (F(t»): FE F '}.

we put

Ord[R,X] «F'(t»))

min Ord[R,X] (F'(t»)

Deg[R,X] «F'(t»))

max Dcg[R,X] (F'(t»).

and

Given any isomorphic to definitions.

c E [l,o(X)],

we observe that

R[X(I{c})]Q[X(c)]Q For any

F E R[X]Q

For any

F E R[X]Q

uEQ

and

u E Q,

by

F(c) [u]

such that

F(c) [u]X(c)u .

we define

supt(F(c»)

{u E

is naturally

and this leads to the following

denote the unique element in F

R[X]Q

Q:

F(C) [u]

O}

and we put Ord [R, X] (F( c) )

min abs(supt(F(c»))

Deg [R,X] (F( c) )

max abs(supt(F(c»))

and

we

28

and we note that:

F

For any

o '" F'

Ord[R,X] (F(c»)

C

R[X]Q

co

cc

Deg[R,X] (F(c»)

we put

Ord[R,X] (F'(c»)

{Ord(R,X] (F(c»): F E F'}

and

Deg[R,X] (F'(c»)

For any

F' c R[X]

{Deg[R,X] (F(c»): FE F ' } .

we put

Ord(R,X] ((F'(c»))

min Ord(R,X] (F'(c»)

Deg(R,X] ((F'(c»))

max Deg[R,X] (F'(c»).

and

29

§10.

Indexing strings

By an indexing string we mean a system

t

consisting of

o (t) E Z

b(t) E Z

for

1" b" o t z )

and T(d,b,t) E Z

for {

l"b"o(t) d E Z

such that o (t) 'I 0 b(t) 'I 0

""

'"' dEZ

for

b

1 "

T(d,b,t) = b(t)

"

0

(t) - 1

for

1 "

b

"

0

(t)

and T(O,b,t)

U)Q and

ker(Info[R,Y,=u] )

ker(Info[R,Y,=uJ*) Iso(R,Y,>u).

and we

46

§16. Let

R

and let

v,e

t

Indeterminate nets with restrictions

be a ring, let

Y

be an indeterminate net over

R,

be a net-restriction.

define the subrings R[Y(t)]Q

and

of

R[Y]Q

by putting {f E R[Y]Q:

supt(f) c Q(9,(Y),t)}

and {f E R[Y]Q: supt(f) c and we define the subrings R[Y(t)]

and

of

RlY]

by putting R[Y(t)] == R[Y(t)]Q n R[Y] and moreover for any

RO

C

R

and

with

R[YUt)] == 0 E R O we put

and

and R lY( t) ] O

R [Y( t)] n R lY] O

and

RO [Y(

]

n ROlY]

.

47

We observe that then R[Y(l)]Q

=

R[Y]Q

and

We also observe that for any

R[Y]

R[Y(l)]

.

b E [l,o(£(Y»], in an obvious

manner we have

R[Y({b})]Q

R[Y(b)]Q

and moreover for any

and

R[Y({b})]

c E [l,b(£(Y»],

R[Y(b)]

in an obvious manner we

have R[Y(b,c)]Q

Given any

u E Q

and

R-submodules Y(t) u(RP)Q

and

R[Y({(b,c)})]

P E

we define the

Y(t) u(RP)

and

R[Y(b,c)].

of

R[Y(t)]Q

and

R ( Y(t)]

respectively by putting

u

Y(t) (RP)Q

and for any

l: jEQ(£ (Y)Pu,t) R O

C

R

with

and

u

Y( t) (RP)

0 E RO we put

and u

Y(t) (RP) n RO[Y(t)]

u

Y(t) (RP)Q nR[Y(t)]

48

and we note that U

ISO(R,Y(t),PU)Q

Y( t) (RP)

Q

ISO(R,Y,PU)Q

n R[Y(t)]Q

ISO(R,Y,PU)Q

n R[Y(t)].

and

ISo(R,Y(t) ,Pu)

Given any

U

Y(t> (RP)

U E Q

and

P E

we define

ISO[R,Y(t),PU]Q: R[Y(t)]Q

+

R[Y]Q

Iso [R, Y( t) ,Pu]: R [Y( t>]

R [Y]

and

+

to be the R-homomorphisms induced by

Iso[R,Y(t) ,pu]Q: R[Y(t)]Q

ISO[R,y,PU]Q

+

Iso(R,Y(t) ,pu)Q

and IsO[R,Y(t),PU]*: R[Y(t)] to be the R-epimorphisms induced by Given any

U E Q

Iso(R,Y(t),Pu) ISO[R,Y,PU]Q"

we define

Info[R,Y(t),=U]Q: Iso(R,Y(t) and

+

and we define

+

R[Y]Q

49

Info[R,Y(t),=U]:

R[Y]

to be the R-hornornorphisrns induced by

ISO[R'Y'=U]Q

and we define

Info[R,Y(t),=U]Q:

ISO{R,Y(t),=U)Q

Info[R,Y(t),=U]*: Iso{R,Y(t)

Iso{R,Y(t),=u)

and

to be the R-epirnorphisrn induced by

ISO[R'Y'=U]Q

and we note

that ker{Info[R,Y(t) ,=u]Q)

ker{Info[R,Y(t) ,=u]Q) IsO{R,Y(t) ,>u)Q

and ker{Info[R,Y(t),=U])

ker{Info[R,Y(t),=U]*) ISo{R,Y(t) ,>u).

Given any

U E Q

and any net-restriction

U

R-subrnodules Y(t,k) (R=)Q

and

U

Y(t,k) (R=)

respectively by putting U

Y(t,k) (R=)Q and

jEQ{£{Y)=u,t,k)

of

k

we define the

R[Y(t)]Q and R[Y(t)]

50

u

u

y(t,k) (R=}Q n R[y(t)J

Y(t,k) (R=) and we define the

ideals

u

and

in

R[y(t)J

Q

and

R[y(t)J

respectively by putting

y(t,k)

u

and

n R[y(t)J

= and for any

P E

as an alternative notation, we put

ISO(R,y(t,k),PU}Q

U

y(t,k) (RP}Q

and IsO(R,y(t,k) ,Pu}

Given any

u E Q

U

y(t,k) (RP)

and given any

.

b E Z

and

r c Z,

we define

the

u u R-subrnodules y(t,b,r) (R=}Q and y(t,b,r) (R=) of R[y(t)J Q and R[y(t)J respectively by putting

= . JEQ(£(Y}=u,t,b,r}

51

and u

n R[Y(t)J

Y(t,b,r) (R=) and we define the

and

ideals

Y( t,b,r)

u

in

R[Y(t)J

Q

and R[Y(t)J

respectively by putting

Y(t,b,r)

u

R[Y(t)J Q

and Y(t,b,r) and for any

u

= Y(t,b,r)

P E

u

n R[Y(t)J

as an alternative notation, we put u

IsO(R,Y(t,b,r) ,pu)Q

Y(t,b,r) (RP)Q

and u

Iso (R, Y( t,b,r) ,Pu) = Y( t,b,r) (RP)

bE

Given any

u E Q

[l,o(,Q,(Y))J

and

and

P E

and given any

s E Z*(o(JI.(Y))), we put u

Y(t,b,s) (RP)Q

u

Y(t,b,s(b» (RP)Q

and u

Y(t,b,s) (RP)

Y( t,b, s (b)

u

(RP)

52

and as an alternative notation we put

ISO(R,y(t,b,s) ,PU)Q

U

y(t,b,s) (RP)Q

and Iso(R,y(t,b,s) ,Pu)

U

y(t,b,s) (RP)

53

§17. Let

Restricted degree and order for indeterminate nets R

be a ring and let

Y

Given any net-subrestriction naturally isomorphic to

F(t) [j]

t, we observe that

R[Y(lt)]Q[Y(t)]Q

the following definitions. by

be an indeterminate net over R.

For any

R[Y]Q

and this leads to

F E R[Y]Q

we denote the unique element in

and j E Q(,Q, (Y) ,t), R[Y(lt)]Q

such

that

l:

F

For any

F E R[Y]Q

jEQ (,Q, (Y) , t)

F(t) [j]y

j

we define

supt(F(t») = {j E Q(9,(Y),t): F(t)[j] IO} and we put Ord [R, Y] (F( t) )

min abs(supt(F(t»))

Deg [R, Y] (F( t) )

max abs(supt(F(t»))

and

and we note that:

0", Ord[R,Y] (F(t»)

F

For any

F'

C

R[Y]Q

Ord[R,Y] (F'(t»)

00

'"

is

Deg [R, Y] (F( t) )

we put {Ord[R,Y] (F(t»): F E F '}

_00

•

54

and Oeg[R,Y] (F'(t» For any

F ' c R[Y]

{Oeg[R,Y] (F(t»: F E F '} .

=

we put

Ord [R, Y] «F I -

R[Y]

by putting

Y]

Sub[R,Y and for any

R

O

C

R

(F)

jEsupt(F) with

0 E R

F[j]y

j

for all

F E R[Y]

we put

O

Sub [R, Y = Y] (R O[Y] ) and we define Sub[R,Y=Y]*: R[Y]

->-

R[Y]

to be the R-algebra-epimorphism induced by Given any

y

where

y E R(o(Y)) { whereas

Sub[R,Y =Y].

in case

y E R(9,(Y))

Y

is a string

in case

Y

is a net

68

we define sub[R,Y =y]: R[Y]

R

->

to be the R-algebra-epimorphism induced by

Sub[R,Y =y]

and we

note that then sub[R,Y=y] = Sub[R,Y=y]* By a pseudomorphism we mean a such that

R'

is a ring and

(set-theoretic) map

g(O) = 0

Given any pseudomorphism

g: R

->

and R'

g: R

->

R'

g(l) = 1.

and given any

Y'

where Y' E R' [Y] (o(Y)) {

whereas

in case

Y' E R' [Y] (9, (Y))

Y

is a string

in case

Y

is a net

we define the pseudomorphism Sub[g,Y =Y']: R[Y]

->

R' [Y]

by putting Sub[g,Y =Y'] (F)

g(F[j])y,j

for all

F E R[Y]

jEsupt(F) and we define Sub[g,Y=Y']*: R[Y]

-e-

g(R)[Y']

to be the surjective map induced by that:

if

g

Sub[g,Y =Y']

is a ring-homomorphism then

and we note

Sub[g,Y =Y']

and

69

Sub [g, Y = Y'] *

are ring-homomorphisms and

Sub [g, Y = Y']

is a

g-algebra-homomorphism. Given any pseudomorphism

y ' E R' (o(Y)) {

whereas

g:R

in case

Y

y' E R'(9, (Y) )

R'

and given any

y'

where

is a string

in case

Y

is a net

we define sub[g,Y =y']: R[Y]

R'

to be the pseudomorphism induced by sub[g,Y=y']*: R[Y]

Sub [g, Y = y']

and we define

sub [g, Y = y']

and we note

g(R)[y']

to be the surjective map induced by that then sub[g,Y=y']* = Sub[g,Y=y'l* and we observe that:

if

g

is a ring-homomorphism then

sub[g,Y=y']

and

sub[g,Y=y']*

sub[g,Y=y']

is a g-algebra-homomorphism.

Given any pseudomorphism

are ring-homomorphisms and

g:R

R'

we define the

pseudomorphism

by putting

2

jEsupt(F)

g(F[j])y

j

for all

F E R[Y]Q

70

and we define Sub[g,Y]: R[Y] -+ R' [Y] to be the pseudomorphism induced by

Sub[g'Y]Q

and we define

Sub[g'Y]Q: R[Y]Q -+ g(R) [Y]Q and Sub[g,Y]*: R[Y] -+ g(R) [Y] to be the surjective maps induced by

Sub[g'Y]Q

and we note that

then Sub[g,Y] = Sub[g,Y=Y] and we observe that: Sub[g'Y]Q

and

if

9

Sub[g'Y]Q

and

Sub[g,Y]* = Sub[g,Y=Y]*

is a ring-homomorphism then are ring-homomorphisms and

Sub[g'Y]Q

is a g-algebra-homomorphism. We define the R-algebra-homomorphism

by putting

Sub[R,Y = O]Q(F)

jEQ

\

(0

(Y) =0)

F [j]

F [j] 1: jEQ(9- (Y)=O)

and we define Sub[R,Y=O]: R[Y] -+R[Y]

in case Y is a string in case Y is a net

71

to be the R-algebra-homomorphism induced by

Sub[R,Y = O]Q

and

Sub[R,Y =O]Q

and

we define Sub[R,Y = 0];: R[Y]Q Sub[R,Y = 0] *: R[Y]

R

-+

R

-+

and sub[R,Y = 0]: R[Y]

-+

R

to be the R-algebra-epimorphisms induced by we note that sub [R, Y = 0] = Sub [R, Y = 0] Given any pseudomorphism

*.

g: R

-+

R', we define the

pseudomorphism

by putting

1:

g (F [j]) in case Y is a string

1:

g (F [j] ) in case Y is a net

jEQ(o(Y)=O)

Sub[g,Y = O]Q(F)

jEQ(Q, (Y)=O)

and we define Sub[g,Y=O]: R[Y]

-+

R'[Y]

sub[g,Y = 0]: R[Y]

-+

R'

and

72

to be the pseudomorphism induced by Sub [g , Y = 0]

Q:

R[Y]Q

-+

Sub [g , Y = 0] Q

and we define

g(R)

Sub [g , Y = 0] * : R[Y]

-+

g (R)

= 0]*: R[Y]

-+

g(R)

and sub[g,Y

to be the surjective maps induced by

Sub [g , Y = 0] Q

and we note

that sub [g , Y

= 0] * = Sub [g , Y = 0] *

and we also observe that:

if

g

is a ring-homomorphism then

the above defined six maps are ring-homomorphisms and out of them the first three are g-algebra-homomorphisms.

