A monotonicity theorem for dp-minimal densely
ordered groups∗
John Goodrick
September 19, 2013
Abstract
Dp-minimality is a common generalization of weak minimality and
weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal
(Fact 2.2), but there are dp-minimal densely ordered groups that are not
weakly o-minimal. We introduce the even more general notion of inpminimality and prove that in an inp-minimal densely ordered group, every
definable unary function is a union of finitely many continuous locally
monotonic functions (Theorem 3.2).
1
Introduction
Since Shelah’s work a few years ago on theories without the independence property ([10]), there has been much work to try to generalize as far as possible the
results from stability theory to the context of theories with NIP. For instance,
some dividing lines analogous to the superstable-nonsuperstable dichotomy were
proposed in [11]. The simplest natural class of theories with NIP studied so far
are the dp-minimal theories (defined below), which were introduced by Onshuus
and Usvyatsov in [9]. For more background on NIP, the reader is encouraged
to look at [2] and [1], but the present paper is self-contained (modulo some
elementary model theory).
Onshuus and Usvyatsov have given a satisfying characterization of which
stable theories are dp-minimal ([9]). This paper grew out of asking a somewhat
opposite question: which ordered structures are dp-minimal?
While perhaps not too much can be said about general dp-minimal ordered
structures, we did discover that in the context of densely ordered abelian groups,
there are some parallels with o-minimality. The most striking result so far is
that any unary definable function is a union of finitely many continuous, locally
monotonic functions (Theorem 3.2). Along the way, we show that definable
nowhere dense subsets of the domain are finite (Lemma 3.3) and that there are
∗ The final version of this paper was published in The Journal of Symbolic Logic, vol. 75
(2010), no. 1, pp. 221–238. The copyright of this article is retained by the Association for
Symbolic Logic.
1
no dense planar graphs (Lemma 3.10). We also show that there are dp-minimal
densely ordered groups which are not weakly o-minimal (Example 2.10).
This paper is organized as follows: the remainder of this section introduces
the basic concepts we need (dp-minimality, inp-minimality, and valuational
cuts). Section 2 gives some examples of dp-minimal and inp-minimal ordered
structures to show that these notions are significantly more general than weak
o-minimality. Finally, section 3 develops a theory of definable unary sets and
functions in a densely ordered group, assuming inp-minimality, which is a hypothesis even more general than dp-minimality. As a culmination of this section,
we prove the local monotonicity theorem.
Much of the inspiration for these results came from the work on o-minimality
and its generalizations, and in particular the local monotonicity theorem for
weakly o-minimal structures (proved in [7] for divisible abelian groups and in
[3] in full generality). Unfortunately, it is not clear to us how to define a useful
notion of “cell decomposition” in our context, so we have had to use different
techniques. We expect that it is possible to develop a topological dimension
theory for definable sets in a densely ordered inp-minimal group, generalizing
what has been done for weakly o-minimal theories and theories with o-minimal
open core, but the old proofs do not seem to generalize straightforwardly to the
inp-minimal context.
One might also ask whether dp-minimality has any nice algebraic consequences. Here are two open questions. The answer to the first is “yes” if we
assume weak o-minimality ([7]).
Question 1.1. Is every dp-minimal group abelian by finite?
Question 1.2. Which fields are dp-minimal in the pure language of rings?
We would like to thank everyone with whom we had fruitful conversations
and correspondence while this paper was in preparation, including Hans Adler,
Alf Dolich, Ehud Hrushovski, Krzysztof Krupinski, Chris Laskowski, David Lippel, Chris Shaw, and Sergei Starchenko. Last but not least, many thanks to the
anonymous referee, whose suggestions significantly improved the exposition of
this paper.
1.1
Conventions and notation
All structures in this article will be assumed to be strictly linearly ordered by
the binary predicate symbol <. All such structures have the order topology, and
all topological terms such as “continuity,” “openness,” etc. refer to this topology,
or to the topology on a subset of M that is induced by this topology.
“Definable” means “definable over parameters,” not just “∅-definable.” If we
do not specify where a parameter or set comes from, then it lives in a highly
saturated monster model C of whatever theory we are considering at the time.
Variables and parameters without bars over them are singletons
from the home sort, contrary to the usual convention in model theory.
2
To save time and space, “interval” will always mean “nonempty
open interval” unless it is otherwise specified.
All groups in this paper are assumed to be abelian with group operation
“+.”
