TRANSACTIONS OF THE
AMERICAN MATHEMATICALSOCIETY
Volume 177, March 1973
LOGIC AND INVARIANTTHEORY. I: INVARIANTTHEORY
OF PROJECTIVE PROPERTIES
BY
WALTER WHITELEY(l)
ABSTRACT.
some
problems
of modern
variant
This
paper
and results
first-order
logic.
significance
for the general
linear
to define
transformations
syntactic
form:
category
cover
all changes
that
an equation
of an arbitrary
first-order
vector
spaces
formulas
in a projective
formula
of dimension
to formulas
in determinants
transformations
is of in-
in two directions.
for a cate-
n.
These
of a particular
or "brackets".
The
is also
introduced
in order
space.
Invariance
for this
to
is investigated.
are extended
and contravariant
Finally,
tended
Klein's
notion
has recently
to cover
invariant
Erlanger
of invariance
Program
is reexamined
Classical
enjoyed
as well
invariant
new vogue,
as some
theory,
with the extensive
rell [2],
of articles
In this
of the basic
Invariant
series
both model
results
theory
of points
"invariant"
Received
with both covariant
survey
in the light
geometric
a series
article
and proof theory,
polynomials
in a projective
space
of coordinates
December
of the excategories.
of ups and downs,
book of Hermann
by Dieudonné
the tools
Weyl
and Car-
of the first-order
to recast
and extend
pred-
some
theory.
considered
by the editors
after
we will employ
theory
of the invariant
under changes
possible
first with the well-known
[lO] and more recently
calculus,
formulas
vectors.
Introduction.
ordinates
the framework
the notion
by equivalence
of coordinates
The results
icate
will reexamine
within
is extended
are characterized
of all semilinear
which
theory,
group
between
homogeneous
fuller
category
invariance
of papers
invariant
paper
gory of linear
formulas
a series
In this
It is extended
invariant
initiates
of classical
and polynomial
and asked
which
for the space?
equations
such
in the co-
expressions
It was noticed
are
quite
13, 1971.
AMS(MOS)subject classifications (1970). Primary 02H15, 15A72, 18F99, 50A20,
50D99; Secondary 02H05, 13L05, 14A25, 20G15, 50A15, 50D30.
Key words and phrases.
Analytic
projective
geometry,
category
of models, invariance for a category,
invariant
first-order
formula, semilinear
transformation,
projective
properties,
vector space, change of coordinates,
general linear group, determinant,
bracket,
Klein's
Erlanger
Program.
(1) Portions
of this work are taken from the author's
Ph.D. thesis written at the
Massachusetts
Institute
of Technology
under the supervision
of Professor
G.-C. Rota.
Preparation
for publication
was supported
in part by NRC grant A8226.
Copyright © 1973, American MathematicalSociety
121
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122
WALTER WHITELEY
early
that a fundamental
determinants
tive space.
nomials
role was played
of subsets
of n points,
The First
that are relative
in the brackets
polynomial
invariants
namely
of Invariant
are expressible
Gram's
Theory
states
as homogeneous
theorem
in terms of conjunctions
of the projec-
polynomials
characterized
of homogeneous
that poly-
invariant
equations
in the
[8, p. 27l].
Our purpose
are invariant
in this paper
under changes
in logic and in geometry,
properties
and theorems
of logical
connectives
orem, stating
nants.
of coodinate
we believe
of classical
that invariant
system.
first-order
synthetic
and thereby
require
in two stages,
logical
theories
This notion of "invariance
nize some other natural
both
Many
by means
full first-order
extension
formulas
logic
of Gram's
the-
in the determi-
first for open or quantifier-
formulas.
has a natural
invariant
which
to ask.
are expressed
is a natural
are homogeneous
on all the classical
questions
geometry
formulas
and then for general
formulas
In view of developments
are natural
the characterization
The development
that a geometry
general
these
Our characterization
We present
free formulas
is to describe
and quantifiers,
for their expression.
logic.
"brackets",
n - 1 is the dimension
Theorem
[10, p. 45], [8, p. 312].
equations
brackets
by the so-called
where
Fundamental
[March
questions
setting
which provides
as well as some theorems
for a'category
of models"
of invariance,
and to extend
is given as the invariant
a unified
view
of mathematical
allows
us to recog-
Klein's
notion
theory of a group of transformations
[6,
p. 130].
In future
papers
tems for analytic
Fundamental
geometric
we will examine
geometry
Theorem
questions
homogeneous
in the invariants
of Invariant
Theory,
and representable
in the proof theory
The analysis
sponding
extends
geometries
projective
explains
for finding
The presentation
a given
and definitions.
for the analytic
fixed field
over an arbitrary
geometry.
K.
by earlier
if K = 0
models
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and
translations
or combinatorial
[4, Preface].
We must begin by defining
or all analytic
corre-
as identities
lattices
workers
theory of projective
form of a
the identities
to investigating
such as modular
By convention,
field
standard
in the ring of brackets.
of theorems
is the form best suited
to other settings
to the Second
to combinatorial
the observed
procedure
[ll].
sys-
with some techniques
a practical
It is also the form anticipated
suitable
results
identities
of identities
1. Languages
languages
domains
of these
geometry:
also yields
and generalizations
and their relationship
and applications
of projective
to the theorems
conjunctions
geometry.
of integral
of theorems
[13], axiom
matroids.
We have found that the combination
wide class
coordinates
the first-order
geometry,
where the field
we are referring
of Pappian
to
synthetic
123
LOGIC AND INVARIANT THEORY. I
1973]
Definition
1.1.
Ln(SAK),
dim n - 1 extending
a field
the simplified
language
K, is the modified
for analytic
two-sorted
geometries
language
of
of type
r = SV, 5; =; e\ e2, ... , e", +, . , -; 0, 1, (cf)feK\
and signature
a:
= is a relation
between
terms
e , • • • , e" ate operations
+ and • are operations
— is an operation
0, 1, and c,,
of sort
5,
V —►
5,
5x5
—»5,
5 —►5,
/ £ K, ate constants
and only variables
of sort
of sort
5,
V are allowed.
Definition 1.2. A standard model of Ln(SAK) is a model
U = \V, S;=;e\
where
5 is a field,
the field
5.