73

§2l. Let

R

Substitutions with restrictions

be a ring.

e i t he r

Y

Let

Y

be given where

is an indeterminate string over

R

{ or

Y

t

and

t'

be given where

t

and

t'

are string-restrictions in case Y is a string

and let

{

is an indeterminate net over

whereas

Given any

t Y

and

R

t' are net-restrictions in case Y is a net.

where

y E R[Y] (o(Y» in case { whereas i E R[Y] (l(Y»

Y

is a string

in case

Y

is a net

we define Sub[R,Y(t) =Y]: R[Y(t)]

R[Y]

->-

to be the R-algebra-homomorphism induced by for any

R

O

C

R

with

Sub[R,y=i]

and

0 E R we put O

Sub[R,Y=i] (RO[Y(t)]) and we define Sub[R,Y(t) =i]*: R[Y(t)]

->-

R[i(t)]

to be the R-algebra-epimorphism induced by define

Sub [R, Y = i],

and we

74

Sub[R,Y =Y(t')]: R[Y]

R[Y]

-+

to be the unique R-algebra-hornornorphisrn such that

Sub[R,Y =Y(t')] (F)

{

F

for all

FE R[YU t')]

Sub [R, Y = Y] (F)

for all

FE R[Y(t')]

and we define Sub[R, Y(t)=Y(t')]: R[Y(t)]

-+

R[Y]

to be the R-algebra-hornornorphism induced by Sub [R, Y = Y( t ')], and we define

SUb[R,Y=Y(t')]*: R[Y]

-+

Sub[R,Y=Y(t')] (R[Y])

and Sub[R, Y(t)=Y(t')J*: R[Y(t)]

-+

Sub[R,Y=Y(t')] (R[Y(t)])

to be the R-algebra-epirnorphisrns induced by Given any

y

where

y E R(o(Y)) { whereas

Sub [R, Y = Y( t ')] .

in case

y E R(£ (Y))

Y

is a string

in case

Y

is a net

we define

sub [R, Y(t) = y]: R [Y( t)]

-+

R

to be the R-algebra-epirnorphisrn induced by

Sub [R, Y = y]

and we

75

note that then sub [R, Y(t)

='

Y]

='

Sub [E, Y(t)

Given any pseudomorphism

Y] *.

g: R

R'

+

and given any

Y'

where Y' E R' [Y] (0 (Y) ) {

whereas

in case

Y

Y' E R' [Y] (Q, (Y»

is a string

in case

Y

is a net

we define Sub[g,Y(tj =Y']: R[Y(t)]

R'[Y]

+

to be the pseudomorphism induced by Sub[g, Y(t) =Y']*: R[Y(t)]

+

to be the surjective map induced by Sub[g,Y =Y'(t')]: R[Y]

+

Sub[g,Y =Y']

and we define

g(R) [Y'(t)] Sub[g,Y =Y'],

and we define

R ' [Y]

to be the pseudomorphism obtained by putting, for all

F E R[Y],

Sub[g, Y =Y'(t'j] (F)

!

Sub[g, Y] (F(suPt(o(y),t')[j])y,j i f Y is a string

j E Z(o(Y),t')

!

Sub[g,Y] (F(sUPt(Q,(y),t'»[j])y,j

j E Z(Q,(Y),t')

and we define

Sub[g, Y(t) =Y'(t')]: R[Y(t)]

+

R ' [Y]

i f Y is a net

76

to be the pseudomorphism induced by

Sub [g, Y = Y' (t') J

and we

define Sub[g,Y=Y'(t')J*: R[YJ

Sub[g,Y=Y'(t')J (R[YJ)

->-

and Sub[g,Y(t) =Y'(t')J*: R[Y(t)J to be the surjective and we observe that: above defined six

->-

maps induced by if

g

Sub[g,Y=Y'(t')J (R[Y(t)J) Sub[g,Y=Y'(t')J

is a ring-homomorphism then the

maps are ring-homomorphisms and out

of them the three unstarred ones are g-algebra-homomorphisms. Given any pseudomorphism

g: R

R'

->-

and given any

y'

where y ' E R' (o(Y» {

whereas

in case

y' E R' (Q,(Y»

Y

is a string

in case

Y

is a net

we define sub[g, Y(t) =y'J: R[Y(t)J to be the

R'

->-

pseudomorphism induced by

sUb[g,Y(t) =y'J*: R[Y(t)J to be the surj ective

->-

Sub[g,Y =y'J

g(R) [y'(t)J

map induced by

Sub [g , Y = y' J

and we note that then sub [g, Y(t) = Y , J *

Sub [g, Y(t) = y' J *

and we define

77

and we observe that:

if

sub [g, Y(t) = y']

sub [g, Y(t) = y'] *

and

and

sub [g, Y( t) = y']

g

is a ring-homomorphism then are ring-homomorphisms

is a g-algebra-homomorphism.

Given any pseudomorphism

g: R

+

R'

we define

and SUb[g,Y(t)]: R[Y(t)]

R'[Y]

+

to be the pseudomorphisms induced by SUb[g,Y(t)]Q: R[Y(t)]Q

Sub[g'Y]Q

and we define

giRl [y(t)]Q

+

and SUb[g,y(t)]*: R[Y(t)]

+

g(R) [Y(t)]

to be the surjective maps induced by

Sub[g'Y]Q

and we note that

then

Sub[g,Y(t)] =Sub[g, Y(t) and we observe that:

if

g




R[Y(T!t')]Q

and SUb[R,Y=O(t')]*: R[Y]

R[Y(T!t'»)

->

to be the R-algebra-epimorphisms induced by

Sub[R,Y = O(t'»)Q"

vJe define Sub[R, Y(t) =O(t')]Q: R[Y(t)]Q

->

R[Y]Q

Sub[R, Y(t) =O(t')]: R[Y(t»)

R[Y]

and

-s-

to be the R-algebra-homomorphisms induced by

Sub [R, Y = O( t ') ) Q

and we define Sub[R, Y(t)=O(t')]Ci: R[Y(t»)Q

->

R[Y(t»)Q

n

R[Y(T!t'»)Q

and Sub[R, Y(t)=O(t')]*: R[Y(t»)

R[Y(t»)

->

to be the R-algebra-epimorphisms induced by Given any pseudomorphism

g: R

Sub[g, Y(t) = O)Q: R[Y(t»)Q Sub[g, Y(t) =0): R[Y(t»)

-> R'

-> R'

R'

->

[Y)Q

[Y)

n R[Y(r't'») Sub [R, Y = O( t'») Q"

we define

80

and sUb[g,Y(t) == 0]: R[Y(t)]

R'

-+

to be the pseudomorphisms induced by Sub[g,Y(t) =O]Q: R[Y(t)]Q

Sub [g , Y == 0] Q

and we def ine

g(R)

-+

SUb[g,Y(t) =0]*: R[Y(t)]

-+

g(R)

sub[g,Y(t) =0]*: R[Y(t)]

-+

g(R)

and

to be the surjective maps induced by

Sub [g, Y = 0] Q

and we not ..

that then sUb[g,Y(t) =0]* == SUb[g,Y(t) ==0]* and we observe that:

if

g

is a ring-homomorphism the above

defined six maps are ring-homomorphisms and out of them the three unstarred one are g-algebra-homomorphisms. Given any pseudomorphism

g: R

-+

R'

we define the

pseudomorphism

to be the composition

Sub[g'Y]Q

and we define

R' [Y]Q

Sub [R' , Y == O( t ') ]

R' [Y] Q

81

Sub[g,Y = O(t')]: R[Y]

R' [Y]

-+

Sub[g,Y(t) = O(t')]Q: R[Y(t)]Q

-+

R' [Y]Q

Sub[g,Y(t) =O(t')]: R[Y(t)]

R'[Y]

and

-+

to be the pseudomorphisms induced by

Sub [g, Y = O( t' ) ] Q

and we

define Sub[g,Y = O(t')](j: R[Y]Q Sub[g,Y=O(t')]*: R[Y]

-+

-+

g(R) [Y(lt')]Q g(R) [Y(lt')]

Sub[g,Y(t) =O(t')](j: R[Y(t)]Q

-+

g(R) [Y(t)]Qng(R) [Y(;;it')]Q

and Sub[g,Y(t) =O(t')]*: R[Y(t)] to be the surjective maps induced by observe that:

if

g

-+

g(R) [Y(t)] ng(R) [Y(lt')] Sub [g , Y = O( t ' ) ] Q

and we

is a ring-homomorphism then the above

defined eight maps are ring-homomorphisms and out of them the four unstarred ones are g-algebra-homomorphisms.

82

§22. Let Let

e

R

Coordinate nets and Monic polynomials

be a ring.

Let

Y

be an indeterminate net over

be a prechip with

Given any net we mean an

(Y) •

B'E

by an

R[Y]-net

R.

Y

Y(B,C)

Y(B,C)

for

Y(B,C)

Y(B,C)

for

with

(R,Y(B'),e)-coordinate= £(Y)

such that:

1 :0: B :0: B'-l 1 { :0: C :0: B (e) )

and B 1 :0: B :0: 0(£ (e)) { 1:0:

c «

and B ' :o:B:o:o(£(e))-l for { T(-l,B,£(e))

Y(B,C)

and

Sub[R,Y=O(B+l)] (Y(B)) 1

is a free R-basis of Y(l,B) (R=)

Given any

B' E [l,o(£(e))]

Coord(R,Y(B') ,e)

I

for B':o:B:o:o(£(e))-l

we put

the set of all (R,Y(B') ,e)-coordinate-nets.

83

B' E

Given any B E [B'

and given any and

u E Q,

we define

Nonmon(R,Y(B' ,B) ,e =u)

_

-

U

YECoord(R,Y(B') ,e)

Sub[R,Y=Y]

-1

u

,

and

Nonmon( (R,Y(B' ,B) ,e =u»

_

U

YECoord(R,Y(B') ,e)

SUb[R,y=y]-l(Y(B',B,T

and we define

Mon(R,Y(B' ,B) ,e =u) =

and

Mon( (R,Y(B' ,B) ,e =u» =

Byan

(R,Y,e)-coordinate-net we mean an

coordinate-net.

Finally we put

(R,Y(l),e)-

84

Coord (R,Y,e)

Coord (R,Y(l) ,e).

Here we have used, and we shall continue to use, the following obvious conventions for sets g: S ..,. S'

(set-theoretic) map If

x

is any S-string then by

such that

o (g (x))

c E [l,o(x)]; well as an by

g (SO)

and

g(x)

we may also regard If

So

of

g(x)

g (x (c))

=

of all S-strings

X

such that

to be a set of

for all

sl-strings then

E So with x (e) E S

we may also regard

Si-strings then by x

1

g(S)-string

to be an S'-string as

S'-strings as well as a set of

is any set of

S' c S'

we denote the

is any set of

g(S)-strings;

and a

1

and

S c Sl

g (x ) (c)

we denote the set {g (x):

be a set of

g-l(SO)

o(x)

Si-string.

c E [l,o(x)]}

So

=

where

S'

and

Sl

Sl-strings.

g(SO)

Si-strings.

g-l(SO)

g(x) E SO;

for all to

If

we denote the set

we may also regard

Similarly for

si-nets, S-webs, sets of

S-nets, sets

of

Sl-nets, sets of

sl-webs, and sets

of

Si-webs, where the concept of webs is to be defined later. We have also used, and we shall continue to use, the following

obvious conventions for an S-module S

Nand

S-submodule

NO

N

where

is a ring.

If

x

x

is a free S-basis of

NO

to mean that the indexed family

of

is an N-string then we say that

is a free S-basis of

NO.

Similarly for N-nets and

N-webs, where the concept of webs is to be

defined later.

85

§23. Let By

R

be a ring and let

gr(R,I)

gr(R,I)

Graded ring of a ring at an ideal I

R.

be an ideal in

we denote the graded ring of

the external direct sum

r

R

at

n l In/I +

I, i.e.,

with

1°

n=O

and for every

n E Z

we define the map

to be the composition of the natural maps

and we put grn(R,I)

We observe that now

gr(R,I)

the (internal) direct sum

r

grn(R,I).

n=O

We put res(R,I)

gro(R,I)

and we define res[R,I]: R

+

res(R,I)

to be the ring-epimorphism induced by we define

gr O [R, I].

For any

fER

R,

86

ord[R,I] (f)

R

and we note that ord[R,I] (f) = For any

f' c R

00

a f E n In. n=l

we define

ord[R,I] (fl)

{ord[R,I] (f): f E fl}

and ord[R,I] ((f'»

min ord[R,I] (f')

and we note that: ord [R,I] ((f'»

00

a

f' c

n

In.

n=l We define the map gr[R,I]: R by putting, for any

gr(R,I)

fER,

gr[R,I] (f)

We note that for any gr[R,I] (f')

f' c R

if

ord[R,I] (f)

n t-

if

ord[R,I] (f) =

00

00

we now have

{gr[R,I] (f): f E f

!

}

and we define gr [R, I] ( (f' ) )

the ideal in

gr(R,I)

generated by

gr[R,I] (fl)

87

and we observe that

gr[R,I] ((fl))

is then a homogeneous ideal in

gr (R). Given any ideal

J

in

R

with

res[(R,I) ,J]: res(R,J)

J c I -+

we define

res(R,I)

to be the unique ring-epimorphism which makes the triangle

R res[R,J]

res[R,I])

1

/ / - , res[ (R,l) ,J]

res(R,J)

commutative.

res(R,I)

88

§24. Let

R

M{R)

be a ring.

Graded ring of a ring We define

the intersection of all maximal ideals in

We put gr (R) and for every

gr{R,M{R») n E Z

and

gr[R]

gr[R,M{R)]

we put gr n (R,M (R) )

and we put

res[R] and for any

res[R,M{R)] fER

f' c R

res(R)

res (R,M (R) )

we put

ord [R] (f) and for any

and

= ord [R,M (R)] (f)

we put

ord [R] (f ') = ord [R,M (R)] (f ') and ord[R] «fl»

ord [R,M (R)] { (f ') )

and gr[R] «fl»

gr[R,M{R)] ( (f'»

R.

89

and for any ideal

J

in

res [ (R) ,J]

R

with

J c M(R)

res [ (R,H (R)) ,J] .

we put

90

§25.