1.2
Basic definitions
Definition 1.3.
1. (Shelah) An independent partition pattern of length κ,
α
κi of formulas
or inp-pattern, is a sequence of pairs h(ϕα (x; y α ),
k ) : α < α
and natural numbers such that there is an array bi : α < κ of witnesses,
n
o
α
that is, for all α < κ, the “row” ϕα (x; bi ) : i < ω is k α -inconsistent, but
n
o
α
every “path” ϕα (x; bη(α) ) : α < κ (for η ∈ ω κ ) is consistent.
2. A theory T is inp-minimal if there is no inp-pattern of length 2 in a single
free variable x.
3. (Onshuus and Usvyatsov) A theory is dp-minimal if it is inp-minimal and
has NIP (i.e. it does not have the Independence Property; see [2]).
Remark 1.4.
1. In the terminology of [1], inp-minimality is equivalent to
every nonalgebraic 1-type having burden 1.
2. In the definition above, a standard Erdős-Rado argument shows that we
α
may assume without loss of ngenerality that each
sequence hbi : i < ωi
o
β
is indiscernible over the set bi : β 6= α, i < ω ; some details of this are
spelled out in [1]. If we are working in the theory of an ordered structure,
α
we may further assume that each sequence hbi : i < ωi is increasing.
3. Our definition of dp-minimality looks slightly different from Onshuus and
Usvyatsov’s original definition in [9], which uses Shelah’s “ict-patterns”
(from [11]) in place of inp-patterns. However, by Proposition 10 of [1], it
does not matter whether we use inp-patterns or ict-patterns in defining
dp-minimality for a theory with NIP, so the two definitions are equivalent.
The following diagram summarizes the logical realtionships between various
notions of minimality. The fact that SU -rank 1 simple theories are inp-minimal
is by the equivalence of forking and dividing in simple theories plus the remark
above, and the fact that weak o-minimality implies dp-minimality is Fact 2.2
below.
SU -rank 1
weakly o-minimal
⇒
inp-minimal
⇑
⇒
dp-minimal
⇑
stable and weakly minimal
3
⇒ NIP
If (M ; <, . . .) is an ordered structure, then M denotes the Dedekind completion of M , which contains endpoints −∞ and ∞. Following [7], we will talk of
sorts in M : these are simply subsets SY ⊆ M of the form
SY := sup(Yb ) : b ∈ M n ,
where Y ⊆ M n+1 is definable and Yb = a ∈ M : (a, b) ∈ Y . Note that these
are sorts in M eq in the usual model-theoretic sense, corresponding to definable
equivalence relations of the form
x ∼ y ⇔ ∀z [z > Yx ↔ z > Yy ] .
The point of sorts is that it is often useful to consider functions that map into
M instead of simply M (such as the limit functions fi : M → M we will define
in section 3), and it is very useful to have a notion of “definability” for such
functions. To be precise: when we speak of a definable function f : M → M ,
we mean that we have a definable function f0 : M → M and a definable set
Y ⊆ M 2 such that f (a) = sup(Yf0 (a) ), so that f maps M into the sort SY .
However, we will not need to mention the sort SY explicitly when talking about
definable functions into M .
As in the weakly o-minimal case, the following concept is also useful.
Definition 1.5. If (M ; <, +, . . .) is an ordered group, then b ∈ M is valuational
if its corresponding downward-closed Dedekind cut C ⊆ M has the property:
there is a positive ∈ M such that for any x ∈ C and any y ∈ M \ C, y − x > .
2
Examples of inp-minimal theories
In this section, we will give some examples of inp-minimal theories, focusing
especially on ordered groups.
We begin with a simple lemma about inp-patterns of intervals which will be
useful for a couple of the examples.
Lemma 2.1. In any densely ordered structure, any inp-pattern in one free
variable consisting of Boolean combinations of intervals must have length 1.
Proof. This result was announced by Onshuus and Usvyatsov ([9], Fact 3.2),
but we sketch a proof here for completeness.
Suppose towards a contradiction that {ϕ(x; ai ) : i < ω} ∪ ψ(x; bi ) : i < ω
forms such an inp-pattern of length 2. Without loss of generality (by replacing the rows by subsequences), each row is 2-inconsistent. By the pigeonhole
principle, there are numbers i0 and j0 such that for infinitely many i < ω, the
i0 th connected component of ϕ(x; ai ) is a subinterval of the j0 th component
of ψ(x; b0 ); again without loss of generality, this hols for every i < ω. Similarly, we can assume that there are i1 and ji such that the i1 th component of
every ϕ(x; ai ) is a subinterval of the j1 th component of ψ(x; b1 ), and the fact
that ψ(x; b0 ) and ψ(x; b1 ) are disjoint implies that i0 6= i1 . We can repeat this
4
inductively, but eventually we run out of numbers ik to select, and we get a
contradiction.