■■■, e", +,-,-;0,
V is a vector
The operations
space
of dim n over
and constants
e , • ■• , e" ate the projections
l,(M/))/eK¡
S and
= is equality
in
are as follows:
of a vector
onto its components
el(vv---,vn)=v.,
+ , •, and — are the domain
operations
on the field
5,
0, 1 are the zero and unit of the field K,
h is an isomorphism of K into the field 5, if K ^ 0.
The collection
spaces
of all standard
of dim n over fields
Remarks.
extending
Two comments
On'e concerns
multiplication
struct
languages
of the vector
*, as well
Ln(AK)
ever, for the standard
as the field
and models
models
VnK, vector
K.
are worth making
the omission
scalar
models of Ln{SAK) is denoted
about
space
this choice
operations
operation
of language.
+ and - and the
. We could
UnK containing
these
we find that for any formula
is,
F and
To eliminate
S(F)
define
+, - and
ate linear
in all standard
identities
hold in all
Only terms of sort
the same property
* for the vectors
models:
F in Ln(AK)
there
5 occur in the relation
~
eliminate
from all equations
substitute
"
an operation
0 for t~
The net result
The second
, and
is the formula
omission
in F, by taking
S{F) in Ln(SAK)
is the variables
models.
Since
these
in F creating
F*.
=, so this eliminates
say by defining
two cases:
l ¿ 0, where we multiply
UnK.
use the fact that the e'
substitutions
in all the fields,
models
of the standard
e'{a *v + v ) = a • e'{v) + e'(v').
UnK , we carry out these
After making
in each
we repeatedly
con-
How-
is a formula S(F) in Ln(SAK) such that F <-»5(F) in all standard
That
easily
operations.
through
all occurences.
0-1 =0,
we can
t = 0, where we
to clear
fractions.
which was promised.
for elements
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of the field
5.
It is a
124
[March
WALTER WHITELEY
well-observed
phenomenon
all reasonable
conditions
field
elements.
of analytic
There
are two ways
given a variable
over the field,
the vector
and replace
space
will be explained
properties
projective
can be expressed
to view this.
say
paper
studied
algebra
to variables
One is the simple
of k with
in this
and linear
resorting
k, we can replace
occurences
in the fourth
traditionally
geometry
without
series
reason
it by a variable
e (x).
that,
x over
The deeper
reason
when we recognize
are "combinatorial"
that
for the
that the
and we can eliminate
all
quantifiers.
In classical
geneous
invariant
coordinates
properties
or properties
sponded
to changes
lineations
which
The First
lineation
of coordinate
which
[T],
were principally
ticular
cases
a series
Definition
by matrices
of three
categories
1.3.
Sl{nK)
questions
1.
Gl{n),
characterizes
col-
or
T, given
by non-
an automorphism
Classical
over the reals
workers
and there-
over the reals.
is in terms
of par-
and their collineations.
to be considered
in this
in the classical
paper.
We
Particular
theory.
with objects
For a vector
corre-
transformations
of invariance
geometry
which
maps are the col-
by taking
[l, p. 44].
geometry
group,
is the category
[T] with determinant
Geometry
A, induced
of projective
two were considered
These
of homo-
on invariant
sense.
transformations
to all coordinates
for these
of models
of the first
of linear
linear
space
as semilinear
with a projective
setting
categories
will define
this
on the general
An appropriate
of Projective
and automorphisms
as classes
centered
for the space.
in the projective
of dim 3 or more,
concerned
fore concentrated
system
are composites
and applying
were viewed
and interest
by maps of the vector
Theorem
for spaces
matrices
of the field
spaces
space
onto lines
Fundamental
transformations
vector
not affected
map lines
maps,
singular
theory
of some projective
VnK and morphisms
v written
given
as an n x 1 matrix
[v], T(v) is [T] x [vl
Definition
given
1.4.
Gl(nK)
by nonsingular
Definition
matrices
1.5.
Pa(nK)
bitrary
compositions
(a{v.),
• • • , a(v )) where
particular
Notice
the models
vector
every
space
morphism
satisfies
is the category
VnK and morphism
Ta:
(i^,
• • • , v)
ar-
—»
of the field underlying
the
which is the domain of Ta.
are not necessarily
structural
isomorphisms
of
.
can now be made specific
significance
A formula
by rewording the classical
of an equation.
F is invariant
T in C with domain
FiTim^,
with objects
a is a field isomorphism
of invariance
of the invariant
1.6.
VnK and morphisms
[T].
that the morphisms
Definition
with objects
of the maps of Gl{nK) and maps
in the category.
The notion
notion
is the category
for a category
M, M satisfies
Fim^
• • • , Tim )).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
of models
C iff for
••• , m ) iff T{M)
1973]
LOGIC AND INVARIANT THEORY. I
Our task is to characterize
invariant
for Pa(nK).
of the categories
In the process
defined
Remarks.
formula
If the category
together
theory
theorem
[7].
a formula
gory is such
that
then persistent
variant
invariant
of models
morphism
syntactic
invariant
Theorem
theory,
form of invariant
vectors
are determinants
We now present
concern
language
invariant
theory.
Definition
is a modified
as
and
with do-
T(m )). If the cateor Pa{nK),
In general,
Gram's
theorem
we have a strong
For equations
a language
where
for such formulas.
of formulas,
F is in-
and the First
hunch
about
the
invariant
for Sl(n)
the basic
terms involving
or Gl(n)
■=■on scalars,
for proving
although
theorems
For simplicity,
because
we will omit the operators
which
these
operations
will be presented
our
+, -,
will appear
in Logic
and
in
and
III.
1.7.
The language
two-sorted
language
Ln{SBK),
the
simplified
of type
T=\V, S; = ;[],+,.,_;
= is a relation
between
[ ] is an operation
+ and
in logic
Gl{nK)
0, l,(cf)feK\
and signature
o:
as those
or "brackets".
as well as
the basic
specifically
to be formulas
is with the significance
* on vectors
such
a
[3, p. 75].
C iff for T a morphism
•••,
with
theory,
for C.