Graded rings at strings and nets and the notions of separatedness and regularity for strings and nets

Let

R

be a ring.

e i t he r { or and let

Y

y

be given where

is an R-string

y

y

Let

is an R-net be given where

Y is an indeterminate string over

R

in case whereas

Y

We put

and for every

1

gr(R'YR) n E Z

n gr [R,y]

=

and

gr[R,y]

we put

n 1 gr [R'YR]

and

n gr (R,y)

and

res(R,y)

and we put 1

res [R,y] and for any

res[R'YR]

fER

we put

ord [R,y] (f) and for any

fl c

y

=

o(y)

is an R-string

is an indeterminate net over R with tty) in case

gr(R,y)

with o(Y)

R

we put

1

ord [R'YR] (f)

y

is an R-net

tty)

91

ord[R,y 1 (f') R]

ord [R, y] (f' ) and

1

ord[R,y] « f ' ) )

ord [R'YR] ( (f '))

and gr[R,y] « f ' ) ) 1

If

then we put

yRcM(R)

1

res [(R) ,y] If

is an ideal in

I

R

res [(R) 'YR]. with

res[ (R,l) ,y] If 1

Y R

Y

then we put res[ (R,l) ,Y

is an R-string with

-1

Y R

C

1

gr[R'YR] « f ' ) ) .

=

or

1

R]. Y

is an R-net with

then we put res [ (R,y) ,y]

-

If

1

YR 1

YR

Y -

t

is an R-string and -

1

C

y( t) R

or

C

y( t)

then we put

y

is an R-net and

-

res [(R,y(t») ,y] If

J

is an ideal in

R

is a net-restriction with

t

- - 1 1 = res [(R,y( t)R) 'YR].

with

res [ (R,y) ,J] We define

is a string-restriction with

then we put 1

res[ (R'YR) ,J].

92

gr [R, Y = y]: res (R,y) [Y] to be the unique

-+

gr (R,y)

res(R,y)-algebra-epimorphism such that 1

{gr[R,Y =y] (Y(c)) = gr [R,y] (y(c)) for all c E [l,o(y)] in case y is an R-string whereas 1

g rl R , Y = y ] (Y(b,c)) =gr [R,y] (y(b,c)) for all (b,c) E supt(1(y)) {

in case y is an R-net. Now let t

{

t

be given where

i s a string-restriction in case

whereas

t

y

is an R-string

is a net-restriction in case

y

is an R-net.

Ne put gr(R,y(t»)

and for every grn[R,y(t)]

n E Z

we put

n

1

gr [R,y( t) R]

and

gr[R,y(t)]

and

gr (R,y( t) )

and

res(R,y(t»)

n

and we put res[R,y(t)]

and for any

1

res[R,y(t)R]

fER

we put

ord [R,y( t)] (f)

1

ord [R,y( t) R] (f)

1

res (R,y( t> R)

93

and for any

fl c R

we put

ord[R,y(t)] (f ")

(f ")

and ((fl»

ord [R,y( t)] ((f I» and gr[R,y(t)] ((fl» = If

c M(R)

then we put

res [ (R) If

If

Y 1

y(t)R

.s; t)] =

is an ideal in

I

R

with

res [ (R, I)

.s: t)]

C

-1

YR

Y C

with

y( t) R

-

-

1 y( t) R 1

then we put 1

rest (R,I) ,y(t)R]' or

-

y

is an R-net with

.s; t)]

C

or -

-

y 1

y( t) R

is an ideal in

t

is a string-restriction with

is an R-net and

t

is a net-restriction

then we put

res [ (R,y( t» J

1

y(t)R c I

.

then we put

is an R-string and

1 y( t) R

If

res [ (R) ,y(

is an R-string with

res [ (R,y) If

((fl».

R

res [ (R,y( t»

.s; t)] with ,J]

J =

C

then we put 1

res [ (R,y( t) R) ,J] .

94

We define gr [R, Y( t)

r

to be the unique

y]: res (R,y( t») [Y( t)]

->-

gr (R,y( t»)

res(R,y(t»)-a1gebra-epimorphism such that

(Y(e))

(y(e)) for all e E supt(o(y),t) in case y is an R-string

whereas

g r [R ,

(Y(b,e))

{

(y(b,e)) for all (b,e) E sup t t z (y) ,t) in case y is an R-net.

DEFINITION 1.

In case

y

is an R-string, we say that

y

is R-separated to mean that

and 1

y( [1,e-1])R

DEFINITION 2.

In case

y

for all e E [l,o(y)].

is an R-string, we say that

is R-regular to mean that

and f o r every { we have

e E I1,o(y)J zy(e)

y(

and every z E R with z

y(

y

95

DEFINITION 3.

In case

y

is an R-string, we say that

y

is R-ultraseparated (resp: R-ultraregular) to mean that for every bijection

H:

[l,o(y)]

to be the R-string with for all

[l,o(y)],

o(H(y»

c E [l,o(y)],

=

o(y)

we have that

upon letting and

H(y)

H(y) (c)

H(y) =

y(H(c»

is R-separated (resp:

R-regular) . DEFINITION 4.

In case

y

is an R-string, we say that

is R-superregular to mean that

y

is R-ultraseparated and

y y

is R-ultraregular. DEFINITION 5.

In case

is R-separated (resp:

y

is an R-string, we say that

y(t)

R-ultraseparated, R-regular, R-ultraregular,

R-superregular) to mean that, upon letting h:

[l,card(supt(o(y) ,t»]

supt(o(y) ,t)

preserving bijection and upon letting that

o(x)

=

card(supt(o(y) ,t»

c E [l,card(supt{o{y) ,t»],

and

x

to be the unique orderto be the R-string such

x(c)

we have that

y(h{c» x

for all

is R-separated

(resp: R-ultraseparated, R-regular, R-ultraregular, R-superregular). DEFINITION 6. g:

In case

y

[1,card{supt(l(y) ,t»]

such that for every upon letting

c

I

is an R-net, let

supt(l{y) ,t) and

c

2

in

be the unique bijection

[1,card(supt(l(y),t»],

g(c with B E [1,o(l(y»] and i i) i) Ci E [1,B i(l(y»] we have: c c2 either B I = B 2 and I C C or B < B now let x be the R-string such that I 2 I 2; o(x)

=

(Bi,C

card(supt(l(y) ,t»

[1,card(supt{l(y) ,t»],

and such that for every upon letting

(B,C)

=

g{c)

c

in with

96

BE [l,o(.Q,(y»]

and

C E [l,B(.Q,(y»)],

we have

with this notation in mind, we say that

y(t)

x(c) = y(B,C); is R-separated

(resp: R-ultraseparated, R-regular, R-ultraregular, R-superregular) to mean that

x

is R-separated (resp: R-ultraseparated, R-regular,

R-ultraregular, R-superregular). DEFINITION 7.

In case

y

is an R-net, we say that

y

is

R-separated (resp: R-ultraseparated, R-regular, R-ultraregular, R-superregular) to mean that

y(l)

is R-separated (resp:

R-ultraseparated, R-regular, R-ultraregular, R-superregular). DEFINITION 8.

9' Rand

that y(t)

We say that gr [R, Y(t) = y]

y(t)

is R-quasiregular to mean

is injective.

is R-ultraquasiregular to mean that

for every

y(t')

We say that is R-quasiregular

t'

)Where

t' c

lwhereas We say that to mean that

supt(o(y),t) in case

t' c

y

is an R-string

y

supt(.Q,(y) ,t) in case

y

is an R-net.

is R-quasiregular (resp: R-ultraquasiregular) y(l)

DEFINITION 9.

is R-quasiregular (resp: R-ultraquasiregular). Assume that

y(t)

is R-quasiregular.

Then

by

grIR, y(t)=Y]*: R

-r

res(R,y(t»

[Y(t)]

we denote the unique surjective map which makes the triangle

97

R· gr[R,y(t)=y]*) res(R,y(t»

gr[R,y(t»)

1

[y(t)]

gr[R,y(t)=y]

gr(R,y(t» commutative.

Also we define

gr[R, y(t)=y]: R

res(R,y(t»

->-

to be the map induced by

[Y]

gr[R, y(t)=y]*.

For any

f' c R

we

define gr [R, y(t>

Y]

* ((fl»

the ideal in

res (R, y( t»

[Y( t)]

generated by

gr[R, y(t)=Y]if ')

and gr [R, y( t) =y] ( (f I

n E Z

For every

)

)

res[R, y(t)=Y] (fl)

generated by

gr [R, y(t)=y] (fl).

we define

n

n

gr [R, y( t) =y]: y( t) R to be the

the ideal in

res (R, y( t»

->-

[Y]

res [R, y( t.)] -homomorphism obtained by putting gr [R, y( t) =y] (f)

for all

o

for all

grn[R,y(t)=y] (f) f E

and we define grn[R, y(t)=y]*:

->-

Iso(res(R, y(t»,Y(t), =n)

98

to be the

res[R,y(t)]-epimorphism induced by

Given any

I

either

I

or

I = x

or

I

grn[R,y(t)

Y].

where is an ideal in where

iCE)

or

I

Y

or

I

y("t)

x

where

where

Y

where

R

with

c I

is an R-string with x

is an R-string and

t

is a string-restriction with y(t)Rcx(t)R

1 -

is an R-net with

-

1

1 -1 y(t)R c YR

y

is an R-net and

t

is a net restriction with y(t)Rcy(t)R

1 -

-

1

we define gr [(R, I) ,y( t)=y]: R

-+

res (R, I) [Y]

to be the composition of the maps R gr[R,y(t)=y] >

res (R, I) [Y] and we define gr[(R,I).y(t)=y]*: R

-+

res(R,I) [Y(t)]

to be the surjective map induced by £1 C

R

we define

gr[(R,I) ,y(t)=Y]

and for any

99

gr[ (R,I) ,y(t)=y] * (fl))

the ideal in

res(R,I) [y(t)]

generated by

gr[(R,I) ,y(t)=Y]*(fl)

and the ideal in

res (R, I) [Y]

generated by

gr[(R,I),y(t)=Y] (f ")

gr [(R, I) ,y( t)=Y] ( (fl))

n E Z

and for any

we define the

n

res[R,I]-homomorphism n

gr [(R,I),y(t)=Y]: y(t)R

-+

res(R,I)[Y]

to be the composition of the maps

n gr [R,y( t)=Y] ) res (R,y( t») [Y]

1

sub[res[(R,I) ,y(t)] ,Y]

res (R,I) [Y] and we define grn[(R, I),y(t)=Y]*: to be the

Iso(res(R,I) ,Y(t), =n)

res [R,I]-epimorphism induced by

c M(R)

If

-+

then we define the map

gr [(R) ,y( t)=Y]: R

res (R) [Y]

-+

by putting gr[ (R) ,y(t)=Y]

gr[ (R,M(R)) ,y(t)=Y]

and we define gr[(R),y(t)=Y]*: R

-+

res(R) [Y(t)]

grn[(R,I) ,y(t)=Y].

100

to be the surjective map induced by fl c

R

gr[(R) ,y(t)=Y]

and for any

we define

gr [(R) ,y(t)=Y] * «fl))

the ideal in

res (R) [Y( t)]

generated by

gr[(R),y(t)=Y]*(f')

and gr[(R),y(t)=Y] «f'))

and for any

n E Z

the ideal in

res(R) [Y]

generated by

gr[(R) ,y(t)=Y] (f')

we define the

n

n

gr [(R) ,y( t)=Y]: y( t) R

res[R]-homomorphism

res (R) [Y]

->-

by putting n gr [(R) ,y( t)=Y]

grn[(R,M(R)) ,y(t)=y]

and we define n

n

gr [(R) ,y(t)=y]*: y(t)R

->-

Iso(res(R) ,y(t), =n)

to be the res[R]-epimorphism induced by DEFINITION 10.

If

Y

grn[(R) ,y(t)=Y].

is R-quasiregular then we verbatim

take over the entire above material of Definition 9 after everywhere deleting LEMMA 1. g E R

If

(t). y(t)

is R-quasiregular and if

are such that ord[R,y(t)]f I

and

00

fER

and

101

g ={Y(C) for some c E supt(o(y) ,t) in case y is an R-string y(b,c)

for some (b,c) ESllpt(9,(y),t) in case y is an R-net

then 1 + ord[R,y(t)] (f).

ord[R,y(t)] (fg)

PROOF.

Obvious. 2.

t*

Assume that

y(t)

is R-quasiregular.

Let

t'

and

be disjoint sets such that t ' U t* = supt(o(y) ,t) in case y is an R-string { whereas t' U t* = supt(9,(y) ,t) in case y is an R-net.

Let

R = res(R,y(t*»)

and

y

The case when

y

res[R,y(t*)] (y).

Then

y(t')

is

R-quasiregular. PROOF. when

y

obvious. card(t*)

is an R-string. Finally, when

0

induction on R-string,

is an R-net follows from the case

Also the case when y

y(t)

be the element in

is

is an R-string, the general case of

follows from the case of card(t*).

card(t*) = 0

card(t*)

So we may suppose that

is R-quasiregular, and [l,o(y)]

{c },

Let there be given any HE !so(R,y(t'), =v)

with

0

y

is an

card(t*) = 1.

such that t*

by

1

v E Z

Let

c

102

such that

-z

sub [R,Y We shall show that then

-

with

-

v+1

z E y( t ') R

•

Sub[res[R,y(t')] ,Y] (H)

o

and this

will complete the proof. We can take H E IsO(R,Y(t'), =v)

v+1

and

z E y(t)R

such that Sub [res [R, y( t*)] ,Y]

(H)

H

and

res [R,y( t*)]

(z)

z.

Now z + fy(c)

sub [R, Y = y] (H)

with

fER

and in view of Lemma 1 we see that ord[R,y(t)] (f) > v - 1 and hence f

sub [R, Y = y] (F)

for some

F E Iso (R, Y( t), =v - 1) .

Let F

H - FY(c).

Then F E Iso (R, Y( t), =v) and sub [R, Y = y] (F) = z E and hence by the R-quasiregularity of Sub[res[R,y(t)] ,Y] (F)

y(t) O.

s; t) ;+1

we must have

103

Because

F

H - FY(c)

=

H E R[Y(t')]

and

with

c

t',

the

above equation yields that Sub[res[R,y(t)] ,Y] (H)

O.