Thus any o-minimal theory is inp-minimal. More generally, one can show:
Fact 2.2. Any weakly o-minimal theory is dp-minimal.
Proof. See [9], Corollary 3.8.
Note that Fact 2.2 gives us examples of dp-minimal densely ordered groups
which are not rosy: for instance, the theory RCVF of real-closed valued fields in
the language of ordered rings plus a predicate for a convex valuational subring
(see [5]).
Lemma 3.3 below says that any nowhere-dense unary definable set in a
densely ordered inp-minimal group is finite. The next example shows that this
fails drastically if we remove the hypothesis that the structure has a group
operation:
Example 2.3. Let P ⊂ [0, 1] be the Cantor middle third set, and let T =
Th (R; <, P ).
Claim 2.4. T eliminates quantifiers in an expanded language with constant
symbols for 0 and 1 and predicates P0 and P1 , where P0 names all the left
boundary points of P (points a ∈ P for which there is an a0 < a with (a0 , a)
disjoint from P ) and P1 names all the right boundary points of P .
Proof. Let M be the standard model (R; <, P ) of T . Let S be the set of open
intervals comprising [0, 1] \ P (M ), and note that S (with the natural induced
ordering) is a countable dense linear ordering without endpoints. Therefore
(S; <) is strongly ω-homogeneous.
Suppose that a0 , . . . , an−1 and b0 , . . . , bn−1 are any two strictly increasing
sequences in (0, 1) such that ai is in P (M ) (or P0 (M ), or P1 (M )) if and only
if bi is. In the case where ai ∈ P0 (M ) (or P1 (M )) and ai is the right (or
left) boundary point of some open interval Ii ∈ S, there is an open interval
Ji ∈ S such that bi is the right (left) boundary point of Ji . In the case where
ai ∈ P (M ) \ (P0 (M ) ∪ P1 (M )), ai is the supremum of a Dedekind cut Ci ⊆ S,
and the point bi is the supremum of a Dedekind cut Di ⊆ S.
By the strong homogeneity of S, there is an order-preserving bijection σ :
S → S such that for every i < n for which Ii is defined, σ(Ii ) = Ji , and
similarly σ(Ci ) = Di whenever Ci and Di are defined. Since any two of the
open intervals comprising [0, 1] \ P (M ) are order-isomorphic, σ can be extended
to an automorphism f : M → M such that for every i < n, f (ai ) = bi .
Now suppose that we
of length 2 consisting of formulas
have an inp-pattern
{ϕ(x; ai ) : i < ω} and ψ(x; bi ) : i < ω . By the claim, we may assume that
_
ϕ(x; ai ) =
ϕ1j (x; ai ) ∧ ϕ2j (x)
j<n
5
where ϕ1j (x; ai ) is an interval and ϕ2j (x) is a Boolean combination of P (x),
P0 (x), and P1 (x), and ψ(x; bi ) can be decomposed similarly. Arguing as in the
proof of Lemma 2.1, we see that there must be i0 and j0 such that for infinitely
many i and j, ϕ1i0 (x; ai ) is a subinterval of ψj10 (x; bj ). But this contradicts the
inconsistency of the ψj (x; bj )’s.
So T is inp-minimal and P is an example of a unary definable set which is
not only infinte and nowhere dense, but does not even locally resemble a finite
union of points and intervals.
Not every dp-minimal ordered abelian group is densely ordered:
Example 2.5. Th(Z; <, +) is dp-minimal. This follows from the fact that any
definable subset of Z is eventually periodic.
It turns out that there are inp-minimal densely ordered groups with definable sets that are dense and codense in the whole structure. One of the most
obvious ways to try to construct such examples is with a dense pair of o-minimal
structures, but this fails:
Proposition 2.6. If (G; <, +, . . .) is an inp-minimal densely ordered group,
then G has no proper definable divisible subgroup which is dense in G.
