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. Chapter 1

 

Let X,Y,Z be sets and x,y,z variables raging over X,Y,Z respectively and φ(x,y,z) and ψ(x,y,z) be conditions on a point (x,y,z)X×Y×Z . We define Φ,ΨX×Y×Z as
Φ={(x,y,z)X×Y×Z:φ(x,y,z) holds }
Ψ={(x,y,z)X×Y×Z:ψ(x,y,z) holds }
Throughout the letures, for given conditions φ and ψ we use the notations
  1. φψ,
  2. φψ,
  3. ¬φ,
  4. zφ(x,y,z),
  5. zφ(x,y,z).

 

The associated sets would correspond to
  1. The union,
  2. the intersection,
  3. the complement,
  4. the projection on the first two variables,
  5. the complement of the projection of the complement.

 

Usually all variables will range over powers of the same set. for example
X=m, Y=n, Z=p

 

Definition: A boolean algebra on a set X is a collection of subsets of X 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 X and .

 

Definition We define a natural partial order in Boolean algebras by
ABAB=A
An atom of a boolean algebra is a minimal element under the previously mentioned order that lies above .

 

Definition: Let A1,,An be a collection of subsets of X . We define the boolean algebra on X generated by A1,,An to be
(A1,,An;X)={ΔΓ(iΔAiiΔX\Ai):Γ({1,,n})}
Where (Y) denotes the power set of Y . That is the collection of subsets of Y . Thus a boolean algebra generated by n elements has at most 2n atoms and has at most 22n elements.

 

1.1. Facts on Structures

 

Definition: A structure on a nonempty set R is a family S=(Sm)m such that for each m0 ;
  1. Sm is a boolean algebra of subsets of Rm ;
  2. if ASm , then R×A and A×R are both in Sm+1 ;
  3. {(x1,,xm)Rm:x1=xm}Sm ;
  4. If ASm+1 , then π(A)Sm , were π:Rm+1Rm is the projection on the first m coordinates.

 

We use the notation (R,S) to denote a structure, and we call the subsets of Rm that are in Sm definable sets.

 

Examples
  • equipped with semialgebraic sets is an example of a structure.
  • Let Ω be an algebrailcally closed field and K a subfield. Take Sm to be the K -constructible subsets of Ωm . That is, finite unions of sets of the form {xΩm:f1(x)=f2(x)==fn(x)=0 g(x)0} where f1,,fn , and g are all polynomials in m 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 ARm and BRn for m,n . We say that a function f:AB is definable if its graph is a definable set.

 

Lemma 1.1.
  1. If ASm and BSn , then a×BSm+n .
  2. For 11jm we have that δi,j:={(x1,,xm)Rm:xi=xj} belong to Sm .
  3. For BSm and i(1),,i(n){1,,m} we have that ARm given by
    (x1,,xm)A(xi(1),,xi(n))B
    is definable.

 

  1. A×Rn and Rm×B are in Sm+n so their intersection belongs to S
  2. A={(x1,,xj-i+1):x1=xj-i+1}Sj-i+1 so take the cross product with Ri-1 on the left and Rm-j .
  3. (x1,,xm)A if and only if y1yn(xi(1)=y1 and and xi(n)=yn and (y1,,yn)B.

 

Lemma 1.2. Let XRm and f:XRn be a map belonging to S .
  1. XSm ;
  2. if AX , ASm , then f(A)sn and fA belongs to S ;
  3. if BSn , then f-1(B)Sm ;
  4. if f is injective, then f-1 belongs to S
  5. if f(X)TRn and g:TRp is a second map belonging to S , then gf:XRp belongs to S .

 

2. O-minimal structures

 

Given ordered sets R1 and R2 and a map f:R1R2 we say that:
f is strictly increasing if x<yf(x<f(y)
f is increasing if x<yf(x)f(y)
f is strictly decreasing if x<yf(x)>f(y)
f is decreasing if x<yf(x)f(y)
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 R is called dense if for all a,bR with a<b there is a cR such that a<c<b .

 

A subset X of a linearly ordered set is called \convex if for all a,bX , cR such that a<c<b then cX .

 

Let (R,<) be a dense linearly ordered nonempty set without endpoints. We add two endpoints -, such that R=R{-,}

 

Notation: An interval is an open set of the form (a,b):={xR:a<x<b} for -a<b 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 R with the interval topology (i.e. the topology in which the intervals form a base). We further equip Rn for n with the product topology ( The interval in which the ``boxes'' (a1,b1)××(an,bn) form a base). It is easy to check that Rn i a hausdorf space with this topology.

 

Given a set A subset of a topological space we denote its topological closure by cl(A) and its topological interior as int(A) .