Theory,
formulas.
we are told they can be forced
a category
for C ate synonymous.
of Invariant
of a theory
coincides
formulas
T(M) \= F(T(ot,),
has an inverse,
for C and invariant
From classical
for such
for a category
every
is a universal
we could mimic the practice
for C iff F and ~] F are persistent
Fundamental
for each
for work on logic
for the category
to quantifier-free
•••,772 ) implies
each
invariant
of all models
When the theory
formulas
is persistent
main M, M f= F(m,,
properties
of models,
is the basis
invariance
the theory,
category
This
is the collection
embeddings,
within
For a general
formulas
only isomorphisms
for C.
for extension.
characterizes
equivalent,
consider:
C contains
If the category
of invariance
of logic
formulas
we will consider
is invariant
with all structural
the notion
as fully as possible,
above.
in the language
as invariant
syntactically,
125
0, 1, and
c
5 only,
S x S —*S,
S —>S,
f £ K, are constants
and only variables
Definition 1.8.
of sort
V" —*S,
• are operations
- is an operation
terms
of sort
of sort
5,
V occur.
A standard model of Ln{SBK) is a model
lV=jy,5;=;det,+,.,-;0,
l,(M/))/6/cl
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language
of brackets,
126
WALTER WHITELEY
where
V is a vector
relation
tion
[v
on S.
space
of dimension
The operations
V"—>S which
carries
• • • v ] with the vectors
The collection
is the opera-
of the matrix
of the matrix,
on S,
In this
of K into S.
of all standard
language
can be viewed
models
the atomic
as polynomials
description
Definition
formula
as columns
operations
= is the equality
det
1 are the zero and unit of S,
h is a field injection
For easier
S and
are as follows:
v ) into the determinant
written
+, •, and - are the usual
0 and
n over the field
and constants
(v .,•••,
[March
appearing
formulas
s and
t
in K.
definitions.
F in Ln(SBK)
in F is an equation
s = t where
[v^ ••• v ] with coefficients
the following
A formula
BnK.
are equations
in brackets
we give
1.9.
is denoted
is homogeneous
homogeneous
iff each
in occurences
atomic
of the brackets.
Definition 1.10. Det: VnK —»BnK is the function such that Det ((7) is the
model
with the same
underlying
vector
space,
field,
det(f, 1 ••• v n ) interpreted
as the determinant
r
function is a surjectiori
onto BnK.
and injection
of the matrix
of K, with
[e'(v.)];
in U.
This
Definition 1.11. Det: Ln{SBK) —*Ln(SAK) is the embedding of formulas
obtained
by extending
the basic
The connection
between
map:
these
[v , •-- v ] —»det [e'(v .)].
maps is illustrated
by the basic
fact that
DetiU) 1=F{vv • ■•, vm) iff U t= Det (EG;,, •••, vj).
2.
Invariant
framework
An equation
equations
open formulas.
of the classical
E in LniSAK)
G.,•••,
n new variables
The following
theorem
G
c,,
In
is invariant
in Ln(SBK)
— , c
such
result
is a translation
into our
of Gram.
for Gl(nK)
with the same
that
iff there are homogeneous
free variables
£ ♦-» Det (Vc,
plus perhaps
• • -Vc (G,& • • • &G )) in
1
n
1
m
all models in VnK.
The classical
techniques
case
considered
can be simplified
K = 0.
invariance
theory
for Sl(nK),
same
amount of information
However,
well
fields
K and to the
to open formulas
formulas
and
invariant
will not, in general,
formulas.
about the structure
of sufficient
formulas
quantified
used
for quantifier-free
It seems
for open
to other
0 but the
PainK).
the methods
our information
sense.
and an algorithm
and
K of characteristic
this theorem
we will characterize
categories.
here will increase
equally
we will extend
Gl{nK)
these
in a practical
to apply
In this section
In the next section
only fields
interest
provide
The methods
of open,
to extract
invariant
this
the
employed
formulas
information
and then move on to the more general
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for
case.
1973]
LOGIC AND INVARIANT THEORY. I
Theorem
there
2.1.
An open
is an open formula
formula
F in Ln(SAK)
G in Ln{SBK),
127
is invariant
for Sl(nK)
with the same free variables
iff
plus
c.,
• • • , c n , such that F <-»Det ((Vc,,in • • • , c )G) in the theory of VnK.
Proof.
Part I.
Assume
F is invariant
for Sl{nK).
We will construct
the re-
quired G.
Take any model
construct
where
V in VnK and any vectors
the matrix
C..
i]
In
c
[c, ••• c ] and the transformation
of c
ji
in the matrix
in V.
From these
T with matrix
In
is the cofactor
If det[c,
c.,•••,,
[C..]
z;
[c, ••• c ].
In
• • • c ] = 1 then det[T] = 1 and T is a morphism
in Sl{nK). With
A
this assumption
V^F(Vl,...
,vj
Then we have
el{T{v))
all considered
as equations
iff T(V)h PiTivJ, • - - , TivJ).
= 1 C. .e*(v) = 1 (cofactor
c.)e'(v)
z;
in the field
= det[c, •••c.
17
1
5 of the model
.vc. , •••c
z—I
V. Also,
z+1
n
],
T(V) has the
same field as V so
7tV)l=F(T(Wl), ••• , 7twm)) iff V h Flrd/j), ••• , TivJ).
Repeated
use of the identity
yr-rti/lt
Dropping
•••,
Vj
the assumption
now produces
iff
on [cj
a formula
G in Ln(SBK)
and
Det(V) [-Gd»lt V , »„, clf •••.c(|).
• • • c ], we find that,
DetWKcj.-cJ/l
for all
c , • ••, c ,
V G(t,1,-..,%,c1,...,cn).
Therefore,
Det(V)l=
Vcj.--
Vc„(l>, ...ej^l
V G(vv ■■■ , cj)
and we see that
V^F(vv
Part II.
Using
... , wJ-.Det(VCl
this
definition
••• Vcn([Cl •••c„]¿l
of G we must verify
V G)).
that
Det(Vc, • • • WcJLci •• - cj ¿ 1 V G))— F.
Assume
tuting
that
these
V t= Det(G
') and take the unit basis
2> , • • • , b
for V. Substi-
for the c .,
Dct(V)(=tè1...èn]^lVG(l;1,...,vai,...,^).
However, Det(VÏ N [b. ••• b ]= 1 and det[è.
i
Det(V)\=G(v.,
n
■•■ ,b )
and
i
••• b.
,zb.
z—lt+l
, • •• ¿> ] = e'(z)
V f=F(v,, • • • , v >.
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n
so
128
WALTER WHITELEY
We conclude
Part III.