Therefore SUb[res[R,y(t')] ,Y] (H)

LEMMA 3.

Assume that

R-quasiregular. PROOF. when

y

Then

y(t)

y(t)

The case when

is an R-string.

O.

is R-separated and

is R-regular. y

is an R-net follows from the case

So we shall suppose that

R-string and we shall make induction on The assertion is obvious when let

card(supt(o(y) ,t))

true for all values of given one. for every y

=

Let 0

c Z

=

min supt(o(y) ,t).

O.

t'

zy(c) =

Then, in view of Lemma 1,

O.

Let

R = res(R,y({c))

supt(o(y) ,t)\{c);

then obviously

card(t') < card(supt(o(y) ,t)),

induction hypothesis it follows that Therefore

y(t)

LEMMA 4. y(t) PROOF.

So now

smaller than the

is R-separated, and by Lemma 2 we also see that

R-quasiregular; since

then

=

and assume that the assertion is

card(supt(o(y) ,t))

and

is an

card(supt(o(y) ,t)).

card(supt(o(y) ,t))

E R we have

res[R,y({c)] (y)

y(t')

0

y

y(t)

y(t') by the

is R-regular.

is R-regular. If

y(t)

is R-ultraseparated and R-quasiregular

is R-superregular. Follows from Lemma 3.

is

104

§26.

Inner products and further notions of separatedness and regularity for strings

Let

G'

semigroup. let

Q'

be a nonnegative ordered additive abelian Let

Q'

be the rational completion of

G', i.e.,

be the unique (upto G'-isomorphisms)

nonnegative ordered additive abelian oversemigroup of that for every (n

u E Q'

depending on For any

u).

we have Let

i E Q(o(O)) inpo(i,O)

nu E G'

for some

G'

t

0

be a G'-string.

0

we define 2:

i(c)O(c)

Ls cs o Ib )

and we note that inpo (i,o) E Q' and: if For any

i E Z(o(O))

i' c Q(o(O)) inpo (i'

For any

Gc Q

then

inpo(i,O) E G'

we put ,0)

Unpo (i, 0): i E .i "} •

we put inpo(G,O)

inpo (G (0

(0)) ,0)

and we note that: if

Gc Z

then

inpo(G,O) c G'

.

.

such

n E Z;

105

For any

G c Q

and any string-restriction

t

we put

inpo(G,D(t») = {inpo(i,D) : iE G(o(D)) with supt(i) csupt(O(D),t)} and we note that: if For any

Gc Q

Gc Z

and any

u E Q'

G(DPu) Recall that u E G'

and every

then

inpo(G,D(t» and

c G'

.

P E

we put

{i E G(o(D)): inpo(i,D)Pu} .

G'

is archimedian means that for every

0

u' E G'

we have

n(u,u')u'

u

for some

n(u,u') E Z. Now let there be given a string-restriction

t.

Let

t

supt(o(D) ,t) n supt(D)

and t'

lc ' E

t:

for every c E t n(c,c')D(c')

we have D(c) for some n(c,c')EZ}

and til We say that We note that

i\t' D(t)

is archimedian to mean that

til

106

(1)

D(t)

is archimedian

D(t)

is archimedian

and we also note that (2)

if

G'

is archimedian then so are

We say that

D

D(t)

and

is archimedian to mean that

D(t).

D(l)

is

archimedian. Now clearly there exists a unique sequence of pairwise disjoint nonempty subsets of m E Z

and

t"

t

U

Ls rs m

such that for

1

q

s;

f o r every

s;

m

t"

t l,t 2, ... ,tm

with

r

it is true that:

c' E t q and every Pinpo (j ,D))

we have

133

and hence for any

()

P E

we have

n

nEZ

iEZ(o(x)=n+l,t') jEZ(o(x),t n )

(22)

n

n

nEZ

iEZ(o(x)=n,t') jEZ(o(x),t n )

iso(R,x(t) ,DPinpo(i+j,D»

iso(R,x(t),DPinpo(i+j,D».

Obviously

for any

n

nEZ (23)

P E

we have

n

iEZ(o(x)=n,t') jEZ(o(x),t n )

n

wEinpo (Z ,D( t»

iso(R,x(t),DPinpo(i+j,D»

iso(R,x(t) ,DPw).

By (22) and (23) we see that

for any

n

(24)

nEZ

P E

we have

n

iEZ(o(x)=n+l,t') jEZ(o(x),t n )

n

wE inpo (Z ,D( t»

iso(R,x(t> ,DPw).

134

By (2),

(4),

(6) and (21) we see that

for any

P E

and

n E Z

we have

iso(R,x(t'),DPA(n+2)D(c 2» (25)

n

c

iEZ (0(x)=n+1,t') jEZ(o(x),t")

iso(R,x(t),DPinpo(i+j,D»

isO(R,x(t'),DPnD(c

c

Upon taking intersections as

n

2).

varies over

Z, by (25) we see

that

for any

n

nE Z (26)

n

nEZ By (8),

n

iE Z (0 (x) =n+1, t' ) jEZ(o(x),t")

nE Z

c

(20),

for any

n

iso(R,x(t),DPinpo(i+j,D»

isO(R,x(t'),DPnD(c

(24) and

nEZ

we have

iSO(R,x(t'),DPA(n+2)D(c 2»

n

c

P E

2»·

(26) we see that

P E

we have

iso(R,x(t') ,DPAnD(c 2»

n

(27)

wE inpo (Z,D( t»

n

nEZ

iso(R,x(t) ,DPw)

iSO(R,X(t'),DPnD(c 2»·

135

The assertions of the Lemma now follow from (9),

(12),

(17)

and (27). LEMMA 3.

Assume that

x(t)

is

(R,O)-separated.

Then given

any f E R\

there exist that

n

wEinpo(Z,O(t) )

iso(R,x(t) ,0 >w)

u E inpo(Z,O(t»)

sub[R,X = x] (F) = f PROOF.

and

and

FE Iso(R,X(t),o 0

144

and assume that the assertion is true for all values of smaller than the given one.

Let

c = max t

By assumption

(1)

x(t)

for every

card(t)

t'

and

is R-regular and hence

Z

I

E R\x(t')R

we have

I

zx (c) rt x( t ') R •

Upon letting

(2)

F

we have F E Iso(R,X(t),Dv)

E Iso(R,X(t'),02V)

and

(8 )

Info [res [R,x( t)] ,X,D = v] (H ) l

'I

0

•

We shall now divide the rest of the argument into two cases according as sub[R,X =x] (H or

2)

E iso(R,x(t'),O 2V)

sub[R,X=x] (H ) rf. iso(R,x(t'),02V). 2

146

First consider the case when

(11)

sub[R,X=x] (H

2)

E iso(R,x(t'),D;";V).

Now there exists

(12)

H

2

E ISO(R,X(t'),D;.,;V)

such that

(13)

sub[R,X =x] (H

2).

Upon letting

by

(7)

and

(12)

we have

(14)

and by

HE Iso(R,X(t'),D;.,;v)

(5)

and

(13)

(15)

we have

sub [R,X = x ] (F)

sub[R,X=x] (H

O

+Hx(c)

b

)

and obviously we have

Info [res [R, x( t l ], X, D = v] (H) = Info [res [R, x( t)], X, D = v] (HI) •

(16)

Now by

(8)

and

(16)

Info[res[R,x(t)] ,X,D =v] (H)

'I

0

147

and hence a fortiori

(17)

By

t- o.

Info[res[R,x(t')] ,X,D =v] (H)

(1),

(14)

and

(17)

we

see that

HX ( C ) b E Iso(R,X(t'),D;"V) {

and hence in view

t-

Info[res[R,x(t')],X,D =v] (HX(c)b)

(6)

H (18)

we

O

< card(t)

0

see that

+ Hx(c)

b

E Iso(R,X(t'),D;"v)

Info[res[R,x(t') ,X,D =v] (H

Now card(t')

and

and hence,

in view of

O

and

+Hx(c)

(3)

and

b

)

(18),

the induction hypothesis we conclude that

(19)

By

sub[R,X=X](H

(2),

(20)

(15)

and

(19)

O

+

Hx(c)b)

t-

O.

we get

sub[R,X =x] (F

l)

t-

sub[R,X =x] (F

2).

Next consider the case when

(21)

sub[R,X=x] (H

2)

e iso(R,x(t'),D;"v).

Now by Lemma 3 there exists

(22)

w E inpo(Z,D(t'»)

t-

O.

by

148

and

(23)

HE

Iso(R,X(t'),D w)

wEinpo(Z,E(t'»)

isO(R,y(t'),E

iso (R,y(t) ,E

Follows from Lemma 2 of §29.

w)

w).

z]

152

LEMMA 3.

If

f E R\

there exists that

LEMMA 4.

is (R,E)-separated then given any

n

iso(R,y(t),E > w)

wEinpo(Z,E(t) )

u E inpo(Z,E(t»

sub[R,Y =y] (F) = f PROOF.

and

F E Iso(R,Y(t),E

u)

such

Info[res[R,y(t)] ,Y,E =u] (F) 'i' O.

and

Follows from Lemma 3 of §29. If

F l E Iso(R,Y(t),E that

y(t)

y(t) u)

is (R,E)-regular and if and

F

2

E Iso(R,Y(t),E

sub [R, Y = y] (F l) = sub [R, Y = y] (F 2)

u E Q' u)

and

are such

then we have

Info[res[R,y(t)] ,Y,E =u] (F I) = Info[res[R,y(t)] ,Y,E =u] (F 2). PROOF. LEMMA 5.

Follows from Lemma 4 of §29. If

y(t)

is (R,E)-regular then for any

u E Q'

we have ker(Info[res[R,y(t)],Y(t),E=U]*)::-J ker(sub[R,Y(t) PROOF.

This is simply a reformulation of Lemma 4.

DEFINITION 1. for any

Assume that

is (R,E)-regular.

Then

u E Q', in view of Lemmas 1 and 5, we can now define

info[R,y(t) =Y,E=u]*: to be the unique triangle

y(t)

+Iso(res(R,y(t»,y(t),E=U)

res [R,y(t)]-epimorphism which makes the following

153

I

iso (R,y( t) ,E;;,u) info [R,y( t) =Y ,E=u] *) Iso (res(R,y (t») ,y( t),E=U)

,ub[R,y(t)=y,E>u]'

Info[res[R,y(t)] ,y(t),E=U]*

ISO(R,y(t),E;;'U)

commutative, and we define

info [R,y( t)=y ,E = u]: iso (R,y( t) ,E ;;,; u) .... res (R,y( t») [Y]

to be the

res[R,y(t)]-homomorphism induced by

info[R,y(t)=Y,E=U]*,

and we note that

info[R,y(t)= Y ,E =u]

info[R,y(t)=Y,E=U]

info [R, y( t) = Y , E=u] *

info[R,y(t)=Y,E=u]*.

and

We observe that given any positive integer

n

II

i=l

f. E iso(R,y(t),E;;, 1.

n

z

i=l

u

n

and given any

i)

and info[R,y(t) =Y,E

(n

i=l

nn f.) i=l 1.

info[R,y(t) ,E =u i] (f i)·

154

For any

fER\

n iso(R,y(t),E>w) wEinpo(Z,E(t) )

in view of Lemmas 1, 3 and 4 we can now define

ord [R,y( t) ,E] (f)

the unique

u' E Q'

f E iso(R,y(t) ,E

such that u') and

info(R,y(t),E=u') (f)

F 0

and we observe that then

ord[R,y(t),E] (f) E inpo(Z,E(t») and for any

u E Q'

we have:

u ,;; ord [R,y( t) ,E Hf) '" f E iso (R,y( t) ,E

u)

whereas:

u < ord [R, Y( t) ,E] (f)

and

f E cc

{ info[R,y(t) =Y,E =u] (f) = O.

We also put

ord [R,y( t) ,E] (f)

We note that for any ord [R,y( t) ,E] (f)

00

for all

fER co

cc

fEn iso (R,y(t),E > w) • wEinpo(Z,E(t»)

we now have: fEn iso(R,y(t),E>W). wEinpo(Z,E(t»)

155

We also observe that for any

fER

we have

ord[R,y(t),E] (f) = ord[R,y(t),E] (f). For any

f' c R

we put

ord[R,y(t),E] (f')

{o rd [R,y(t) ,E](f):

f E f "},

We define the map info[R,y(t)=Y,E]: R by putting, for all

-+

res(R,y(t») [Y]

fER,

info[R,y(t) = Y,E] (0 = {infO[R,y(t)=Y,E=ord[R,y(t),E] (f)] (f) if ord[R,y(t),E] (f)

o

if

ord[R,y(t),E] (f) =

1-

00.

