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English Pages 245 [244] Year 1982
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;;