Ln(SBK),
that
[c
Assuming
• •• c ] 4 1 V G(v,,
that
E«->Det(Vc
that
F is invariant
we will verify
[March
• • • , c ) is the required
••• Vc
formula.
G) for some open
for Sl(nK)
G in
by checking
this for
Det(Vc, 1 ••• Vc n G).
Take a model
S common
V and a morphism
to V and
detLTty,)
Substituting
T(V),
we have
T with domain
det[T]
V7. Working within
the field
= 1 and
••• T(yn)] = det([T]xLy1
••• yj) = det[y,
••• yj.
into G we find
VhD«(G(t»1....,t/ii|,c1,...,c.)
iff
T(V)\=Det(G(r(Wl), ••• , TivJ, T(c,), ... , TicJ)).
However,
T is surjective,
so, as
c,,
• - • , c^ cover
V, T{c{),
• • • , T(cn)
cover
T(V). Thus,
Det(V)hVc,
... Vc^GG;,, • • • , c,, • • • , cn
iff
DetiT(V))^ Vc, •■• VcnG(T(i;i), ••• , TG;J, c,, ••• , c¿.
Therefore,
F is invariant
Before
formulas
proceeding
theorem
D
to invariance
in the language
Lagrange's
here,
for Sl(nK^.
of integral
for roots
for GlinK).
domains
of polynomial
as it has been presented
we need a lemma
which
plays
equations.
elsewhere,
together
about
open
a role analogous
to
This lemma will just be stated
with some applications
[12,
3.1].
Lemma 2.2.
// F is an open formula
of the theory
of integral
domains
and
K is an infinite field, then K 1=F{a., • • ■, a , t) for infinitely many t iff
K N F (a.,
•••, a
) where
F
CÎ. p .t') - 0, by the formula
comes
from
Yl(p . = 0).
Here
F by replacing
each
equation,
Yl stands
for the conjunction
technique
for homogenizing
of the
formulas.
This
tions
lemma is an extension
by taking
ficient
conditions
Theorem
there
their
for invariance
2.3.
parts.
plus
We can now give necessary
equaand suf-
for Gl(nK).
An open formula
is a quantifier-free,
free variables
of the usual
homogeneous
homogeneous
n new variables
F in Ln(SAK)
formula
c.,••-,
is invariant
for Gl(nK)
G in Ln(SBK),,with
c , such that
iff
the same
F«-»Det(Vc1 • •• Vc G)
in the theory of VnK.
Proof.
Part
VnK and extend
open,
for any
I.
Assume
F is invariant
it to a model
v.,•••,
v
for Gl(nK).
inf (V) with an infinite
Take
field
in V
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
any model
inf (S).
Since
V in
F is
1973]
LOGIC AND INVARIANT THEORY. I.
Vr-rt»!,
Take
••',%)
the transformation
det[T] = sn.
iff inf(V)hFG;„...
in Gl(nK)
This holds
many
s.
for all
Applying
el\v.)
s • I , s 4. 0 in
inf(S)
means
F
We can now apply
to F
det [c,
that
iff
v ) where
makes
F
e'(v.)
for infinitely
F
homogeneous
comes
in
VFF'(v,,---,v).
the transformation
T which
comes
from the vectors
• • • c ] / 0, as in the proof of Theorem
that if V N F(v.,
Det(l/)h[c,
This
the
holds
an open formula
■•• , v )
• • • , c , such
Treating
which
inf (V) r2 F (»,,•••,
of sl in F.
is also
= s • e'(v).
this is a formula
Lemma 2.2, we get
the coefficients
and since
this we derive
iff T(inf(l/)) |=F(T(i/,),
• • • ,' T{v771)).
1 *
we find that
V\=Fiv,,
c,,
matrix
s 4 0 in inf (S) and e'{T{v))
of the field,
from separating
the
with
, vj.
Therefore,
inf(V) h F{v,,
... ,' i/J71
I '
as elements
129
• • • ,v
) then,
■■■cj = 0 VG'(V
for all
c.,
•••, c
■••,fm,cl,...
2.1.
From
in V,
,cn).
Thus,
V h Pivv • • • , vj
- Det (Vc, • - - Vcn([c, . - - cj = 0 V G'^,,
• • • , cj)).
If Det(V) N Vc,1 ••• Vc n ([c.1 ••• c n ] = 0 V G'(v.,1 --^c « )) then,, takingo
the unit basis for V, b,, - • • , b , we have Det (V) t= [b, • • ■ b ] 4 0 and thereL
n
In
fore, after substitution
det [è. • • • b. ,vb .
I
i—1
therefore
V M F(v,,
1
This
F
formula
produced
Part
II.
Det ((Vc,,
suffices
z+l
Det(V)
M G (v.,
-••,v
, ¿1, • • • , è ). Since
• • • b ]= el{v), we conclude
that V 1= F '(y,,
n
•••,f
m
I'm
).
is homogeneous
in the brackets
because
the necessary
homogeneity
Assume
is a homogeneous
open
any morphism
in Gl(nK),
•••,
to prove
cn)G).
that
there
Take
Vc,
the homogeneity
• • • Vc G is invariant
x [y, • •• yjt) = det[T]
G such
under
that
• det [y, • • • y ].
Since
F «-»
T, with domain
Also,
while
c. ranges
over
covers
Det(r(v)) r Vc, • • • VcnciKi/j),..
TicJ).
T(V) so
Det (V) h Vc, ... Vc^GG;,, ... , cn)
iff
•, r(%), c,,...,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
It
• • • T(y )] =
G is homogeneous
cn) iff Det (T(V))\= G(T(v,),-••,
V, T{c.)
V.
T.
the brackets and det[T] 4 0, we have
Det(V) h G(f ,,•••,
of
in G .
In the field S common to V and T(v) we find that det[T(y,)
det([T]
• • •, v ) and
cj.
in
130
WALTER WHITELEY
We conclude that V \= Fiv v ■■• , vj
For the category
a necessary
Pa(nK)
and sufficient
K, the prime
fields
fields
geometric
2.4.
class
of fields
it this way.
fields
are the prime
algebra
Fortunately,
of a field
fields
we can prove
these
An open formula
for
involving
K
ex-
do include
and the empty
that these
are the only
the principal
cases
of
Part
Part II.