 

Given functions f,g:XR on a set XRm we put
(f,g):={(x,r)X×R:f(x)<r<g(x)}
[f,g]:={(x,r)X×R:f(x)rg(x)}
We write f<g to indicate that f(x)<g(x) for all xX .

 

Let (R,<) be a dense linearly ordered nonempty set without endpoint. An o-minimal structure on (R,<) is a structure S on R such that it satisfies the following two conditions.
  1. {(x,y)R2:x<y}S2
  2. The sets in S1 are exactly the finite unions of points and intervals.
If the conditions above are satisfied we say that (R;<,S) is an o-minimal structure .

 

Unles stated otherwise we will assume from now on that R is always equipped with an o-minim al structure and 'definable' stands for 'definable with parameters'.

 

Lemma 2.1. Let AR be definable. Then :
  1. inf(A) and sup(A) exist in R . (This property is called Dedekind Completeness).
  2. the boundary mboxbd(A):={xR: each interval containing x intersects both A and R-A}=Cl(A)-int(A). is finite. Further more, if a1<<al are the points of the boundary of A , then each of the intervals (ai,ai+1) is completely contained, or disjoint from A , for i=0,,k and ao=- and c.

 

Note that the first numeral in the previous lemma does not imply that every subset of R has a supremum (or infimum).

 

Lemma 2.2.
  1. If ARm is definable, then so are its closure and interior.
  2. If ABRm are definable and A is open in B , then there is a definable open URm with UB=A .
  1. (x1,,xm)A iff
    z1zmy1ym(zi<xi<yiw1wm(zi<wi<yi and (w1,,wm)A))
    The interior gets taken care of in a similar way.
  2. Let U be the union of boxes such that the intersection of the box with B is contained in A . Note that a point (x1,,xm) is in U iff there are a1,,am and b1,,bm such that ai<xi<bi and for all w1,,wm such that ai<wi<bi we have that (w1,,wm) is a p[oint in B then it is a point in A .

 

Definition: we say that XR is definably connected if X is not the union of two nonempty disjoint open subsets of X .

 

Example: is not connected with the order topology, but it is definably connected in the structure ;<,{q}q

 

Lemma 2.3.
  1. The definable connected subsets of R are the following; the empty set, the intervals, the sets [a,b),(a,b],[a,b] .
  2. The image of a definably connected set under a definably connected set under a definably continuous map f:XRn is definably connected.
  3. If X and Y are definable subsets of Rm X l(X) and X isdefinablyconnected,then Yisdefinablyconnected.
  4. If Xand Y aredefinablyconnectedsubsetsof R^m and X emptyset, then XY 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 Y=AB definably open sets. Then either AX= or BX= . Without loss of generality let AX= so Acl(X)= so AY= .
For 4: Let XY=AB definably connected sets. Thus either
i) AX= and AY= , or
ii) BX= and BY= , or
iii) AX= and BY= , or   BX= and AY= .

 

Corollary 2.1. For a definably continuous f:[a,b]R the Intermediate Value Theorem holds. That is; If f(a)<c<f(b) , then there is d with a<d<b such that f(d)=c .

 

Exercise: Let S be a definable subset of Rm show the following;
  1. {xRm:Sx is open } is definable,
  2. {(x,y)Rm+n;yint(Sx)} is definable. (Note that this is not the same as the interior of S .)

 

3. O-minimal ordered groups and rings

Definition: An ordered group is a group G; equipped with a linear order < such that x<y(xz<yzzx<zy)

 

Examples:
  1. ;<,+
  2. \{0};<,×

 

Proposition: Suppose that R;<,S is an o-minimal structure and S contains a binary operation on R such that R,<,. is an ordered group. Then R;. is abelian, divisible and torsion-free.

 

To prove this proposition we begin by stating a lemma.

 

Lemma 3.1. The only definable subsets of R that are also subgroups are {1} and R .
First show thatany definable subgroup of R must be convex. Otherwise you may find an ncreasing sequence (an) where the odd number indexed elements lie outside of G and the even indexed elements lie inside G . Assume then that G is different from {1} and that there is sR\G . Take g>1 with gG , then svg lies in the convex hull of any group containing g but it is nopt in G , contradicting the first part.

 

The proof of the proposition goes as follows:
Abelian: Fix rR and consider Cr:={xR:xr=rx} . this is clearly a definable subgroup containing r . Thus by the previous lemma Cr=R .

 

Torsion free: This is true for all ordered groups.

 

Divisible: Fix n> . The set {xn:xR} is a definable subgroup of R and it is not trivial since R; is torsion free.

 

Remark: Let R;<,+ be a non-tricial ordered abelian group. Then +:R2R and -:RR 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
  1. 0<1
  2. < is a translation invariant. That is, x<y gives x+z<x+y for all x,y,zR
  3. < is invariant under multiplication of positive elements. That is x<y and z>0 gives xz<yz .