Proof. Suppose to the contrary that H ≤ G is definable and divisible. Then if
1
)a ∈ H, then by clearing denominators we get
n, m ∈ N, a ∈ G, and ( n1 − m
that (m − n)a ∈ H, which implies that a ∈ H since H is divisible. So if we pick
a ∈ G \ H, we see that infinitely many distinct cosets of H have representatives
in the interval (0, a), and hence infinitely many H-cosets are represented in any
nonempty interval. Thus we can construct an inp-pattern of length 2 using
pairwise-disjoint intervals for the first row and cosets of H for the second row.
What does work for constructing inp-minimal theories with dense predicates
is to expand an o-minimal group by a “generic predicate” in the sense of Chatzidakis and Pillay ([4]):
Proposition 2.7. For every o-minimal theory T of a densely ordered group,
there is an expansion T 0 of T to the language L(T )∪{P } (where P is a new unary
predicate) such that T 0 is inp-minimal and proves “P is dense and codense.”
Proof. Let T = Th(M ), and let T 0 be a model completion of T in L(T ) ∪
{P }, as guaranteed by Theorem 2.4 of [4] (since o-minimality implies that T
eliminates ∃∞ ). Then T 0 contains the statement “P is dense and codense” by
Theorem 2.4(ii) of [4]. Every unary parameter-definable formula ϕ(x; a) in T 0
is a Boolean combination of formulas of the form x ∈ I ∧ P (f (x)) where I is an
interval and f is a continuous, strictly monotone T -definable function. Therefore
the closure of any T 0 -definable set is a Boolean combination of intervals, and by
Lemma 2.1, the inp-minimality of T 0 is implied by the following claim:
6
Claim 2.8. Suppose that hϕ(x; ai ) : i < ωi is any k-inconsistent sequence of
formulas in T 0 of the form
_
ϕ(x; ai ) =
x ∈ Iaji ∧ P (faji (x)),
j<n
where Iaji is an interval and faji is a continuous, strictly monotone T -definable
T
S
function. Then i∈ω j<n Iaji = ∅.
Proof. There are intervals Jaji such that ϕ(x; ai ) is equivalent to


∃y P (y) ∧ 

_
y ∈ Jaji ∧ faji (x) = y  .
j<n
T
S
So if i∈ω j<n Iaji were nonempty, then there would be a j < n and an infinite
T
subset S ⊆ ω such that i∈S Iaji is nonempty, and the set {ϕ(x; ai ) : i ∈ S}
would be consistent by Theorem 2.4 of [4], a contradiction.
Note that the theory T 0 above is not dp-minimal: since it is an expansion of
an infinite group, Proposition 2.10 of [4] implies that T 0 has the independence
property.
Question 2.9. Is there a dense, codense subset P ⊆ R such that Th(R; <, +, P )
is dp-minimal?
Here is an example of a dp-minimal densely ordered group with a dense,
codense definable set (thanks to David Lippel for suggesting this):
Example 2.10. Let Z(2) be the set of all rational numbers whose denominators
are powers of 2, given the induced ordering as a subset of Q. Then (Z(2) ; <, +)
is a densely ordered group, and the definable set of all all elements divisible by
3 is dense and codense. After adding unary predicates for n-divisibilty for each
n ∈ N, the theory eliminates quantifiers. Since it is simple to check that none of
the atomic formulas in this expanded language have the independence property,
it follows that no formula in T has the independence property (see Remark 9 of
[2]).
Finally, suppose that {ϕ(x; ai ) : i < ω} ∪ ψ(x; bi ) : i < ω is an inp-pattern
of length 2. By quantifier elimination, we may assume that
ϕ(x; ai ) = ϕ1 (x; ai ) ∧ ϕ2 (x; ai )
where ϕ1 (x; ai ) is a Boolean combination of intervals and ϕ2 (x; ai ) is a Boolean
combination of congruences modulo n (for some n ∈ ω). By the pigeonhole
principle, there is an infinite S ⊆ ω such that the set {ϕ2 (x; ai ) : i ∈ S} is
consistent, and in fact the set of realizations of this partial type must be dense.
7
So if the set {ϕ(x; ai ) : i < ω} is k1 -inconsistent, the sequence {ϕ1 (x; ai ) : i ∈ S}
is k1 -inconsistent as well. Similarly, we can modify the ψ(x; bi )’s to obtain an
inp-pattern of length 2 consisting of Boolean combinations of intervals. This
contradicts Lemma 2.1.