I. Since
Assume
By Theorem
that
is the required
I
G is invariant
G is invariant
is essentially
0 and
only the constants
homogeneous
1, with the same
for Pa{nK)
formula
free
it is invariant
G with the same
••• Vc
n
G) with
G
variables
induced
V. Consider
for GlinK)
constants.
G as in the theorem.
for all morphisms
under morphisms
5 on a model
in GlinK).
It remains
by an isomorphism,
any atomic
formula
of the model.
e'iT(v))
Thus
= aie'iv))
and
a(f) = / since
to
a, of the
of Det(G),
of the form S f. • p .= 0 where the p. ate monomials
Now in the field:
field
involving
is an open,
F is invariant
F<-> Det(Vc,
2.3,
field
e'(c).
iff there
only the constants
and by Theorem 2.3 there
underlying
Pa(nK)
F in Ln(SAK),
c , such that F «-»Det (Vc. • • • Vc G) in the theory of VnK.
Proof.
prime
of fixed
of universal
1, is invariant
plus c.,•••,
and
for only a small
a
we have
importance.
in Ln(SBK),
check
geometry
map on K.
examples
[5, p. 184].
Theorem
0 and
for invariance
We approach
• • • , TivJ).
of projective
A field K is fixed iff any automorphism
field 0 . From results
fixed
condition
K, is the identity
The obvious
iff TÍV) * FiTiv^
of all collineations
and 0.
Definition 2.1.
tending
[March
which
in the eliv)
/ is in the
we have
Lf, ■U^Tiy)) -T.fi •n«(e*'(y))
=!>(/,) •UaieKy))
=«(£/. . Re^y))
and a[0]=0.
Det(T(V))
Therefore
t= GiTiv.),i
this concludes
the proof.
The existence
variance
single
G of this form is always
what coefficients
and a field
occur in F.
condition,
in V iff
in a single
sufficient
However,
for the init remains
even for open formulas.
of any characteristic
as well as for open formulas
we have verified
variable
over any field.
a
For a
the necessity,
In general
it
unknown.
3. Invariant
izations
• • •, c ) for all c.
D
that this is a necessary
equation
remains
N G(fj,
Tic n )) for all Tic)z in TÍV). Since T is surjective
'
of a formula
of F, no matter
conjecture
Det(V)
••-,
for general
formulas
with quantifiers.
formulas
which
directly
In all categories
extend
the results
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we have
of §2.
character-
1973]
LOGIC AND INVARIANT THEORY. I
Theorem
formula
3.1.
A formula
G in LniSBK),
plus e., •••,
Proof.
consider
F in LniSAK)
c , such that:
Part I.
F «-►Det(G)
Assume
that
results
front,
replacing
placing
from putting
the universal
the existential
s , each
function
lie outside
depending
quantifier
form of a formula
VhF(ß,,
••• ,aj
Take
• • • , v^y^
iff there
is a
bound variables
a model
V and
• • •, y , Sj, •••,sq)
form, with all its quantifiers
by new variables
by Skolem
on the variables
of the existential
of the Skolem
normal
quantifiers
quantifiers
for SlinK).
in V, Fsiv^,
F in prenex
for SlinK)
and the same
in the theory of VnK.
F is invariant
the Skolem form of the formula
which
is invariant
with the same free variables
131
functions
v,,
y.,
• • •, v
in the prefix
• • •, y
in the model
and those
of F.
in the
and reV: ,s.,
•••,
y. which
The principal
property
is this
iff V±Fsiav-..
,am,bv-..
,bp,sv
... ,sq)
for all b,, • • •, fc in V.
We can now operate
VrFiav
... ,aj
on Fs
much as we did on the open formulas in Theorem 2.1.
iff TiV) *rFiTia ,),-••,
iff T(V)hF*(r(öl),
TiaJ)
••• ,TiaJ,bv
... ,bp,sv...
,sq)
iff TiV) \=F'iTia^), ... , TiaJ, 7(7" H^)), • • • ,
TiT-Hbp)),TiT-lis1)),...,TiT-Hs€))).
Since
T is a bijection
ered as new Skolem
and
T
functions
(x) is a point
in T(V).
Thus
in TiV),
7
is.)
can be consid-
we have
TiV)h F'iTiaJ, .... TiaJ, TidJ, .... TidJ, Tis>),..., TisJ)
for all d,1 , ■• ■, d n in V.
Using
the same
corresponding
trick
as in Theorem
2.1, we introduce
the new c.,•••,
c
to T and find that
Det(V)
f= Gsiav
■■• , am, dv
• ■• , dp, c v • • • , c n, s [, ■ ■■ , s q)
for all d * in Det ÍV) and all c.7 such that [c.l ••• c tt ] = 1. The s¿r depend
also
*
on the new variables
c ,,•••,
in
G(«j, •••,a,Cj,»",c
V\*Fiav
...
The construction
,aj—
DetiVcj
of G assures
That V 1=Det(Vc.
usual
c . Thus,
), for all c..
i
Det ÍV) U [c,
'In
We conclude
••• Vcn([cx
us that
••• c ] / 1 V
that
• • • C>1]^ 1 VG(í,,
G has the correct
••• , C)j))).
free and bound variables.
• • • Vcn ([c.L • • • c n ] ¿ 1 V G)) -►F follows from the
substitution.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
132
WALTER WHITELE Y
Part II.
Assume
F«->Det(G).
Take
LMarch
any model
V and morphism
T in
Sl{nK), with domain V. Using the identity in the field S common to V and T(V),
det [T(y,) ••• TiyJ] = det[y, ••• yj,
Det(V)h=G(«,,
••• ,am)
iff
we have
Det (V) Gs(a ,,•••,
b., ■■■, sq)
iff Det (T(V))h Gs(T(ai), ... , T{b.), • • • , T(sq))
iff
since
T is surjective
Thus
and
F is invariant
Theorem
3.2.
a homogeneous
T(s.)
is a new Skolem
for Sl(nK).
A formula
formula
same bound variables
Det (V) )= G(T(a,),...,
F in Ln(SAK)
c.,•-•,
function.
O
G in Ln(SBK),
plus
T(am)),
is invariant
with the same
c , such that
for
Gl(nK)
iff there
free variables
F *-* Det(G)
is
and the
in the theory
of
VnK.
Proof.
