Lecture in O-minimality
By Santiago Camacho for MA595, U Illinois, Fall 2015
We will use this format to publish lectures in o-minimality online.
This page may be readable in various web browsers but it will look its best in firefox.
We follow very closely the Book "Tame topology and o-minimal structures" by Lou van den Dries.
1.1. Chapter 1
Let
be sets and
variables raging over
respectively and
and
be conditions on a point
. We define
as
|
|
Throughout the letures, for given conditions
and
we use the notations
-
-
-
-
-
The associated sets would correspond to
- The union,
- the intersection,
- the complement,
- the projection on the first two variables,
- the complement of the projection of the complement.
Usually all variables will range over powers of the same set. for example
Definition: A
boolean algebra on a set
is a collection of subsets of
that is closed under finite unions and complements.
It is easy to see that Boolean algebras are closed under finite intersections and they always contain both
and
.
Definition We define a natural partial order
in Boolean algebras by
An
atom of a boolean algebra is a minimal element under the previously mentioned order that lies above
.
Definition: Let
be a collection of subsets of
.
We define the boolean algebra on
generated by
to be
|
Where
denotes the
power set of . That is the collection of subsets of
.
Thus a boolean algebra generated by
elements has at most
atoms and has at most
elements.
1.11.1. Facts on Structures
Definition: A
structure on a nonempty set
is a family
such that for each
;
- is a boolean algebra of subsets of ;
- if , then and are both in ;
- ;
- If , then , were is the projection on the first coordinates.
We use the notation
to denote a structure, and we call the subsets of
that are in
definable sets.
Examples
- equipped with semialgebraic sets is an example of a structure.
- Let be an algebrailcally closed field and a subfield. Take to be the -constructible subsets of .
That is, finite unions of sets of the form where , and are all polynomials in variables.
In order to check that these are in fact structures the main challenge is to prove closure under projections.
Usually technics that help with this kind of issue lay in the realm of
quantifier elimination.
Nevertheless there are also times in which the problem of checking wheter a collection of sets form a structure lies in checking that said collection of sets is closed under taking complements.
Sometimes this type of hurdles can be overcommend by technics such as
Gabrielov's theorem.
Notation: Let
and
for
. We say that a function
is
if its graph is a definable set.
1.1Lemma 1.1.
- If and , then .
- For we have that belong to .
- For and we have that given by
|
is definable.
- and are in so their intersection belongs to S
- so take the cross product with on the left and .
- if and only if and and and
1.2Lemma 1.2.
Let
and
be a map belonging to
.
- ;
- if , , then and belongs to ;
- if , then ;
- if is injective, then belongs to
- if and is a second map belonging to , then belongs to .
2.2. O-minimal structures
Given ordered sets
and
and a map
we say that:
f is
strictly increasing if
f is
increasing if
f is
strictly decreasing if
f is
decreasing if
f is
strictly monotone if it is strictly increasing or strictly decreasing, and
f is
monotone if it is increasing or decreasing.
A linearly ordered set
is called
dense if for all
with
there is a
such that
.
A subset
of a linearly ordered set is called
if for all
,
such that
then
.
Let
be a dense linearly ordered nonempty set without endpoints. We add two endpoints
such that
Notation: An
interval is an open set of the form
for
The notation for intervals is the same as the notation for ordered pairs, but hopefully the readers of this notes will be able to identify when are we reffering to each of them in context.
We equip
with the interval topology (i.e. the topology in which the intervals form a base).
We further equip
for
with the product topology ( The interval in which the ``boxes''
form a base).
It is easy to check that
i a hausdorf space with this topology.
Given a set
subset of a topological space we denote its
topological closure by
and its
topological interior as
.
Given functions
on a set
we put
|
|
We write
to indicate that
for all
.
Let
be a dense linearly ordered nonempty set without endpoint. An
o-minimal structure on is a structure
on
such that it satisfies the following two conditions.
-
- The sets in are exactly the finite unions of points and intervals.
If the conditions above are satisfied we say that
is an
o-minimal structure .
Unles stated otherwise we will assume from now on that
is always equipped with an o-minim al structure and 'definable' stands for 'definable with parameters'.
2.1Lemma 2.1.
Let
be definable. Then :
- and exist in . (This property is called Dedekind Completeness).
- the boundary is finite.
Further more, if are the points of the boundary of , then each of the intervals is completely contained, or disjoint from , for and and c.
Note that the first numeral in the previous lemma does not imply that every subset of
has a supremum (or infimum).
2.2Lemma 2.2.
- If is definable, then so are its closure and interior.
- If are definable and is open in , then there is a definable open with .
- iff
|
The interior gets taken care of in a similar way.
- Let be the union of boxes such that the intersection of the box with is contained in . Note that a point is in iff there are and such that and for all such that we have that is a p[oint in then it is a point in .
Definition: we say that
is
definably connected if
is not the union of two nonempty disjoint open subsets of
.
Example: is not connected with the order topology, but it is definably connected in the structure
2.3Lemma 2.3.
- The definable connected subsets of are the following; the empty set, the intervals, the sets .
- The image of a definably connected set under a definably connected set under a definably continuous map is definably connected.
- If and are definable subsets of X l(X) X Y
- If Y R^m X emptyset, then is definably connected.
Item 1 is an easy consequence of o-minimality. The contrapositive of item 2 can be proven almost immediately.
For 3: Let definably open sets. Then either or . Without loss of generality let so so .