There is another way that one could to try to construct a dp-minimal densely
ordered group that is not weakly o-minimal: take a densely ordered abelian
group and add a unary predicate for an open set with infinitely many connected
components. We believe that this is possible (at least if one chooses the group
and the open subset carefully), but a detailed verification of this would be too
long to include in the current paper.
3
Monotonicity in inp-minimal densely ordered
groups
In this section, we consider the question of what kinds of unary functions f :
M → M can be definable an inp-minimal densely ordered group M . In general,
f may be discontinuous at every point in M : for instance, if X ⊆ M is a
definable dense codense subset (such as in Example 2.10) and b, c are any two
points in M , then we could have f (a) = b for every a ∈ X and f (a) = c for
every a ∈ M \ X. However, things look much nicer if we consider instead the
topological closure of graph(f ) in M ×M , which we call Γ(f ). We show that Γ(f )
is nowhere dense (Lemma 3.10), that it has no isolated points (Lemma 3.16), and
that it is the union of the graphs of finitely many continuous partial functions
(Corollary 3.26). A general strategy for proving such results will be to note
that “bad” behavior of f or of Γ(f ) often projects down to a definable set in
M , and from the assumption that this set is infinite we can usually construct
an inp-pattern of length 2 in a single free variable. We use this analysis of Γ(f )
to adapt an argument of Macpherson, Marker, and Steinhorn from [7] to prove
our monotonicity result (Theorem 3.2).
The first definition is needed for the statement of our monotonicity theorem.
Definition 3.1. Let f : M → M be a partially defined function on a densely
ordered set.
1. f is locally increasing if for all x ∈ dom(f ) there is an interval J ⊆ M
such that x ∈ Int(J) and f (dom(f ) ∩ J) is strictly increasing;
2. f is locally decreasing if for all x ∈ dom(f ) there is an interval J ⊆ M
such that x ∈ Int(J) and f (dom(f ) ∩ J) is strictly decreasing;
3. f is locally constant if for all x ∈ dom(f ) there is an interval J ⊆ M such
that x ∈ Int(J) and f (dom(f ) ∩ J) is constant.
Here is our monotonicity theorem:
Theorem 3.2. Let (M ; <, +, . . .) be an inp-minimal densely ordered group.
Then if f (x) : M → M is any definable unary function, there exists a finite
8
partition M = X0 ∪ . . . ∪ Xn−1 of M into definable subsets such that for each
i < n, f Xi is continuous and either locally increasing, locally decreasing, or
locally constant.
Note that unlike in the weakly o-minimal case, the sets Xi in Theorem 3.2
cannot be expected to be convex.
Throughout this section, we will assume that M = (M ; <, +, . . .) is an inpminimal densely ordered abelian group, unless stated otherwise. Also, we will
assume that M is ℵ1 -saturated: it is simple to check that all our results are
invariant under passing to elementary substructures, and saturation will be
useful for our proofs.
3.1
Basic results on inp-minimal densely ordered groups
In this subsection, we collect some basic results about definable sets and functions which will be useful in proving the monotonicity theorem in the following
subsection. Some of these lemmas may be of independent interest, especially
our first lemma.
Lemma 3.3. 1. If (M ; <, +, . . .) is a inp-minimal densely ordered group, X ⊆
M is definable, and X is nowhere dense, then X is finite.
2. If X ⊆ M is definable (that is, it is a definable subset of a definable sort),
X contains no valuational cuts, and X is nowhere dense, then X is finite.
Proof. 1. Suppose to the contrary that X ⊆ M is infinite, definable, and
nowhere dense. Pick a sequence of distinct points hai : i < ωi in X.
By induction on j < ω, we construct:
1. Sequences hIij : i < ωi of pairwise disjoint intervals such that ai ∈ Iij ;
2. Intervals Jij = (bji − dji , bji + dji ) ⊆ Iij which are disjoint from X;
3. Elements cj ∈ M .
We also require:
4. Iij+1 ⊆ Iij \ Jij ;
5. cj ≤ dji for every i < ω;
c
6. The length of Iij+1 is less than or equal to 2j .
0
Note in the base case j = 0 we can pick Ii as in 1 using the ℵ1 -saturation
of M . Then the Ji0 exist since X is nowhere dense, and 5 is easy. For the
induction step, pick Iij+1 satisfying 4 and 6 (using ℵ1 -saturation again) and
pick Jij+1 ⊆ Iij+1 disjoint from X using the fact that X is nowhere dense.