We define info[R,y(t)

Y,E]*: R

-+

res(R,y(t») [Y(t)]

to be the surjective map induced by

info[R,y(t)=Y,E].

that info[R,y(t)=Y,E]*

info[R,y(t)=Y,E]*

and A

info[R,y(t)=Y,E] = info[R,y(t)=Y,E]. For any

f' c R

we define

We observe

00

156

info[R,y(t)=Y,E]*((f'»

the ideal in

res (R, y( t»

[Y( t)]

generated by

info[R,y(t)=Y,E]*(f')

and A

info [R,y(t)=Y ,E] ((f'»

Given any

I

the ideal in

res(R,y(t»

[Y]

generated by

info[R,y(t)=Y,E] (fl).

where

either I is an ideal in R with or I

X where x

or I

x(t)

where

c I

is an R-string with

x

c

is an R-string and

t

is a stringA

1

__ 1

restriction with y(t)R c x(t)R or I

Y is an R-net with

or I

yet)

Y (A)l t R

-1 YR

C

t

where y is an R-net and A

1

with y(t)R

is a net-restriction C

_ -

1

y(t)R

we define info[(R,I),y(t)=Y,E]: R

-+

res(R,I)[Y]

to be the composition of the maps R info[R,y(t)=Y,E]

1

> res(R,y(t»

[Y]

Suh[res[(R,I) ,y(t)] ,Y]

res (R,l) [Y] and we define

157

info[(R,l),y(t)=Y,E]*: R

->-

res(R,l) [Y(t)]

to be the surjective map induced by

info[(R,l) ,y(t)=Y,E]

and

we observe that info[(R,l) ,y(t)=Y,E]*

info[(R,l) ,y(t)=Y,E]*

and info[(R,l),y(t)=Y,E]=info[(R,l) ,y(t)=Y,E] and for any

ff c R

we define

info[ (R,l) ,y(t)=Y,E] * «fl»

the ideal in

res(R,l) [Y(t)]

generated by info[(R,l),y(t)=Y,E]*(f ') and info[ (R,l) ,y(t)=Y,E] «fl»

the ideal in

res (R, I) [Y]

generated by info [(R, I) ,y( t)=Y ,E] (ff) and for any

u E Q'

we define the

res[R,l]-homomorphism

info[(R,l),y(t)=Y,E=u]: iso(R,y(t),E-

res(R,l)[Y]

to be the composition of the maps

iso (R, y( t) , E res(R,y(t» [Y] Sub{res{(R,I)

res(R,l)[Y]

,y(tll

,Y]

158

and we define info[(R,I) ,y(t)=Y,E=u]*: to be the

Iso(res(R,I) ,Y(t),E=u)

res [R,I]-epimorphism induced by

info[(R,I) ,y(t)=Y,E=u],

and we observe that info[(R,I),y(t)=Y,E=U]

info[(R,I) ,y(t)=Y,E=u]

and info[(R,I) ,y(t)=Y,E=u]* If

A

1

y(t)R

C

M(R)

info[(R,I) ,y(t)=Y,E=U]*.

then we define t:he map

info[(R),y(t)=Y,E]: R

res(R)[Y]

by putting info[(R) ,y(t)=Y,E]

info[(R,M(R)),y(t)=Y,E]

and we define info[(R),y(t)=Y,E]*: R

-+

res(R) [Y(t)]

to be the surjective map induced by

info[(R) ,y(t)=Y,E]

observe that info[(R) ,y(t)=Y,E]*

info[(R) ,y(t)=Y,E]*

and info [(R) ,y( t)=Y ,E] and for any

f' c R

info[(R) ,y(t)=Y,E]

we define

and we

159

info [(R) ,y(t)=Y ,E] * ((fl))

the ideal in

res(R) [Y(t)]

generated by

info[(R) ,y(t)=Y,E]*(f')

and info[(R),y(t)=Y,E] ((fl))

and for any

u E Q'

the ideal in

res(R) [Y]

generated by

info[(R) ,y(t)=Y,E] (f')

we define the

res[R]-homomorphism

info[(R),y(t)=Y,E=u]: iso(R,y(t),E;o:u)

->-

res(R) [Y]

by putting info[(R) ,y(t)=Y,E=u]

info[(R,M(R)) ,y(t)

Y,E

u]

and we define info[ (R) ,y(t)=Y,E=u] *: iso(R,y(t) ,E;o:u) to be the res [R]-epimorphism induced by

->-

Iso(res(R) ,Y(t) ,E=u)

info[(R) ,y(t)=Y,E=U],

and

we observe that info[(R) ,y(t)=Y,E=u]

info[(R) ,y(t)=Y,E=U]

and info[(R) ,y(t)=Y,E=u]*.

info[(R) ,y(t)=Y,E=U]* DEFINITION 2.

If

Y

is

(R,E)-regular then we verbatim take

over the entire above material of Definition 1 after everywhere deleting supt (E)) •

(t)

and replacing

t

by

E

(or, equivalently, by

160

LEMMA 6. PROOF.

If

is R-regular then

y(t)

is R-quasiregular.

Follows from Lemma 5 of §29.

LEM}1A 7. y(t)

y(t)

If

y(t)

is R-separated and

then

is R-regular. PROOF.

This is only a repetition of Lemma 3 of §25.

LEMMA 8.

y(t) '" y( t)

is R-ultraseparated and R-regular

'" y( t)

is R-ultraseparated and R-ultraregular

'" '" PROOF.

is R-superregular

y( t)

is R-ultraseparated and R-ultraquasiregular

y(t)

is R-ultraseparated and R-quasiregular.

Follows from Lemmas 6 and 7.

161

§3l.

Protochips and parachips

Recall that for any prechip e[B]

e(B,B,O)

for

1

e

we have put B

o(Z(e»

and e [[B]] Ls bs o (Z (e) ) Oscs b (Z (e) )

e(B,b,c)

By a protochip we mean a pre chip

e[[B]] f 0

for

1

B

for

1

B

e

1

B

such that

o(Z(e».

By a parachip we mean a protochip e[B] f 0

for

o(Z(e»

e

such that

- 2 .

o(Z(e».

162

§32.

N-support of an indexing string for

Let

be an indexing string. supt

{(b,c)

2

N

6

Recall that

E z(2): 1

b

and 1

c

We define

{(b,c)

E Z (2): 1

b

and 0

0

c

b

and {(B,b,e)

E z(3): 1

B

(b,c)

and

and E supt

{ (B I 13 I 13 .e: E Z (4): B E [1 I

2

0

) ]

and

(B,B,C) E supt 3 and {(B,B,C,b,c)

E Z

(5)

:

E supt

(B,B,C) (b,c)

and

3

E

and

{ (B I B,B I C I b ,e) E Z

(6)

: B E [1

I

0

]

(B,B,C,b,c) E

and

163

§33.

Pres cales

By a pre scale we mean a system indexing string

2(E)

E

consisting of:

called the index of

an

E, and for every

(B,B,B,C,b,c) E supt (2 (E)) 6

a nonnegative rational number called the every

,....,

A

A

(B,B,B,C,b,c)

th

A

A

E(B,B,B,C,b,c)

primary component of

E,

and for

(B,B,B,C,b,c) E supt 6(2(E)) a nonnegative rational number

called the

A

A

(B,B,B,C,b,c)

th

E((B,B,B,C,b,c))

secondary component of

E.

We define denom(E)

{o

t-

n E Z:

nE (B, B, B, C, b , c) ) E Z for all (B,B,B,C,b,c) E sUPt 6(2(E))}

and

{o

denom( (E))

t- n E Z: nE((B,i,B,C,b,c)) E Z for all (B,B,B,C,b,c) E supt 6(2(E))}

and for any set denom (E' )

and

E'

of prescales we put

n

EEE'

denom(E)

164

n

denom( (E'»

For any

denom( (E).

EEE'

B E [l,o(£(E»)]

denom(E,B)

{a

we define

I n E Z: nE(B,B,B,C,b,c) E Z for all (B,B,C,b,c) E supt s (£ (E»}

and

{a

denom( (E,B»

I n E Z: nE((B,B,B,C,b,c»E Z

for all

For any £(uE) = £(E) and

u E Q,

by

uE

(B,B,B,C,b,c) E supt

we denote the prescale with

(uE) (B,B,B,C,b,c) = UE(B,B,B,C,b,c)

such that

(uE) ((B,B,B,C,b,c)

E

=

uE ((B,B,B,C,b,c»

6(£(E».

for all

Likewise, given any indexing

string £, we may regard the set of all prescales whose index is £

as an additive abelian semigroup with componentwise addition. Finally, given any

E supt

we denote the Q-net whose index is component is by whose

4(£(E»,

£(E) and whose

for all

by

E(B,B,B,C)

(b,c) th

(b,c) E supt(£(E», and

we denote the Q-net whose index is £(E) (b,c)th

component is

(b,c) E supt(£(E».

E((B,B,B,C,b,c»

for all

and

165

§34. Given any prechip which

=

Derived prescales

e, by

e*

we denote the prescale for

and whose components are defined as follows.

Firstly we put

e*(B,B,E,C,b,c) =

o

e*((B,B,C,b,c»

if

(B,B,R,C,b,c) E supt

and b < B.

6

Secondly we put

e*(B,B,B,C,b,c)

if (B,B,E,C,b,c) E supt if

(B,B,B,C,b,c) E supt

6 6

and b

B > B

and b

B oS B > B.

Thirdly, given any wi th

B oS B s

B

to begin with we put

e*(B,B,E,C,b,c)

o

i f B < b oS

o

i f B oS b < Rand

o

i f b = Band

(e )

0

o

if B < b

Band

o

if B = b

B oS

1

if B = b =

0

0

(e) ) (e) )

COoS c oS b

(e) )

c = C f

and

(e )

0 oS

(e) )

oS b

r-;

i f b = 13

and then, by decreasing induction on

0 oS c oS b

c f C f

1

fb

and 0 s c oS b

C

0

- 2 and C = 0 oS - 1 and C = 0 oS

b, we define

C

C

oS b

oS b

(e) )

166

e*(B,B,B,e,b,c) b+L;b',;o(t(e)) Os c l s b ' (t(e))

e*(B,B,B,e,b' ,c')e(b+l,b' ,c')

if

BO ,; b < Band {

,; c ,; b(t(e)).

Finally we define e* ((B,B,B,e,b,c)) b,; b ' ,; 0 ( t (e) ) Os c t s b ' (t(e))

e* (B,B,B,e,b'

rC'

)e(b,b' ,c')

(B,B,B,e,b,c) if

{

and

B,; b.

E s up t ; (

(e))

167

§35. Given any prescale

supt(E)

Supports of prescales E

we define

{(B/B/E/C) E supt

E

4

(B/B/B/C/b/c) t

(Q,

(E»:

0 for some

(b s c ) E supt

2

(Q,

(E»}

and supt ( (E) )

{(B/B/E/C) E supt

E

4

(Q,

t

«B/B/E/C/b,c»

(E»:} 0 for some

(b,c) E supt (Q, (E» 2

and for every

B E [l,o(Q,(E»]

we define

supt(E/B)

{ (B/E/C) E supt

3

(Q,

(E»:

(B/B,E/C)

E supt (E)}

and supt ( (E/B»

{(B/E/C)

E su Pt (Q, (E»: 3

(B,B/E/C)

E supt «(E»}.

168

§36.

Protoscales

By a protoscale we mean a prescale A

A

E(B,B,B,C,B,O) Given any prescale such that

for all

I

to be the prescale

we have

A

A

E(B,B,B,C,b,c) E*(B,B,B,C,b,c)

E*

and such that for every

=

A

such that

(B,B,B,C) E supt (E) .

E, we define

(B,B,B,C,b,c) E

A

E

if

E(B,B,B,C,B,O)

o

E(B,B,B,C,B,O) f 0

E(B,B,B,C,B,O)

if

0

and E(B,B,B,C,b,c) E* «B,B,B,C,b,c))

o We note that if

E

if

E (B,B,B,C,B, 0)) f

if

E «B,B,B,C,B, 0))

E «B,B,B,C,B, 0))

0

0

is any pre scale then obviously

E*

is

a protoscale. It follows that if and we have

e

=

is any prechip then and for every

e** is a protoscale

(B,B,B,C,b,c) E supt 6

we have e*(B,B,B,C,b,c) e**(B,B,B,C,b,c)

if

e*(B,B,B,C,B,O)

o

if

A

A

e* (B,B,B,C,B, 0)

e*(B,B,B,C,B,O) f

0

0

169

and

e**«B,B,B,C,b,c»

e* (B,B,B,C,b,c) e* «(B,B,B,C,B, 0»

o

if

if

e* «(B,B,B,C,B,O»

e* ((B,B,B,C,B,O»

=

0 .

t- 0

170

§37.

Inner products for protoscales

Given any protoscale and

G c

G (E (B)

Q

u E Q

and

u)

=

E

and given any

B E [1,o(£(E»]

we define

{j E G (£ (E»:

inpo (j,E (B,B',i3,C» for all

u

(B',E,C) E supt (E,B)}

and

{j

G(E((B»

E G (£ (E»:

inpo (j,E ((B,B,B,C») for all

and for any

G(E(B)Pu)

P E {>,=}

{j

(B,g,C)

E

suPt((E,B»}

we define

E G(E(B)

u):

inpo (j,E (B,o (£ (E»,o (£ (E», 0) )Pu}

and

G (E ((B) )Pu)

{j

E G (E ( (B) )

u) :

inpo (j , E ( (B, 0 (£ (E) )

,0

u

(£ (E) ) ,0) ) ) Pu }

171

§38.

Scales and isobars

By a scale we mean a proto scale

(1)

E ((B,B,B,C,b,c)) = {

E

such that

° = E (B,B,B,C,b,c)

for all those (B,B,B,C,b,c) E supt

6(£(E))

for which b < B

and

(2)

E ( (B, O(£ (E ) ) , O (£ (E ) ) , O, b , C) ) '!{

for all those (B,b,c) E supt

We note that for any scale

E

° '!-

3(£(E))

E(B,o(£(E)),o(£(E)),O,b,c) for which

b

B.

we obviously have

(O (£ (E) ) , O (£ ( E) ) , O) E supt(E,B) n supt((E,B)) {

for all

Now let

BE E

[l,o(£(E))].

be a scale, let

indeterminate net over

R

with

R

be a ring, let

£(Y) = £(E), and let

Y

be an u E Q.

For any and

BE [l,o(£(E)) -1]

B' E [1, B]

we define the

ideals

Y(B'

and

Y(B'

in

R[Y(B')]Q

by putting

A

A

n

(B,B,C)Esupt(E,B)

Iso(R,Y(B') ,E(B,B,B,C) .:U)Q

172

and

A

A

n

(B,B,C)Esupt ((E,B» and we define the

ideals

and

in

R[Y(B')]

by putting

n

Y(B'

Y(B' ,B)

R[Y(B')]

and

Y( B ' , B)

(R,

and given any

R

O

n

Y(B'

)

C

R

with

0 E R

O

R[Y(B')]

we put

and ISO({R

O

n

=

We observe that, in view of

(1),

RO[Y(B')].

173

for any integers B,B' ,B" with 1;5; B" ;5; B' ;5; B;5; 0(9, (e)) -1 we have:

n

R[Y(B')]Q

n

R[Y(B')]Q

Y(B",B

and

for any integers B,B' ,B" with 1;5; B" ;5; B' ;5; B;5; 0(9, (e)) -1 we have:

n (4)

R[Y(B')] = Y(B'

Y(B"

n R[Y(B')]

I

l

Y(B" ,B)

Y(B' ,B)

= (R,Ez) )R[Y(B")].