T
We modify the proof of Theorem
with matrix
[c, i • • • c n ]~ . That is,
sion of det \_c^ • • • c ] by the zth column)
Now we find
After repeated
e'iTiv))
3.1 by applying
T ij.. = (coefficient
*det[c,
ij
identities
of c.11 in the expan-
• ■• c ], provided
= 1 T.. • e'{v) = det [c,
use of these
the transformation
dette,
• • • c] 4 0.
• • • v • • • c ] -f det [c,
1
n
and multiplication
• • • c ].
1
n
by an appropriate
power
of det[c, 1 ••• c n ] to clear fractions,
we find F has beer, transformed
into a
homogeneous formula G. As before we can verify that F<-> det(Vc, • • • \/cJ[cl •■•cJ = 0VG))
which
gives
us the required
The procedure
identical
for checking
to that for Theorem
for homogeneity
Corollary
then
homogeneous
that such
G are all invariant
3.1 employing
from the proof of Theorem
3.3.
// F is a formula
F is invariant
with the same
G .
for
constants,
Pa(nK)
Skolem
2.3.
forms and using
iff there
is
the identity
□
with only constants
0 and
is a homogeneous
and free variables,
for Gl{nK)
such
that
1 in Ln(SAK)
formula
G in Ln(SBK),
F *-* Det (G) in the theory
of VnK.
Proof.
Simply
The procedure
mulas
in Theorem
combine
used in Theorem
2.3 but it obscures
of F which we obtained
on the structure
practicable.
4.
duality
Covariant
of G.
in that case
In both cases
and contravariant
and to tensors,
were concerned
the proof of Theorem
significant
with languages
3.2.
D
3.2 could have been used on the open forthe information
on the homogeneous
and thus provides
the construction
vectors.
portions
which
2.4 with Theorem
less
parts
information
of G is constructive
For reasons
of the classical
included
useful
coordinates
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
related
and
to geometric
invariant
theory
of "contravariant
1973]
LOGIC AND INVARIANT THEORY. I
vectors",
vectors
A, B, • • • , as well as covariant
represent
indicate
hyperplanes
in the projective
how the preceding
this broader
vectors,
discussion
133
v, x, • • •.
geometry.
can be modified
Geometrically
In this section
to apply
equally
such
we will
well to
context.
Definition
4.1.
The language
ant and contravariant
vectors
LniCAK)
for analytic
is a modified
geometry
three-sorted
t= \V, H, 5; =; e1, ■■■ , e", ep ...
language
, en, +,-,-;
with covariof type
0, 1, {cf)feK\
and signature
a:
= is a relation
on terms
5
V—> 5,
e .,••.,
H —»5,
e
+ and
ate operations
• are operations
- is an operation
0, 1 and c.,
Definition 4.2.
—»5,
f £ K, ate constants
of sorts
of sort
5,
V and H occur in the language.
A standard model of LniCAK.) is a model
C = \V, H, 5;=;
5 is a field,
5x5
5—>5,
and only variables
where
of sort
e , • • • , e" ate operations
e\
...
V and
= is the equality
, en, ev • • • ,«„,+,
H ate vector
relation
spaces
•,-;
of dimension
from
V into
5,
e.,
from H into
5,
■• - , e
are the projections
0 and
are the usual
1 are the zero
and i is a field
The collection
operations
on 5,
and unit of 5,
injection
of K into
of all the standard
5.
models
is denoted
When we come to set up the transformations
order
to correspond
with matrix
to a projectivity
[T] which
takes
must act contagradiently
vector.
This
vectors,
while
B.
v in V into
to preserve
For this reason
the elements
on the models
Definition
4.2.
as simple
the elements
for
[v] a column
[ß] x [T]
a property
in
vector
for [ß]
as the point
of H ate called
covariant
of categories
SlinCK)
is the category
C are all transformations
with
= l,
where
[T] x [v]
we find that,
a transformation
a row
v is in
contravariant
vectors.
by introducing
the appropriate
CnK.
for a model
det[T]
of coordinates,
B in H into
of V ate called
We can now set up the series
morphisms
CnK.
on the models
or change
on H and take
is necessary
the hyperspace
n over 5, and
on 5,
e , • • • , en ate the projections
+, • and -
0, l,<M/))/eKi
T acts
induced
cogradiently
with objects
by matrices
CnK and the morphisms
[T] over the field
on V and contragradiently
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
5
on H.
134
WALTER WHITELEY
Definition
a model
4.3.
C given
Gl{nCK)
by matrices
induced
T(x,,
4.4.
■• ■ , a(xn))
as all morphisms
in each case,
is the category
of the fields,
• • • , xn) = (a(x,),
The definition
4 0, where
which
with objects
a, such
and
of Gl{nCK)
of a formula
a notion
CnK and morphisms
T acts
for
cogradiently
on H.
Pa(nCK)
by isomorphisms
as well
with objects
[T] over S, det[T]
on V and contragradiently
Definition
is the category
[March
that
T(A l, ■. • , A") = (a(A 1), ■■■ , a(An)),
and compositions
being
extends
CnK and morphisms
invariant
of such
for any of these
the definition
morphisms.
categories
for formulas
gives,
with only covari-
ant vectors.
Once more, from the classical
form of invariant
Definition
variant
theory
we have
a hunch
about
the syntactic
formulas.
4.5.
vectors,
The language
is a modified
t = \V,H,S;
LniSCK),
three-sorted
=;[],[
a language
language
]',(),+,
for covariant
and contra-
of type
.,-;0,
1, (cf)feK\
and signature
a:
= is a relation
on terms
[ ] is an operation
[ ]
H" —»S,
( ) is an operation
- are
H x V —*S,
operations
— is an operation
0, 1 and
S x S —> S,
S x S —» S,
c., f e K, are constants
and only variables
Definition
S
V"—» S,
is an operation
+ and
of type
4.6.
of sorts
A standard
of sort
V and
S,
H occur
D = \V, H, S; = ; det, det', (),+,.,-;
where
V and H are vector
= is the equality
det is the operator
the matrix,
det
spaces
relation
is the operator
in the language.