Notice that (bj+1
− cj+1 , bj+1
+ cj+1 ) contains no points in X (by the definii
i
j+1
j+1
tion of cj+1 ) but (bi − cj , bi + cj ) intersects X (since bj+1
and ai are both
i
c
in the interval Iij+1 , which has length at most 2j ). So hcj : j < ωi is a strictly
decreasing sequence, and we get an inp-pattern of length 2 from the formulas
defining the intervals Ii0 and the formulas
ϕj (x; cj , cj+1 ) := ∃y ∈ X [x − cj < y < x + cj ]∧¬∃y ∈ X [x − cj+1 < y < x + cj+1 ] .
9
2. The proof of 2 is the same as the proof of 1, observing that if X contains
no valuational cuts, then for any a ∈ X and any postive ∈ M , there is an open
interval in M containing a with radius less than . The elements bji , cj , and dji
above can still be taken from M .
Recall (from [8]) that the open core M ◦ of a structure M is the structure
with the same underlying set as M whose basic definable relations are all the
definable open subsets of cartesian powers of M , and M has o-minimal open
core if M ◦ is o-minimal. It turns out that any expansion of (R; <, +) in which
every unary nowhere dense definable set is finite has o-minimal open core ([6],
Proposition 2.14), so by the preceding lemma we get:
Corollary 3.4. If M = (R; <, +, . . .) is any inp-minimal expansion of the ordered additive group of real numbers, then M has o-minimal open core.
Corollary 3.5. Any inp-minimal densely ordered group (M ; <, +, . . .) has uniform finiteness. That is, for any formula ϕ(x; y) (in T , not T eq ), there is an
n < ω such that for every a ∈ M , if ϕ(a; y) has at least n realizations in M ,
then it has infinitely many realizations in M .
Proof. It suffices to consider the case where y is a single variable y. So suppose
there is a formula ϕ(x; y) and parameters ai ∈ M such that i ≤ |ϕ(ai ; M )| < ℵ0 .
Then by compactness, there is an elementary extension M 0 of M and a ∈ M 0
such that ϕ(a; M 0 ) is infinite but each realization of the formula is isolated from
every other realization by an open interval, contradicting Lemma 3.3.
Next we will introduce some terminology and notation for limits of functions
in ordered structures.
Definition 3.6. Let f : A → A be any partial function from a densely ordered
set A into its Dedekind completion A, and let a be any element of A.
1. limx→a f (x) = b ∈ A : (a, b) is an accumulation point of graph(f ) . In
other words, b ∈ limx→a f (x) if for every interval (c1 , c2 ) in A containing
a and every interval (d1 , d2 ) in A containing b, there is a c ∈ (c1 , c2 ) such
that f (c) ∈ (d1 , d2 ).
2. The element c ∈ A is in limx→a− f (x) if for every a1 ∈ A and c1 , c2 ∈ B
such that c ∈ (c1 , c2 ) and a1 < a, there is some b ∈ (a1 , a) such that
f (b) ∈ (c1 , c2 ). The set limx→a+ f (x) is defined similarly.
3. f is continuous at a if a ∈ dom(f ) and limx→a f (x) = {f (a)}.
Remark 3.7. limx→a f (x) = limx→a− f (x) ∪ limx→a+ f (x).
Note that the following result applies to arbitrary densely ordered structures,
not just inp-minimal densely ordered groups.
10
Lemma 3.8. Let A be any densely ordered structure and f : A → A any partial
function. If a ∈ A, J is an interval in A, and for every a0 < a, there exists some
a1 ∈ (a0 , a) such that ai ∈ dom(f ) and f (a1 ) ∈ J, then limx→a− f (x) ∩ J 6= ∅,
where J is the topological closure of J as computed in A. A similar statement
holds for limx→a+ f (x).
In particular, for any a ∈ A, limx→a f (x) is nonempty.
Proof. Note that the second sentence implies the first since we can take J to
be (−∞, +∞). The arguments for the existence of an element in limx→a− f (x)
and for the existence of an element in limx→a+ f (x) are identical, so we focus
on the former case.
Claim 3.9.
1
Without loss of generality,
cf({b ∈ M : b < a}) = ℵ0 .