For any

BE

[l,o(9,(e))-l]

and

B' E [l,B]

and

P E {=,>}

we define the

R-submodules by putting

and Y(B'

of R[Y(B')]Q

174

Y} we have:

176

§39.

Properties of derived prescales

This section is by way of details of proofs of assertions to be made in the next section.

The reader may decide how much

of these details he wishes to read. Let

e

be a protochip.

Recall that by the definition of a protochip

(1)

e(B,b,c) E Q

(2)

e(B,b,c)

(3)

e(B,b,c)

(4)

e (B,b, 0)

(5)

e[B] = e(B,B,O)

(6)

e[ [B]]

=

° ° °

(B,b,c) E supt

for every

if

l,;;b,;;B

o (1 (e»

if

l,;; b < B ,;;

0

if

l,;; B

for

Ls bs o (1 (e) ) 0,;; cs b (1 (e) )

3(1(e»

and 1 ,;; c ,;; b (1 (e) ) and

(1 (e) )

0,;; c ,;; b (1 (e) )

< b ,;; 0 (1 (e )

l,;; B,;; o(l(e» e(B,b,c)

t-

°

for

l,;; B,;; o(l(e»

and

for

l,;; B,;; o(l(e», by e(B) we are denoting the Q-net

whose index is

(7 )

e (B,b,c)

for

l(e) l,;; b ,;;

and whose (b,c) 0

(1 (e )

and

th

component is

l,;; c ,;; b (1 (e ) .

vie repeat the two "firstly" equations of §34 by saying that

(8)

if {

(B,B,B,C,b,c) E supt (1 (e ) 6

then e*(B,B,B,C,b,c) =

°

=

is such that

e*((B,B,B,C,b,c»

b < B

177

and we repeat the "decreasing induction" equation of §34 by saying that

i f (B,B,B,C,b,c) E supt

B s: B s: Band

(9)

then

6

(9, (e )

is such that

B s: b < B

e*(B,B,B,C,b,c) l; e* (B, 13, B, C, b' , c' ) e (b+l, b' , c' ) b+ Is: b 's: 0 (9, (e) ) Os:c "s b ' (9, (e )

and we repeat the "finally" equation of §34 by saying that

if (B,B,B,C,b,c) E supt then

(10)

is such that B s: b

6(9,(e»

e*((B,B,B,C,b,c» l; bs b "s o (9, (e) ) Os c l s b ' (9, (e )

e* (B, B, B, C, b' , c' ) e (b, b' , c' ) .

By (8) and (10) we see that

(11)

supt((e*»

c

supt(e*).

By the first and the second and the fifth and the sixth of the "begin with" equations of §34 we see that

A

A

if (B,B,B,C,b,c) E supt A

(12.1)

C = 0

and

B s: band

either B s: B < B then

is such that

6(9,(e»

or

e*(B,13,B,C,b,c)

B s: B

O.

A

B s: 0 (9, (e) )-2

178

In view of

(9) and

(12.1), by decreasing induction on

b

we see

that

i f (B,B,B,C, b, c) E sup t ; (9- (e) ) is such that A

C

(12.2)

= 0

and

A

B ,;; B < B

either

and

or

B ,;; B

(12.1) and

C

=

0

and either

(11) and

6(9-(e))

B,;; B < B

then e*(B,13,B,C,b,c)

In view of

,;;

0(9- (e))-2

(12.2) we see that

if (B,13,B,C,b,c) E supt (12.3)

B

= 0

e*(B,B,B,C,b,c)

then

By (8),

13

B ,;; b
-

u Y(B,B) ((R,E=))Q

and

u

Info[[R,Y(B,B),E=u]]*: Y(B,B) ((R,E2))

to be the R-epimorphisms induced by

->-

u

Y(B,B) ((R,E=))

ISO[[R,Y(B,B),E=U]]Q

and

we observe that

ker (Info [[R, Y(B,B),E = u]] Q)

ker (Info [ [R, Y(B,B) ,E = u]]

5)

u

Y (B,B) «(R,E»)Q

and ker (Info [[R, Y(B,B),E = u]] *)

ker(Info[ [R,Y(B,B),E =u]])

u

Y(B,B) (R,E»)

Given any ring-homomorphisms

g:R

->-

R'

u

Info[g,Y(B,B),E =u]Q: Y(B,B) (R,E2)Q

we define

->-

R' [Y]Q

and u

Info [g, Y(B,B) ,E = u]: Y(B,B) (R,E2)

->-

•

R' [Y]

211

to be the g-homomorphisms induced by Info[g,Y,E(B,o(£(E»

,o(£(E»,O) =u]Q

and we observe that their

images are

u

Y(B,B) (g (R) ,E=) Q

and

u

Y(B,B) (g (R) ,E=)

respectively and we define

Info[g,Y(B,B),E=u];j:

-+

and Info [g, Y(B,B),E = u ] *: Y(B,B)

u

-+

u

Y(B,B) (g (R) ,E=)

to be the g-epimorphisms induced by Info[g,Y,E(B,o(£(E»

,o(£(E»

,0) =u]Q

and we also define

Info[[g,Y(B,B),E=U]]Q: Y(B,B)

u

-+

R' [Y]Q

and Info[ [g,Y(B,B),E =u]]: Y(B,B)

U

-+

R' [Y]

to be the g-homomorphisms induced by Info[g,Y,E«B,o(£(E»,o(£(E»,O»

=u]Q

and we observe that their

images are U

Y(B,B) ( (g (R) ,E=) ) Q respectively and we define

and

u

Y(B,B) «g(F..) ,E=»

212

Info[[g,Y(B,B),E

u]]Q:

-+

Info[[g,Y(B,B),E = u]]*:

-+

and ,E=))

to be the g-epimorphisms induced by Info[g,Y,E((B,o(£(E)) ,0(£(E)) ,0)) = u]Q

and we note that we have

the following four commutative diagrams whereby the first is Info[[R,Y(B,B),E

u

u]]

Y(B,B) ((R,E?-))Q

Ul

*

Q

u) )Q

..('.';

tr

< ..-..

>< ..-..

'./y)

-

'Ii'

to to

-

'-'"

to

'" v/; "-

to

-

'-'"

0

t>1

t>1

Iv

0*

0*

Info [ [g (R) , Y(B,B),E = u]] Q ,E?-))Q

>

u) )Q

whereas the second is obtained from the first by everywhere replacing [[

]] and ((

)) by

and

) respectively while

the third is obtained from the first by everywhere deleting

Q

and finally the fourth is obtained from the second by everywhere deleting

Q.

Given any R-net

y

with

£(y)

£(E)

we define

213

u

sub[R,Y(B' ,B) =y,EPu]: Y(B' ,B) (R,EP)

R

-+

and u

sub[[R,Y(B',B) =y,EPuJJ: Y(B',B) «R,EP»

to be the R-homomorphisms induced by

isO(R,y(B',B),EPu)

R

-+

sub [R, Y = y]

and we put

sub [R, Y = y] (Y(B' ,B)

and

iso ( (R,y(B' ,B) ,EPu»

sub[R,Y=y] (Y(B',B)«R,EP»)

and we define

sUb[R,Y(B',B) =y,EPu]*:

-+

iso(R,y(B',B),EPu)

and sub[[R,Y(B',B) =y,EPu]]*:

to be the R-epimorphisms induced by

-+

sub [R, Y = y]

and we observe

that isO(R,Y(B',B) ,EPu) (1)

iso(R,y(B,B),EPu)

and iso «R,y(B' ,B) ,EPu»

iso «R,y(B,B) ,EPu»

iso«R,y(B',B),EPu»

.

214

§42. Let

E

Initial forms for scales and regular nets be a scale.

Let

R

be a ring and let

indeterminate net over

R

B E [l,o(£(E))-l]

B' E [l,B].

an R-net with

and

such that

£(y) = £(E)

such that

£(Y) = £(E). Let

u E Q.

y(B)

Y

be an

Let Let

y

be

is R-regular.

We define

info[R,y(B' ,B)=Y,E=u]: to be the

+

res(R,y(B») [Y]

res[R,y(B)]-homomorphism induced by

info[R,y(B) =Y,E(B,o(£(E)),o(£(E)),O) =u] and we observe that

im(info[R,y(B' ,B) =Y,E =u]) = Iso (res (R,y(B») , Y(B,B) ,E = u) and we define

info[R,y(B',B) =Y,E=u]*:

1

Iso (res (R,y(B»), Y(B,B),E = u) to be the

res[R,y(B)]-epimorphism induced by

info [R,y(B' ,B) = Y,E = u]

and we note that the following triangle

215

iso (R,y(B' ,B) ,E;;,u)

info [R,y(B' ,B)=Y ,E=U])

sub[R,Y(B,B)=y,E;;,U]*

res (R,y(B) ) [Y]

Info[res[R,y(B)] ,Y(B,B) ,E=u]

Iso(R,Y(B,B) ,E;;,u)

is corrunutative. We define

info[[R,y(B',B)=Y,E=u]]: iso((R,y(B',B),E;;'U))

to be the

+

res(R,y(B») [Y]

res[R,y(B)]-homomorphism induced by

info[R,y(B)=Y,E( (B,o(9, (E)) ,0(9, (E)) ,a)

)=u]

and we observe that im(info[[R,y(B' ,B)=Y,E=U]]) = Iso((res(R,y(B»),Y(B,B),E=U)) and we define

info [ [R,y(B I ,B)=Y ,E=u]] *: iso ( (R,y(B I ,B) ,E;;,u))

1

Iso((res(R,y(B»),Y(B,B),E=U)) to be the

res[R,y(B)]-epimorphism induced by

info[[R,y(B' ,B)=Y,E=u]]

and we note that the following triangle

216

iso{ (R,y(B' ,B),E;;,U))

info[ [R,y(B' ,B)=Y,E=u] ])res{R,y(B») [Y]

sub [[R, Y( B, B) =y I E;;,u] ] *

Info [ [res [R, y( B) ] , Y( B, B) , E=u] ]

Iso{ (R,Y(B,B) ,E;;,u))

is commutative. Given any

I

where

either I is an ideal in

R

with

1

y(B)R

or

I

x where x is an R-string with

or

I

x(t)

where

x

C

I C

t

is an R-string and

is a string-restriction 1

- - 1

with Y(B)R c x(t)R or

I

y where

or

I

y(t)

y

1

is an R-net with y(B)R

C

-1

YR

where y is an R-net and t is a net-restriction 1

with Y(B)R

-

-

1

y(t)R

C

we define

info[(R,I),y(B',B)=Y,E=U]: iso(R,y(B',B),E;;,u)

-+

res(R,I)[Y]

and info[[(R,I),y(B',B)=Y,E=U]]: iso((R,y(B',B),E;;,U)) to be the

res[R,I]-homomorphisms induced by

-+

res(R,I)[Y]

217

info [ (R, I) , Y( B) , E (B, 0 (9, (E) )

,0 (9,

(E) ) ,0) =u]

and info [(R, I), Y(B),E ((B,o (9, (e»,o (9, (E», 0) )=u] respectively and we observe that

im(info[(R,I) ,Y(B' ,B)=Y,E=u]) = Iso(res(R,I),Y(B,B),E=u)

and im(info[[(R,I),y(B',B)=Y,E=U]]) = Iso((res(R,I),Y(B,B),E=U» and we define

info[(R,I) ,y(B',B)=Y,E=u]*:

1

Iso(res(R,I) ,Y(B,B),E=u) and info[[(R,I),y(B',B)=Y,E=u]]*:

1

Iso((res(R,I),Y(B,B),E=u» to be the

res[R,I]-epimorphisms induced by

infa[ (R,I) ,Y(B' ,B)=Y,E=u]

218

and info[ [(R,I) ,y(B' ,B)=Y,E=u]] respectively and we note that the following two triangles

info[ (R,I) ,y(B' ,B)=y,E=U])

iso(R,y(B',B)

res(R,I) [Y]

info[R,y(B',B)=Y,E=U]

res (R,y(B»

,Y]

[Y]

iso«R,y(B',B)

info[[(R,I) ,y(B',B)=y,E=U]]>

info [[R,y{B' ,B) = y,E=uJJ

res(R,y(B»

res(R,I) [Y]

Sub [res [ [R,I) ,y{ B») ,y)

[Y]

are commutative. If

c M(R)

then we put

info[ (R) ,y(B' ,B)=Y,E=u]

and

info[(R,M(R)

,y(B',B)=Y,E=u]

219

info[(R),y(B',B)=Y,E=u]*

info[(R,M(R»

,y(B',B)=Y,E=u]*

and we put

info [[ (R) ,y(B' ,B)=Y ,E=u]]

info[ [(R,M(R»

,y(B' ,B)=Y,E=u]]

and

info[[(R),y(B',B)=Y,E=U]]*

info[ [(R,M(R»

,y(B' ,B)=Y,E=u]] *.

220

§43. Let

e

Isobars for protochips

be a protochip.

an indeterminate net over BE Let

[l,o(9-(E»-l] y

and

be an R-net with

R

Let with

B' E [l,B].

be a ring and let

R

9- (Y)

9- (e).

Let

u E Q

Let and

9- (y) = 9- (e).

We define

iso(R,y(B',B),ePu)

sub [R, Y=y] (Y(B' ,B)

iso((R,y(B',B),ePu»

= sub[R,Y=y]

ep»

and

and we note that by §40 we have

isO(R,y(B',B),ePu) (1)

iSO(R,y(B' ,B),e**Pu)

and iso( (R,y(B' ,B) ,ePu»

iso((R,y(B',B),e**Pu»

.

P E

Y

be

221

§44.