model of Ln(SCK) is a model
0, 1, <M/)>/eXi
of dimension
n over the field
5 and
on S,
det [v. • • • v ], with the vectors
written
det [A, • •• A ], with the vectors
as columns
written
of
as rows of the
matrix,
( ) is the operation
column
vector,
the field;
[A] x [v],
and the resulting
it is the dot product
+, •, and - are the usual
where
lxl
A is written
matrix
as a row vector,
is identified
of the two vectors,
operations
on the field
S,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
with
v asa
the element
of
T
1973]
0 and
1 are the zero and identity
h is an embedding
The collection
usually
of S,
of K into S.
of all standard
As a convention
letters,
models
we will denote
is denoted
constants
and variables
from the end of the alphabet:
of sort
side whether
we are in V or H, we drop the distinctions
We extend
the operation
on basic
linearly
■..yn]
to all terms
The corresponding
operator
the theorems
formulas
nomial
degree
which has,
has total
degree
geneous
of degree
total
the total
in the covariant
variables,
degree
except
ac-
degree
of
variables,
for the zero poly-
-<*>. For example
1 + (Ay) + [A
is to define
of the differing
vectors,
is the total degree
0 and a term like
defined.
all that remains
Because
and contravariant
a term like
(Ay)
• • • A ] • [y, • • • y ] is homo-
0.
of homogeneous
and formulas,
replacing
Ln(SAK), Ln(SCK) replacing
terms extended
all of the theorems
We will give two examples
in the proofs.
4.1.
sections
in the contravariant
as always,
. e\y)
from CnK to DnK is similarly
equations
Theorem
by defining
= det{ei(A.)),
for Ln(SCK).
on covariant
With this definition
required
to Ln(CAK)
= det(e;'(y;)),
of the earlier
in all its variables
the total
[ ] ,
and formulas.
Det
appropriately
tion of the morphisms
a polynomial
[ ] and
terms as follows
Det(G4y))=£e.U)
To extend
between
all primes.
Det ([A, ...An)]
homogeneous
of the al-
from the sort of terms in-
Det to be from Ln(SCK)
Det([y,
and extending
V by small
from the beginning
Since it will be clear
by omitting
the operator
usually
of sort
x, •• • etc., and constants
• • • , B, • •• etc.
and det,
letters,
y, f ,,••■,
phabet: A, A
as well as det
H by capital
DnK.
and variables
less
135
LOGIC AND INVARIANT THEORY. I
is an open formula
haps
n new covariant
with
and illustrate
is a direct
F in Ln(CAK)
G in Ln(SCK),
variables
homogeneous
Ln(SBK) and CnK replacing
example
An open formula
there
of §§2 and 3 apply
of this translation
The first
to define
VnK.
the modifications
translation
of Theorem
2.1.
is invariant
¡or Sl(nCK)
iff
with the same free variables
c , ••• , c , such
Ln(CAK)
that
F«-*Det(Vc
plus per-
...
Vc G)
in all models of CnK.
Proof.
covariant
Part I. Assume
vectors
[T..],
where
then
det[TJ=
r
T..
F is invariant
c,,..-,Cjj
is the cofactor
and take a model
in C. Consider
of el(c.)
-\
1
1 and T is in Sl(nK).
C in CnK and
the transformation
in [c
• •• c ].
I
In this case
71
we have
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
T with matrix
If det [c,
■■■ c ]= 1
±71
136
WALTER WHITELEY
[March
C (=F(i/j, •■• , vm, Av ••• , Ap) iff
TiC) h FiTivx), ... , TivJ, TÍA!>,...,
In the field
5 of C and
TiC),
since
[c,
eJ'(r(y)) = Det[c1
e .(7(A)) = £
Therefore,
this procedure
C r-Fifp
••• cf_iyc.+1
of T~
, we have
'
■•• cj,
e'(c.) ■ ey(A) = Det (A c.).
carries
F into a formula
G
in Ln(SCK)
such
that
•■• >vm. Ai. ••• . Ap)
^Det(Vcj
Part II.
In
7(Ap)).
• • • c ] is the matrix
With G
...Vcn([cj
defined
...«B]|i
as above
h n fot c,,1 • • • , c n and find that
1 V G'^j,
■•• , Ap))).
we can substitute
a unit basis
fe., • • • ,
,
Det [¿j ••■ &¿_,y&¿+1 ••• fej = e'(y),
Det (A fe.) = e¿(A),
Det [fcj ••■ fen] = 1.
Thus, as always,
C N Det(Vcj
.--Vc^tcj
••• cj¿l
V G'G/j, •••, Ap))) -»
Ffc^,...,^).
Part III.
We will simply
note the identities
required
Det [7(y J ■■• 7(yn)] = det [7] • det [y , • • • yJ = Det [y j ■• • yj,
Det (7(A y)) = [A] x [7" !] x [7] x [y] = [A] x [y] = Det (A y),
Det [TU i) • • ■TiAj] = det [Aj . - • Aj . det [7" l] = Det [Aj ... Aj.
The rest of the proof follows
The second
Theorem
there
example
4.2.
is an open,
variables
Proof.
model
An open formula
homogeneous
plus perhaps
F «-» Det(Vc,
1
Theorem 2.1.
will illustrate
O
the role of our definition
F in LniCAK)
formula
n new covariant
is invariant
G in LniSCK),
variables
c.,
of homogeneity.
for GlinK)
with the same
iff
free
• • • , c , such
that
• • • Vc n G) in all models in CnK.
Part I.
Assume
that
C in CnK and consider
F is invariant
the transformation
for GlinK).
Take
7 with matrix
any infinite
si
, s 4 0.
Then
C\*Fivv
... ,vm, Av ... , Ap)
iff
7(C) h FiTiv^l, ... , TivJ, TiA{),...,
T(Aj).
In the infinite field 5 of C and 7(C) we find e*(7(y)) = s • e'iy) and e {TÍA))
= s~
• e \A).
After multiplying
by some power
of s to clear
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lingering
s~
,
1973]
LOGIC AND INVARIANT THEORY. I
we obtain
a formula
the coefficients
after application
The rest
geneity
F"
satisfied
of sr in this
by an infinite
formula
137
set of s in the field
are homogeneous
polynomials
of Lemma 2.2, we have an open, homogeneous
of the proof follows
is correct
to verify
that of Theorem
S.
Since
in our sense,
formula
F .