Proof. First, add a function symbol for f to the language of M to ensure that
f is definable. Then note that for any sort SY in M , the statement “there is a
b ∈ SY ∩ J such that b ∈ limx→a− f (x) ∩ J” is first-order, so we may as well pass
to an elementary extension of M . Now build an elementary chain {Mi : i < ω}
such that M0 = M and for each i, there is an element ai+1 ∈ Mi+1 such that
ai+1 < a but ai+1 is strictly above
S any element of Mi that is less than a. Then
the conclusion we want holds in i<ω Mi .
Given the claim, we can pick an increasing sequence hai : i < ωi that converges to ai . By hypothesis, we can pick elements ci ∈ dom(f ) such that
ai < ci < a and f (ci ) ∈ J. Then the element lim supi<ω f (ci ) of M is in
limx→a− f (x) ∩ J.
From now until the end of this section, we will go back to working in an
inp-minimal densely ordered group M .
Lemma 3.10. Let f : M → M be any definable function. Then there is a
number N (f ) ∈ ω such that for any a ∈ M , | limx→a f (x)| ≤ N (f ).
Proof. Suppose the lemma fails. By compactness and ω-saturation of M , there
is an element a ∈ M such that limx→a f (x) is an infinite set. Without loss of
generality, there is an infinite, increasing sequence L1 < L2 < . . . of elements
of limx→a f (x). By the definition of limits, we can pick elements c1i and c2j
such that c11 < L1 < c12 < c21 < L2 < c22 < . . . and elements b1i ∈ M \ {a}
such that f (b1i ) ∈ (ci1 , ci2 ). Let d1 be an element of M such that for all i < ω,
0 < d1 < |a − b1i |. Now pick b2i ∈ (a − d1 , a + d1 ) such that f (b2i ) ∈ (ci1 , ci2 )
and pick a positive d2 less than every |a − b2i |, and so on. Now we contradict
inp-minimality with the formulas ϕ(x; a, di , di+1 ) := di+1 < |x − a| < di and
ψ(x; cj1 , cj2 ) := cj1 < f (x) < cj2 .
1 Thanks to the anonymous referee for suggesting this claim, which greatly simplified the
proof of the lemma.
11
Definition 3.11. Let f : M → M be a function.
1. Γ(f ) := (a, b) ∈ M × M : b ∈ limx→a f (x) .
2. For i ∈ ω \ {0}, fi (a) is the ith greatest element of the set
b ∈ M : (a, b) ∈ π1−1 (a) ∩ Γ(f )
if this set has at least i elements, and fi (a) is undefined otherwise.
Lemma 3.12. For any definable f : M → M and i ∈ {1, . . . , N (f )}, the
function fi : M → M is a definable partial function.
Proof. By Lemma 3.10, and the fact that for each i ≤ N (f ), one can express
with a formula the property “limx→a f (x) has no more than i elements below
x.”
Definition 3.13. 1. Suppose a, b ∈ M . We define d(a, b) by three cases: if
a < b, then d(a, b) = inf {b0 − a0 : a0 , b0 ∈ M and (a, b) ⊆ (a0 , b0 )}; if a > b, then
d(a, b) = d(b, a); and d(a, a) = 0.
2
2. If (a0 , b0 ), (a1 , b0 ) ∈ M , then d((a0 , b0 ), (a1 , b1 )) = d(a0 , a1 ) + d(b0 , b1 ).
It is easy to check that d as in 1 above satisfies the usual properties that
define a metric (positive-definiteness, symmetry, and the triangle inequality),
although it is technically not a metric since it takes values in M instead of in
R.
Lemma 3.14. If f : M → M is definable and there are no valuational cuts in
the image of f , then the set graph(f ) \ Γ(f ) is finite.
Proof. First note that if (a, b) ∈ graph(f ) \ Γ(f ), then (a, b) is an isolated point
of graph(f ). As usual, we assume that
X := {a ∈ M : (a, f (a)) is an isolated point of graph(f )}
is infinite, which implies (by Lemma 3.3) that there is an interval I such that
X is dense in I.
Next, pick a sequence hIi : i < ωi of pairwise disjoint subintervals of I.
Claim 3.15. For every i < ω and every > 0, there are two points a0 , a1 ∈ Ii ∩X
such that d((a0 , f (a0 )), (a1 , f (a1 ))) < .