Initial forms for protochips and monic polynomials

Let

e

be a protochip.

an indeterminate net over B E

and

R-net with

Q,(y) =

Q,(e)

R

Let

R

with

B' E [1,B]. such that

be a ring and let Q,(Y)

Let y(B)

Q,(e). u E Q.

nonmon(R,y(B' ,B),e =u) {f E iso(R,y(B',B),e",u): info[R,y(B' ,B) =Y,e** =u] (f) E Nonmon (res (R,y(B»), Y(B,B),e = U)}

and

nonmon( (R,y(B' ,B),e =u)) {f E iso((R,y(B',B),e",u)): info[[R,y(B',B),e** =u]] (f) E Nonmon((res (R, y( B) ) , Y( B, B) , e =u))}

and we define

mon(R,y(B' ,B),e =u) {f E iso(R,y(B',B),e",u): info[R,y(B' ,B) =Y,e** =u] (f) E MOn(reS(R,y(B»),Y(B,B),e=U)} and

be

Let Let

y

is R-regular.

We define

Y

be an

222

mon ( (R, Y(B' ,B) ,e = u) ) {f E iso«R,y(B',B),e:2:u»: info[ [R,y(B' ,B) =Y,e** =u]] (f) E MOn«reS(R,y(B»,Y(B,B),e=U»}

Given any

I

where

c

either I is an ideal in R with -

or I =x where

x is

I 1

an R-string with y(B)R

or I =x(t) where x is an R-string and t

-1

x

C

is a string-restriction

with

-

or I

y where y is an R-net with

or I

y(t)

where

y

R

C

1

-

-

1

x(t)R

-1

Y(B)R C YR

is an R-net and t

is a net-restriction 1

with y(B)R

we define

nonmon( (R,I) ,y(B' ,E) ,e =u) {f E isO(R,y(B',B),e:2:u): info[ (R,I) ,y(B' ,B) =Y,e** =u] (f) E Nonmon (res (R, I) ,Y(B,B) ,e = u)} and nonmon( «R,I) ,y(B' ,B) ,e =u» {f E iso«R,y(B',B),e:2:u»: info[ [(R,I) ,y(B' ,B) =Y,e** =u]] (f) E Nonmon «res (R, I) ,Y( B,B) ,e = U»}

C

-

-

1

y(t)R

223

and we define

mon( (R,I) ,y(B' ,B),e =u)

{f

E

info[ (R,I) ,y(B' ,B) =Y,e** =u] (f) E Mon(res(R,I) ,Y(B,B),e =u)}

and mon( ((R,I) ,y(B' ,B),c =u))

{f

E

info[ [(R,I) ,y(B' ,B) =Y,e** =u]] (f) E MOn((reS(R,I),Y(B,B),e=U))}

c M(R)

If

then we define

nonmon( (R ) ,y(B' ,B),e =u) = nonmon( (R,M(R)) ,y(B' ,B),e =u) and nonmon (( (R) ,y(B' ,B),e = u)) = nonmon( ((R,M(R)) ,y(B' ,B),e =u)) and we define

mon( (R) ,y(B' ,B),e =u) = mont (R,M(R)) ,y(B' ,B),e =u)

224

and

mont «R) ,y(B' ,B),e =u)) = mont «R,M(R)) ,y(B' ,B),e =u)).

225

Index of definitions

archimedian net 113, 114 - string 108, 109 coordinate net

85, 86

indeterminate net 42 - string 17 indexing string 32 net 34 net-restriction 36 net-subrestriction 36 parachip 164 prechip 60 prescale 166 protochip 164 protoscale 171 pseudomorphism 71 quasi regular regular -- with regular -- with

99

net 99 restriction string 97 restriction

99 98

{R,D)-preseparated string 109 {R,D)-regular string 110 {R,D)-separated string 110 {R,E)-preseparated net 114 {R,E)-regular net 115 {R,E)-separated net 114 scale 174 separated net 99 -- with restriction 99 separated string 97 -- with restriction 98 string 8 string-restriction 10 string-subrestriction 10 superregular net 99 -- with restriction 99 superregular string 98 -- with restriction 98 ultraquasiregular 99 ultraregular net 99 -- with restriction 99 ultraregular string 98 -- with restriction 98 ultraseparated net 99 -- with-restriction 99 ultraseparated string 98 -- with restriction 98

226

Index of notations

abs(i), i abs(j), j [a,b] 6 b(£),

string net 35

denom(i),

i

denorn(i '), denorn(j),

32

£ indexing string

7

string

9

i' j

denom(e),

set of strings

j'

set of nets

e prechip

denom(e '),

e'

18

Deg[R,X] (F '), F' Deg [R, X] ( (F I

)

),

set of prechips

Deg[R,X] (F(t»), Deg[R,X] (F'(t»),

19

F E R[X]Q

29

F'

R[X]Q

C

Deg[R,X]«F'(t»)), F ' Deg [R, X] (F ( c»),

Deg[R,X] (F'(c»), F'

30

Deg[R,Y] (F '), F '

31

Deg[R,Y] «F')), F'

R[Y]

45

56

Deg[R,Y] (F'(t»),

F' C Deg[R,Y]«F'(t»)), F' Deg[R,Y] (F(b»), Deg[R,Y] (F'(b»),

R[Y]Q

F E R[Y]Q F'

Deg[R,Y] «F ' (b»)),

F'

59

Deg[R,Y] (F(b,c»), F E R[Y]Q Deg[R,Y] (F'(b,c»), F'

R[Y]Q

C

Deg[R,Y] «F'(b,c»)), F'

C

Deg[R,Y,E] (F), F E R[Y]Q Deg[R,Y,E] (F'),

F'

C

Deg[R,Y,E]«F')), F ' denorn(u), u E Q,

7

59

R[Y]

116

R[Y]Q C

R[Y]

E prescale

r lj l ,

58

R[Y]

C

E*,

i

string t

117 118

j

E Z

t

F(b) [j],

b E Z

58

F(b,c) [j]

8

G(o"it) G(o,t,k)

30

net-subrestriction

59 G(o,t)

18

43

net

F(t) [ j ] ,

G(o)

171

string-subrestriction

F(c) [u], c

58

R[Y]Q

C

e prechip

F(t) I i ] ,

57

57

168

e*,

F[i],

57

R[Y]

C

61

e(B,b)

Deg[R,Y] (F(t»), F E R[Y]Q

C

60 60

e(B)

44

R[Y]Q

C

e [[B]]

44

F E R[Y]Q

60

e[B]

31

Deg [R, X] ( (F I (c) ) ), F ' e R [X] Deg[R,Y] (F),

167

denom«E,B))

30

R[X]Q

C

166

167

denom(E,B) 30

R[X]

C

FER [X] Q

61

166

166 denom(E '), E ' set of prescales 167 denom«E')), E' set of prescales

19

R[X]Q

C

denom«E)), E prescale

F ' e R [X]

35

61

denorn(E), E pre scale Deg[R,X] (F), F E R[X]Q

9

35

net

denorn(j'),

85

Coord(R,Y(B'),e) Coord(R,Y,e) 87

Q,

denom(u'), u' c

9

11 11 12

57

56

29

227

G(OPU)

grn(R,y(t»

13

G(Opu,t)

13

G (OPU, t, k) G (9-)

gr[R,y(t)]«f '» gr[R,y(t)=y] 97

13

gr[R,y(t)=y]*

34

38

G(9-,t)

gr[R,y(t)=y]

38

G(9-,;it)

39 39

G(9-Pu,t,k) G(9-Pu,t,b,r) G(DPu)

108

G(EPu)

113

39

G(E«B»Pu), P E {=,>}

88

grn(R,l) gr[R,l]

101

gr [ (R) ,y( t ) =Y]

173

102

173

102

gr[(R),y(t)=y]*

gr[(R),y(t)=Y]*«f ' » gr[(R) ,y(t)=y] «fl»

103

88

grn[(R) ,y(t)=Y]*

)

89

)

91

103

91 91

gr(R,y) grn[R,y]

21

lnfo[R,X,=U]Q lnfo[R,X,=u]

91

22

lnfo[R,X,=U]Q

22

lnfo[R,X,=u]*

22

lnfo[R,X(t) ,=u]Q

93

»

94

gr[R,y]«f ' gr[R,y=y] 95

lnfo[R,X(t) ,=u]Q

26

lnfo[R,X(t),=u]*

26

lnfo[R,y,=U]Q

gr(R,y(t»

95

gr[R, y(t)] n [R, y( t)]

95 95

26 26

lnfo[R,X(t) ,=u]

93

102 102

102

grn[(R) ,y(t)=y]

gr[R] « f l »

gr

101

gr [(R, I) ,y(t)=y]

89

grn[R] gr[R]

gr[(R,l),y(t)=y]

88

gr [R, I] ( (f I gr (R)

100

grn[(R,l),y(t)=Y]*

173

G(E(B)Pu), P E {=,>}

grn[R,l]

100

gr[(R,l),y(t)=Y]*«f'»

173

G(E(B);"u)

100 100

gr [(R,l) ,y(t)=Y] «f'» n[ gr (R,l) ,y(t)=Y] 102

G(E«B»;"u)

gr(R,l)

100

grn[R,y(t)=y]*

39

G(9-Pu,t)

99

grn[R,y(t)=y]

38

G(9-, t,b,r)

96

gr[R,y(t)=y]*«f ' » gr[R,t(t)=y] « f l »

38

G(9-,t,k) G (nu)

95

lnfo[R,Y,=U]

47 47

lnfo[R,Y,=U]Q

47

lnfo[R,y,=u]*

47

103 103

228

lnfo[R,Y(t) ,=u]Q lnfo[R,Y(t),=u]

lnfo[R,Y,E=U]Q

info[(R) ,y(t)=Y,E=u]

52 52

lnfo[R,Y(B,B),E=U]Q

123

lnfo[R,Y(B,B),E=U]

123

lnfo[R,Y,E=u]*

123

162

162

info[(R),y(t)=Y,E=u]*

123

lnfO[R,Y,E=U]Q

162

info[(R),y(t)=Y,E]((f'))

52

lnfo[R,Y(t),=U]Q lnfo[R,Y(t), =u]* lnfo[R,Y,E=u]

info[(R),y(t)=Y,E]*((f'))

51

162 211

211

lnfo[R,Y(B,B),E=U]Q

212

lnfo[R,Y(B,B),E=U]* 212 lnfo[[R, Y(B,B),E=U]]Q

lnfo[R,Y(t),E=U]Q 124 lnfo[R,Y(t),E=u] 124

lnfo[[R,Y(B,B),E=u]]

212 212

lnfo[R,Y(t),E=u]Q

124

lnfo[[R,Y(B,B),E=u]]Q

213

lnfo[R,Y(t),E=u]*

124

lnfo[[R,Y(B,B),E=u]]*

213

lnfo[g,Y,E=u]Q lnfo[g,Y,E=u]

lnfo[g,Y(B,B),E=U]Q

125

lnfo[g,Y(B,B),E=u]

126

lnfo[g,Y,E=U]Q

124

lnfo[g,Y,E=u]*

125

213 213

lnfo[g,Y(B,B),E=U]Q

214

lnfo[g,Y(B,B),E=U]*

214

lnfo[[g,Y(B,B),E=u)]Q

lnfo[g,Y(t),E=U]Q 126 lnfo[g,Y(t),E=u] 126

214

lnfo[[g,Y(B,B),E=u]]

214

lnfO[g,Y(t),E=u]Q

126

lnfo[[g,Y(B,B),E=U]]Q

215

lnfo[g,Y(t),E=u]*

127

lnfo[[g,Y(B,B),E=u]]*

215

info[R,y(B' ,B)=Y,E=u]

217

info[R,y(t)=Y,E=u]*

155

info[R,y(t)=Y,E=u] info[R,y(t)=Y,E]

info[R,y(B' ,B)=Y,E=u]*

156 158

info[R,y(t)=Y,E]*

159

info[R,y(t)=Y,E] ((f'))

159 159

info[ (R,l) ,y(t)=Y,E]*

160

info[(R,l) ,y(t)=Y,E]*((f ')) 160 info [(R,l) ,y(t)=Y ,E] ((f')) 160 info[(R,l) ,y(t)=Y,E=u] 160 info[(R,l),y(t)=Y,E=u]* info[(R),y(t)=Y,E] info[(R),y(t)=Y,E]*

218

info[[R,y(B' ,B)=Y,E=u]]*

158

info[R,y(t)=Y,E]*((f')) info[(R,l),y(t)=Y,E]

217

info[[R,y(B',B)=Y,E=u]]

161 161

161

218

info[(R,l) ,y(B',B)=Y,E=u]

219

info[[(R,I),y(B',B)=Y,E=U]]

219

info[(R,l),y(B',B)=Y,E=u]* 220 info[[(R,l),y(B',B)=Y,E=u]]* 220 info[(R),y(B',B)=Y,E=u]

221

info[(R),y(B',B)=Y,E=u]* info[[(R) ,y(B',B)=Y,E=U]] info[[(R) ,y(B',B)=Y,E=u]]* inpo(i,D), D string

107

inpo(G,D), D string

107

222 222 222

229

inpo(G,D(t», D string inpo(j,E), E net

112

inpo(G,E), E net

112

inpo(G,E(t», E net ISO(Ro,X,PU)Q

113

20

ISO(Ro'X,PU)

20

ISO[R,X,PU]Q Iso[R,X,Pu]

108

21 21

ISO[R,X,PU]Q

21

Iso[R,X,Pu]*

21

ISO(Ro,X(t),pu)Q

25

ISO(Ro,X(t),Pu)

25

Iso[R,X(t),pu]Q

26

Iso[R,X(t),Pu]

26

ISO[R,X(t),pu]Q

26

Iso[R,X(t),Pu]*

26

ISO(R,X(t,k),pu)Q

28

Iso(R,X(t,k),Pu) ISO(Ro,Y,PU)Q ISO(Ro'Y'PU)

28

45 45

ISO[R,Y,PU]Q 47 Iso[R,Y,Pu] 47 ISO[R,Y,PU]Q Iso[R,Y,Pu]*

47 47

ISO(Ro,Y(t),pu)Q

50

ISO(Ro,Y(t),pu)

50

ISO[R,Y(t),pu]Q Iso[R,Y(t),Pu]