 

Observations: The following are useful facts about ordered rings.
  1. R;,<,+ is an ordered group,
  2. The ring has no zero diisors,
  3. x20 for all xR .
  4. kk1:R is a strictly increasing ring embeddng.
  1. This follows directly from the definition of ordered ring.
  2. Suppose a and b are n on-zero elements such that ab=0 . We may assume that a>0 . Then we have two cases. b>0 implies ab>0 and b<0 implies ab<0 , both clear contradictions with ab=0 .
  3. Case; 1 x=0 , so x2=0 . Case 2; x>0 so x2=xx>0 . Case 3; X<0 , so -x2=-xx<0 so x2>0 .
  4. Properties 1 and 2 of ordered rings give you the ``Strictly increasing'' part. It remains to show that 0 maps to 0R . This follows from R;+ 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 xx-1 preserves the sign of x . In the order topology on R (and product topology on powers of R ) we get that and ^-1 are both continuous.

 

Definition: An ordered field is an ordered division ring with commutative multiplication.

 

Examples: (;+,×,<), (;+,×,<), (((t));+,×,<)

 

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 f(X)R[X] , and a<b elements in R are such that f(a)<0<f(b) , then there is c(a,b) such that f(c)=0 .

 

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 (R;<,S) be an o-minimal structure and let S contain binary operations +:R2R and ×:R2R so that (R;<,+,×) is an ordered ring. Then (R;<,+,×) is in fact a real closed field.

 

  1. [Division Ring] Let rR then rR={rx:xR} forms an additive subgroup of R . Thus rR={0} if r=0 and R otherwise. In particular rx=1 for some xR .
  2. [Commutativity of × ] The positive cone of R , Pos(R)={xR;x>0} is a definable subset of R , and (Pos(R);×) is an ordered group. In fact it is an o-minimal group, since everything definable in (Pos(R);×) is definable in (R;<,+,×) . Thus as a group it is commutative, and the extention of the operation to R preserves this property.
  3. [I.V.P.] Let P be a polynomial with coefficients in R in one indeterminate. By definable completeness, the definable subset {x(a,b):P(x)<0} of (a,b) has a supremum. Since + and × are continuous with respect to the order topology, this supremum c satisfies that P(c)=0
This concludes our proof

 

Note that this directly shows that (;<,+,×) is not o-minimal, since X2-2 has no root in .

 

4. Model Theoretic Structures

 

Definition: A language is a disjoint union of three (possibly empty) sets
=LcLFLP
Lc will be a called set of constant symbols ,

 

LF will be called a set of function symbols , each with an associated arity. That is for each fLF we have nf and we call f an nf -ary function symbol.

 

LP will be called a set of relations or predicates or propositional symbols (all interchangeable terms). Each element in LP has an associated arity.

 

Examples:
  • ab{0,+,-} The language of Abelian groups,
  • ab(<)={<} The language of ordered abelian groups,
  • graph={E} the language of graphs,
  • ring={0,1,+,×,-} the language of rings.

 

We say that a language ʹ expands if ʹ . For example ring expands ab .

 

Given a language we come up with a set of symbols we call variables and denote by Var that is disjoint from

 

We construct - terms in the following way
  • The elements of Var are terms,
  • the elements of Lc are -terms
  • the element ft1tn for fLF of arity n and t1,,tn terms, is considered an -term.

 

Examples:
  1. 0 in ab ,
  2. +0x for xVar in ab (usually written as 0+x ),
  3. (1+1)×x in ring ,
  4. 0+1 in ring

 

We construct - formulas as follows
  • If t1,t2 are -terms, then t1=t2 is an -formula,
  • if t1,,tn are -terms and P is an n -ary predicate, then Pt1tn is an -formula,
  • if φ is an -formula, then ¬φ is an -formula,
  • if φ1 and φ2 are -formulas,then wedge isan L -formula,
  • if φ is an -formula and x is a variable, then xφ is an -formula.

 

Examples:
  1. x1=x2 in all languages
  2. <1+0x ( 1<0+x ) for xVar in ring ,
  3. xx×x-1=0 in ring ,
  4. ¬x¬(0+x=x) in ab .

 

Definition: We say that a variable x occuring in a formula φ is bounded in φ if any of the following occurs
  1. φ is xψ for some formula ψ ,
  2. φ is ¬ψ for some formula ψ in which x is bounded,
  3. φ is φ1φ2 for formulas φ1 and φ2 in which x appears in at least one, and is bounded in both.
Definition: Let φ be an -formula, we call φ a sentence if for every variable x 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 =LcLFLP , an - structure is a tuple =(R;(c)cLc,(f)fLF,(P)PLP) such that
  • R 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),
  • c is an element of R for each cLc ,
  • f is a function f^ R :R^{n_f} ,
  • P is a subset of RnP .