2.3 and the definition
that all such homogeneous
of homo-
G must be invariant
for
Gl(nK). a
The reader can now translate
ceding
sections.
and prove any other of the theorems
The sharp eyed reader
of G corresponding
to an invariant
may have noticed
formula
the operation
[A, • ■• A ] in the language
this operation
both to emphasize
covariant
variables,
doing proofs
present
of Invariant
5. Klein's
basis
in later
formulas,
tences,
program
invariant
built
for reasons
related
it has
and
always
to the First
been
Fundamental
Theory.
can be restated
of models.
by formulas
and
work when we will be calculating
historically,
of
to include
of the contravariant
Erlanger program and other categories.
of Klein's
of a category
the existence
We have chosen
duality
and because,
in the work on invariants,
Theorem
Ln(SCK).
of the pre-
for the creation
F, we never needed
the essential
particularly
in these
that,
so that a geometry
The geometric
properties
for the category.
up from the invariant
It is now clear
The geometric
formulas,
is given
are those
that the
by means
properties
theorems
expressed
are those
sen-
which are true in all the models
in
the category.
Driven
algebraic
by a strong
languages
models,
say vector
algebraic
structure.
morphisms,
"bad"
intution,
than other
spaces,
models
the geometric
that essential
The classical
differential
structure
and language
of geometry:
severe.
any discussion
invariance
a category
under
the col-
algebraic
structure
and projective,
category
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for
Even
In our
of first-order
for a category
geometry.
were
Then,
with combinatorial
and is not even a pseudogroup.
of invariance
geometric
[9, pp. 31-41].
associated
for a more abstract
true that the concept
of analytic
the smaller
affine,
of
out of the
were groupoids.
were pseudogroups
embeddings
with
rich in
which is invariant
euclidean,
chosen
though
to separate
the morphisms,
unnecessarily
which includes
problems
with the necessary
and struggle
and the more of the tangled
IV we will examine
It remains
in morphisms,
the categories
seems
paper II we examine
more severe
They begin with a category
the category
The richer
remains
examples
geometry,
In our paper
tions.
enrich
in such a way that the categories
this restriction
geometry
they
transformations.
endure
is very meagre
the collineations,
lection of formulas which
which must be forgotten.
all stated
which
Then
for example
geometers
mathematicians.
valua-
is basic
to
.
138
WALTER WHITELEY
6.
Conclusion.
formulas
We have not been totally
invariant
acterized
under our fullest
the invariant
of formulas
placed
the language
of geometry
which
carry the zero vector
LniSBK)
translate
in the theory
terms of the revised
and the same
formulas
niques.
This
there
is usually
for considering
all theorems
properties
does
formulas,
F and
of F is stated
though
in the two kinds
constructive,
except
a simplification
invariance
are invariant
of categories.
in the brackets,
the complexity
free
in
the quantified
equivalent
as are the homogenizing
for quantifier
in the algebraic
and
all of the theo-
of the formulas
a formula
not increase
alternations,
not
the zero
are bijections
that the same formulas
for invariant
of creating
morphisms
Even better,
while
This means
and should
by removing
LniSAK)
models.
models,
point
projective
are morphisms
revised
is thoroughly
here to study
in the languages
with equivalence
different
translation
by the quantifier
we have
firmly inside
their restrictions
in these
forms occur
formula,
In addition
categories
directly,
that the process
to an invariant
class
examined,
all the original
to the zero vector,
categories.
may represent
We recall
Since
of the revised
standard
a much broader
a projective
revised
All of the operations
ate well defined
Det(G)
to create
of the models.
category.
rems and proofs
chosen
does not represent
from each
the revised
to the models
feasible
vector
of
char-
in this language.
zero vector
It is perfectly
all
we have
considered.
for each of the categories
may be an objection
properties—the
However,
K is a fixed field,
and thus laid the foundation
as expressions
There
in characterizing
PainK).
than previously
properties,
of brackets
successful
category,
where
and a fuller category
all invariant
appear.
formulas
[March
of formula
formulas.
expressions
tech-
as measured
In practice
as we move into
the brackets.
The whole
process
of defining
and its endomorphisms.
characterize
the invariant
gle model
of any cardinality.
Finally,
the work in Grassman's
therefore
formulas.
the prolonged
tend to translate
within
invariance
All the techniques
calculus
the class
to a single
The techniques
experience
as homogeneous
can be applied
apply
of analytic
[4], shows
formulas
to a single
infinite
of §3 would
geometry
that synthetic
with coefficients
model
model,
apply
to
to a sin-
and, in particular,
geometric
properties,
0 and
1 and are
Press,
New York,
we have characterized.
BIBLIOGRAPHY
1.
R. Baer,
Linear
algebra
and projective
geometry,
Academic
1952. MR 14, 675.
2.
J. A. Dieudonne
Math. 4 (1970), 1-80.
and J. B. Carrell,
Invariant
theory,
old and new,
MR 41 #186.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Advant
es in
3.
S. Feferman,
Lectures
on proof
theory,
1967), Springer, Berlin, 1968, pp. 1-107.
4. H. G. Forder,
5.
B. Jonsson,
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139
LOGIC AND INVARIANT THEORY. I
1973]
The calculus
Algebraic
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in Logic
(Leeds
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extensions
Chelsea,
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F. Klein,
Elementary
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Geometry,
Dover,
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H. W. Turnbull,
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O. Veblen
Univ. Press,
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H. Weyl,
Princeton
Univ.
logic
as invariant-theory,
of determinants,
matrices,
and invariants,
3rd ed.,
The foundations
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Cambridge
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The classical
Press,
11. W. Whiteley,
program:
MR 24 #A123.
and J. Whitehead,
Cambridge,
Erlanger
MR 8, 3.
groups.
Princeton,
N. J.,
Logic and invariant
Their
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and representation,
2nd ed.,
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theory, Ph.D. Thesis,
M. I. T., Cambridge,
Mass., 1971.
12.
-,
Homogeneous
13.
-,
Logic
14.
N. White,
University,
sets
and homogeneous
and invariant
The bracket
Cambridge,
Mass.,
theory.
ring and
II:
formulas,
Homogeneous
combinatorial
geometry,
Rend.
Mat. (to appear).
coordinates
Ph.D.
(to appear).
Thesis,
Harvard
1971.
DEPARTMENT OF MATHEMATICS,LAKEHEAD UNIVERSITY, THUNDER BAY, ONTARIO.
CANADA
Current
Lambert,
address:
Quebec,
Department
of Mathematics,
Champlain
Canada
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Regional
College,
St.