51 51

ISO[R,Y(t),pu]Q Iso[R,Y(t),Pu]* ISo(R,Y(t,k),pu)Q Iso(R,Y(t,k),Pu)

51 51 53 53

Iso(R,Y(t,b,r),Pu)Q' r c Z 54 Iso(R,Y(t,b,r),Pu), r c Z 54 Iso(R,Y(t,b,s),pu)Q' s E

55

230

Iso(R,Y(t,b,s),Pu), s E

55

Z*(o(Q,(Y)))

ISO(RO,Y(B',B),epU)Q' e prechip

63

ISO(Ro,Y(B',B),ePu), e prechip

63

e prechip

64

ISO((Ro,Y(BI ,B),ePu)), e prechip

64

ISO(Ro,Y(B',B,s),epu)Q' s E ISO(Ro,Y(B',B,s),epu), S E

Z*(o(Q,(e)))

Z*(o(Q,(e)))

66 66

ISO((Ro,Y(B',B,s),epu))Q S E

Z*(o(Q,(e)))

67

ISO((Ro,Y(B',B,s),epu)), s E

Z*(o(Q,(e)))

67

IsO(Ro,Y(B',B,s'),epu)Q' s' ISO(Ro,Y(B',B,s'),ePu), s'

Z*(o(Q,(e)))

C

Z*(o(Q,(e)))

C

ISO((Ro,Y(B' ,B,s'),epu))Q'

s'

ISO((Ro,Y(B',B,s'),epu)), s'

C

IsO(R,Y,EPU)Q' E string or net Iso(R,Y,EPu), E string or net

68 68

Z*(o(Q,(e)))

C

Z*(o(Q,(e)))

68 68

118 118

ISO(Ro,Y,EPU)Q' E string or net 119 ISO(Ro,Y,EPU), E string or net 119 ISO(Ro,Y(t),EPU)Q' E string or net IsO(Ro,Y(t),EPU), E string or net

119 119

IsolR,Y,EPU]Q' E string or net 121 Iso[R,Y,EPu], E string or net 122 IsO[R,Y,EPu]Q' E string or net Iso[R,Y,EPu]*, E string or net

122 122

IsO[R,Y(t),EPu]Q' E string or net 122 Iso[R,Y(t),EPu], E string or net 122 ISO(R,Y(t),EPU]Q' E string or net Iso[R,Y(t),EPu]*, E string or net iso(R,y,EPu)

127

iso(R,y(t),EPu) IsO(Ro,Y(B' ISO((Ro,Y(B'

122 122

128 E scale 175 E scale 175 E scale 175 E scale

175

IsO(Ro,Y(B',B),EPU)Q' E scale, P E {=,>}

177

231

ISO«RO,Y(B',B),EPU}}Q' E scale, P E {=,>}

177

Iso(R ,Y(B',B),EPu}, E scale, P E {=,>} 177 o ISO«Ro,Y(B',B),EPu», E scale, P E {=,>} 178 Iso[R,Y(B' ,B),EPU}Q' E scale Iso[R,Y(B' ,B) ,EPul, E scale

' 0 Iso[R,Y(B' ,B),EPul*, ISO[R,Y(B',B),EPU 1

209 209

E scale

210

E scale

210

Iso [ [R, Y( B' ,B) ,EPull Q' E scale Iso[[R,Y(B' ,B),EPull, E scale

210 210

ISO[[R,Y(B',B),EPullO' E scale

211

Iso[[R,Y(B' ,B) ,EPull*, E scale

211

iso(R,y(B' ,B),EPu}, E scale

216

iso«R,y(B' ,B),EPu}}, E scale

216

iso(R,y(B' ,B),ePu}, e protochip iso«R,y(B' ,B),ePu}}, e protochip £(y)

34

£(e)

60

£(E)

166

Mon(R,Y(B' ,B) ,e=u}

86

Mon ( (R, Y(B' ,B) , e=u}} mon(R,y(B' ,B),e=u}

86

224

mon«R,y(B',B),e=u}}

225

mont (R,I) ,y(B' ,B)

226

mon«R,I} ,y(B',B),e=u}} mon H k) ,y(B' ,B) ,e=u}

226

226

mon(KR} ,y(B',B) ,e=u}}

227

Nonpremon(RO,Y(B' ,B) ,e=u}

68

Nonpremon«Ro,Y(B' ,B),e=u}} Nonmon(R,Y(B' ,B),e=u} Nonmon«R,Y(B',B),e=u}} nonmon(R,y(B',B),e=u}

86 86

224

69

223 223

232

nonmon«R,y(B',B),e=u))

ord[R,y] « f ' » , fl c:. R

224

nonmon «R, l) ,y(B I ,B) ,e=u) nonmon«R) ,y(B' ,B) ,e=u)

ord[R,y(t>] (f), fER

225

nonmon( «R,l) ,y(B ' ,B) ,e=u»

ord[R,y(t)] «f'»), f'

226

ord[R,y(t) ,E] (i)

19

R[X]

19

ord[R,y(t) ,B) (i ')

Ord[R,X] (F(t»), F E R[X]Q

29

o(x) o (t)

Ord[R,X]«F ')), F'

c;

29

Ord[R,X] (F'(t»), F ' c; R[X]Q Ord[R,X] «F'(t»), F' c; R[X] Ord[R,X] (F(c»), F E R[X]Q

c;

31

Ord[R,Y] «F'(b»), F'

c;

R [X)

56

R

o

ord[R,y], fER

93

ord[R,y] (£'),

c;

£1

R[X(t>]

24 24

R [XUt)]

58

24

R [X( t)]Q 24

58

o

59 59

R 24 o[X\7it)]Q Ro[X(t)] 24 24

89

42

Ro[Y]Q R [Y]

o

45 45

R[Y(t)]Q 91 94

49

R[Y\7it)]Q R[Y(t)]

R

24

R[XUt)]Q

R[Y]Q

89

91

ord [R] ( (f' ) ), f' c. R

19

o

91

ord[R] (f'), f'c:. R

19

R [X\7it)]

ord[R,l] « f ' » , f' c. R ord[R] (f), fER

17

R[X(t)]Q

58

R[Y]

89

c;

«Ro,Y(B',B),e=u»)

o

Ord[R,Y] «F'(b,c»), F' c:. R[Y] ord[R,ll(f'), f'

(Ro,Y(B',B),e=u)

56

Ord[R,Y] (F'(b, c»),F' c. R[Y]Q ord[R,l] (f), fER

32

Premon

R [X]Q

Ord[R,Y] (F(b,c», F E R[Y]Q

158

Premon

R[X]Q

F' c:. R[Y]Q

118

31

Ord[R,Yl(F'(t»), F ' c; R[Y] 57 Ord[R,Y] (F(b»), F E R[Y]Q 57 Ord[R,Y]

R[Y]

157

6

44

R[Y]Q

117

Q

8

44

Ord[R,Y] (F(t»), F E R[Y]Q Ord[R,Y] (F'(t», F'

Q

30

44

Ord[R,Y] (F '), F' c; R[Y]Q Ord[R,Y]«F')), F' c:. R[Y]

116

30

Ord[R,X] (F'(c»), F ' c; R[X]Q Ord[R,X]«F'(C»), F' c; R[X] Ord[R,Y] (F), F E R[Y]Q

R[Y] c;

96

R

c;

c;

Ord[R,Y,E]«F'», F'

18

Ord[R,X] (F'), F' c:. R[X]Q

96

Ord[R,Y,E] (F), F E R[Y]Q

226

Ord[R,Y,E] (F'), F' Ord[R,X] (F), F E R[X]Q

95

ord[R,y(t)](f'), £' c:. R

225

nonmon«(R) ,y(B',B) ,e=u»)

94

R[YUt>]

49 49 49

68

69

233

RO[Y(t)]Q

supt (j) , j net

49 49

R [Y( t) ]

supt(9"t)

35

36

o Ro[YUt)]Q 49 Ro [YUt)] 49 R [Y] 70

supt(9"t,b,r)

Ro[Y(t)] res(R,I)

76 88

supt (F) , F E R[Y]Q 43 supt (F( t) ) , F E R[Y]Q 56

res[R,I]

88

supt(F(b») , F E R[Y]Q 57 supt(F(b,c») , F E R[Y]Q 58 supt 165 N(9,) supt (E) , E pre scale 170

o

res[(R,I),J] res [R]

supt(9",t)

90

91

res [(R) ,J]

92

res[R,y]

93

supt ( (E) )

res (R,y)

93

supt(E,B)

res [(R) ,y]

res[(R,y(t»),y] res [(R,y) ,J]

95

res(R,y(t»)

95

res [(R) ,y( t)]

70 71

71

Sub[g,Y=Y']*

71

sub[g,Y=y'] 96 96 96

Sub[g'Y]Q Sub[g,Y]*

96

supt(i) , i string

73 73

Sub[R,Y=O]Q 73 Sub[R,Y=O] 73

8

10 11

Sub[R,Y=O]Q

74

Sub[R,Y=O]*

74

sub[R,Y=O]

11

supt(F) , F E R[X]Q

72

Sub[g'Y]Q 72 Sub[g,Y] 73

res [(R,y(t}) ,y(t)] res[(R,y(t»),J]

72

sub[g,Y=y']*

96

res [(R,y) ,y(t)]

supt(o ,t,z)

70

Sub [g, Y=Y']

res[(R,I),y(t)]

supt (0 ,;it)

170

sub[R,Y=y]

94

94

res[R,y(t)]

supt (0, t)

E prescale

Sub[R,Y=Y]*

94

37

170

,

Sub [R,Y=Y]

94

res[R,y),y]

37

supt«E,B))

94

res[(R,I),y]

37

supt(9"t,k)

Sub[g,Y=O]Q

18

supt(F(t»), F E R[X]Q

29

supt(F(c»), F E R[X]Q supt(9,) 33

30

74 74

Sub[g,Y=O]

74

sub[g,Y=O]

74 75

Sub[g,Y=O]Q

170

234

Sub[g,Y=O]*

75

sub[g,Y=O]*

75

Sub[R,Y(t)=y]

Sub[g,Y(t)=O]Q 82 SUb[g,Y(t)=O] 82 76

sUb[g,Y(t)=O]

SUb[R,Y(t)=y]*

76

SUb[R,y=y(t')]

77

Sub[R,Y(t)=y(t')] SUb[R,y=y(t')]*

77 77 77

77

SUb[g,Y(t)=Y']

78

Sub[g,Y=Y'(t')]

78

Sub[g,Y=O(t')]

SUb[g,Y=O(t')]O SUb[g,Y=O\t')]*

SUb[g,Y(t)=Y'(t')] SUb[g,Y=Y'(t')]*

78 79

SUb[g,Y(t)=Y'(t')]*

79

79 79

83 84 84 84 84 84

SUb[g,Y(t)=O(t')]O SUb[g,Y(t)=O(t')]*

84 84

Sub[g,Y,EPU]Q' E string or net 120 Sub[g,Y,EPu], E string or net 120 Sub[g,Y,EPU]O' E string or net Sub[g,Y,EPu]*, E string or net

SUb[g,Y(t)]Q 80 Sub[g,Y(t)] 80 SUb[g,Y(t)]O SUb[g,Y(t)]*

83

SUb[g,Y(t)=O(t')]

SUb[g,Y(t)=Y']*

sUb]g,Y(t)=y']*

sUb[g,Y(t)=O]*

SUb[g,Y(t)=O(t')]Q

78

sUb[g,Y(t)=y']

83 83

Sub[g,Y=O(t')]Q

SUb[R,Y(t)=Y(t')]X sub[R,Y(t)=y]

83

SUb[g,Y(t)=U]O SUb[g,Y(t)=O]*

120 120

SUb[g,Y(t) ,EPu]Q' E string or net 120 Sub[g,Y(t),EPu], E string or net 121

80 80

Sub[g,Y(t),EPU]O' E string or net Sub[g,Y(t) ,EPu]*, E string or net

Sub[R,Y(t)=O]Q 80 SUb[R,Y(t)=O] 81

sub[R,Y=y,EPu], E string or net

121 121 127

SUb[R,Y(t)=O]O 81 SUb[R,Y(t)=O]* 81 sUb[R,Y(t)=O] 81

sUb[R,Y(t)=y,EPu], E string or net

Sub[R,Y=O(t')]Q

Sub[g,Y(B' ,B) ,EPU]Q' E scale

Sub[R,Y=O(t')]

sub[R,Y=y,EPu]*, E string or net

sUb[R,Y(t)=y,EPu]*, E string or net

81 81

Sub[R,Y=O(t')]Q SUb[R,Y=O(t')]* Sub[R,Y(t)=O(t')]

SUb[[g,Y(B' ,B) ,EPU]]Q' E scale

82 82

SUb[R,Y(t)=O(t')]Q

127

208 208

Sub[g,Y(B' ,B),EPu], E scale 208 Sub[[g,Y(B' ,B) ,EPu]], E scale 208 82 82

SUb[R,Y(t)=O(t')]O 82 Sub[R,Y(t)=O(t')]* 82

Sub[g,Y(B',B) ,EPU]O' E scale

209

Sub[[g,Y(B' ,B) ,EPU]]O' E scale 209 Sub[g,Y(B' ,B) ,EPu]*, E scale 209 Sub[[g,Y(B' ,B) ,EPu]]*, E scale

209

128 128

235

sUb[R,Y(B',B)=y,EPuj, E scale

216

y(b)

sub[[R,Y(B' ,B)=y,EPuJJ,E scale

216

sUb[R,Y(B' ,B)=y,EPuJ*, E scale

216

sub[[R,Y(B' ,B)=y,EPuJj*, E scale T(d,b,,Q,)

32

T(-d,b,,Q,) T* [,Q,] T[,Q,j

x x

R u

R

yj

40

yj R

40

u YR

40

216

33 33

T[,Q"BJ

i

32

34

41 33

u Y(RP)Q u Y(RP)

14

45 45

u Y(t) (RP)Q

14

u Y( t) (RP)

50 50

15 u

y(t,k) (R=)Q

u

20

X(RP)Q u

u Y( t,k) (R=)

20

X(RP)

u

Y( t,k) (R;;