For 4: Let definably connected sets. Thus either
i) and , or
ii) and , or
iii) and , or and .
2.1Corollary 2.1.
For a definably continuous
the Intermediate Value Theorem holds. That is; If
, then there is
with
such that
.
Exercise: Let
be a definable subset of
show the following;
- is definable,
- is definable. (Note that this is not the same as the interior of .)
3.3. O-minimal ordered groups and rings
Definition: An
ordered group is a group
equipped with a linear order
such that
Examples:
Proposition: Suppose that
is an o-minimal structure and
contains a binary operation
on
such that
is an ordered group. Then
is abelian, divisible and torsion-free.
To prove this proposition we begin by stating a lemma.
3.1Lemma 3.1.
The only definable subsets of
that are also subgroups are
and
.
First show thatany definable subgroup of must be convex.
Otherwise you may find an ncreasing sequence where the odd number indexed elements lie outside of and the even indexed elements lie inside .
Assume then that is different from and that there is . Take with , then lies in the convex hull of any group containing but it is nopt in , contradicting the first part.
The proof of the proposition goes as follows:
Abelian: Fix
and consider
. this is clearly a definable subgroup containing
.
Thus by the previous lemma
.
Torsion free: This is true for all ordered groups.
Divisible: Fix
. The set
is a definable subgroup of
and it is not trivial since
is torsion free.
Remark: Let
be a non-tricial ordered abelian group. Then
and
are both continuous operations in the order topology.
Definition: An
ordered ring is a ring (associative with multiplicative identity 1) equipped with a liner order
such that
-
- is a translation invariant. That is, gives for all
- is invariant under multiplication of positive elements. That is and gives .
Observations: The following are useful facts about ordered rings.
- is an ordered group,
- The ring has no zero diisors,
- for all .
- is a strictly increasing ring embeddng.
- This follows directly from the definition of ordered ring.
- Suppose and are n on-zero elements such that . We may assume that .
Then we have two cases. implies and implies , both clear contradictions with .
- Case; 1 , so . Case 2; so .
Case 3; , so so .
- Properties 1 and 2 of ordered rings give you the ``Strictly increasing'' part. It remains to show that maps to .
This follows from being torsion free.
This concludes the proof.
If moreover the ring is a
division ring , that is there is a multiplicative right inverse for all non-zero elements, then this inverse is unique, and it is also a left inverse.
Moreover the map
preserves the sign of
. In the order topology on
(and product topology on powers of
) we get that
and
are both continuous.
Definition: An
ordered field is an ordered division ring with commutative multiplication.
Examples:
Non-examples since there is no order preserving the field operations;
since it is not a field.
Definition: A
real closed field is an ordered field such that if
, and
elements in
are such that
, then there is
such that
.
The reader should note that this is in fact the Intermediate Value Property. There are many other characterizations of "real closed fields", and they all relate to properties that the field of Real numbers satisfy, hence the name.
Proposition: Let
be an o-minimal structure and let
contain binary operations
and
so that
is an ordered ring.
Then
is in fact a real closed field.
- [Division Ring] Let then forms an additive subgroup of . Thus if and otherwise.
In particular for some .
- [Commutativity of ] The positive cone of , is a definable subset of , and is an ordered group.
In fact it is an o-minimal group, since everything definable in is definable in .
Thus as a group it is commutative, and the extention of the operation to preserves this property.
- [I.V.P.] Let be a polynomial with coefficients in in one indeterminate. By definable completeness, the definable subset of has a supremum.
Since and are continuous with respect to the order topology, this supremum satisfies that
This concludes our proof
Note that this directly shows that
is not o-minimal, since
has no root in
.
4.4. Model Theoretic Structures
Definition: A
language is a disjoint union of three (possibly empty) sets
will be a called set of
constant symbols ,
will be called a set of
function symbols , each with an associated arity.
That is for each
we have
and we call
an
-ary function symbol.
will be called a set of
relations or
predicates or
propositional symbols (all interchangeable terms). Each element in
has an associated arity.
Examples:
- The language of Abelian groups,
- The language of ordered abelian groups,
- the language of graphs,
- the language of rings.
We say that a language
expands
if
. For example
expands
.
Given a language we come up with a set of symbols we call variables and denote by
that is disjoint from
We construct
-
terms in the following way
- The elements of are terms,
- the elements of are -terms
- the element for of arity and terms, is considered an -term.
Examples:
- in ,
- for in (usually written as ),
- in ,
- in
We construct
-
formulas as follows
- If are -terms, then is an -formula,
- if are -terms and is an -ary predicate, then is an -formula,
- if is an -formula, then is an -formula,
- if and are wedge L
- if is an -formula and is a variable, then is an -formula.
Examples:
- in all languages
- ( ) for in ,
- in ,
- in .
Definition: We say that a variable
occuring in a formula
is
bounded in if any of the following occurs
- is for some formula ,
- is for some formula in which is bounded,
- is for formulas and in which appears in at least one, and is bounded in both.
Definition: Let
be an
-formula, we call
a
sentence if for every variable
appearing in it is bounded.
Examples: (3) and (4) in the previous set of examples are sentences in their respective languages.
Definition: Given a language
, an
-
structure is a tuple
such that
- is a set (With care this can be taken to be a set theoretic proper class, but this is not necessary for this set of notes),
- is an element of for each ,
- is a function f^ R :R^{n_f}
- is a subset of .