Proof. If not, then there is some > 0 such that for any a ∈ Ii , there is
an interval Ja of length less than such that (a, f (a)) is the only point of
graph (f (Ii ∩ X)) that lies in the box (a − , a + ) × Ja . (Note that we
are using the fact that there are arbitrarily small open intervals around f (a)
since f (a) is not a valuational cut.) So if we fix any element a of Ii and let
Y = f (X ∩ Ii ∩ (a − , a + )), then for any b ∈ Y , Jb ∩ Y = {b}. But then
Y is an infinite definable set all of whose points are isolated, contradicting
Lemma 3.3.
12
Now let
ϕ(x; w) = ∃(y, z) ∈ graph(f ) [y 6= x ∧ d((x, f (x)), (y, z)) < w] .
By the claim above, we can construct an inp-pattern of length 2 using the
intervals hIi : i < ωi and the formulas
ψ(x; i , i+1 ) := ϕ(x; i ) ∧ ¬ϕ(x; i+1 )
for some decreasing sequence hi : i < ωi of positive elements of M .
Lemma 3.16. If f : M → M is definable and there are no valuational cuts in
the image of f , the set Γ(f ) is closed in M × M and contains no isolated points.
Proof. If (a, b) ∈ Γ(f ) is an isolated point, then it must be an accumulation
point of the set
S = {(c, d) ∈ graph(f ) : (c, d) is an isolated point in graph(f )} .
But this means that the set S must be infinite, contradicting Lemma 3.14.
Theorem 3.17. Suppose f : M → M is definable and i ≤ N (f ). Then there
are at most finitely many a ∈ M such that fi (a) ∈ M is a valuational cut.
Proof. Fix i ≤ N (f ), and for any positive ∈ M , let
C = b ∈ M : ∀c ∈ M [d(c, b) > ]
and let
D = {a ∈ M : fi (a) ∈ C } .
Note that since fi is a definable partial function (by Lemma 3.12) and C is a
definable set, D is definable as well.
Claim 3.18. For any positive ∈ M , D is finite.
Proof. If not, then there is some interval J in M such that D is dense in J. By
Lemma 3.14, the graph of f has only finitely many isolated points, so without
loss of generality, for every a ∈ J, (a, f (a)) is a limit point of graph(f ).
Pick any point (a0 , b0 ) ∈ graph(f ) with a0 ∈ J, and pick some open box
B0 = I00 × I01 around (a0 , b0 ) such that the intervals I00 and I01 each have radius
no more than . The nonisolation of (a0 , b0 ) in the graph of f implies that
there is a subinterval I0 ⊆ I00 such that for densely many a ∈ I0 , f (a) ∈ I01 .
The density of D implies that there is a point a1 ∈ I0 such that f (a1 ) ∈
/ I01 ,
0
1
0
and we can find an open box B1 = I1 × I1 disjoint from B0 such thatI1 ⊆ I0 ,
(a1 , f (a1 )) ∈ B1 , and the sides of B1 each have radius less than .
Repeating inductively, we get a sequence of subintervals I0 ⊇ I1 ⊇ . . . and
pairwise disjoint intervals I01 , I11 , . . . of radius less than such that for densely
many a ∈ Ij , f (a) ∈ Ij1 . If we choose a sequence of pairwise disjoint intervals
13
T
J0 , J1 , . . . contained in j<ω Ij , then we get a contradiction to inp-minimality
via the formulas “x ∈ Jj ” and “f (x) ∈ Ij1 .”
Now for any valuational cut b ∈ M , there is some positive ∈ M such that
b ∈ C . So if there were infinitely many a ∈ M such that fi (a) is a valuational
cut, then by compactness there would be some positive ∈ M such that D is
infinite, contradicting the Claim above.
Lemma 3.19. Suppose that f, g : M → M are definable partial functions such
that for every x ∈ dom(f ) ∩ dom(g), f (x) 6= g(x). Then for every interval I,
there is a subinterval J ⊆ I and a positive ∈ M such that
∀x ∈ J ∩ dom(f ) ∩ dom(g) [d(f (x), g(x)) ≥ ] .
Proof. Suppose that the lemma fails for some interval I: that is, for any subinterval J of I and any > 0, there is x ∈ J ∩dom(f )∩dom(g) with d(f (x), g(x)) <
. Pick a sequence hIi : i < ωi of pairwise disjoint subintervals of I and pick any
positive 0 ∈ M . By hypothesis, we can pick elements a0i ∈ Ii ∩dom(f )∩dom(g)
such that 0 < d(f (a0i ), g(a0i )) < 0 . Continuing inductively, we can pick a sequence 0 > 1 > . . . > 0 and elements aij ∈ Ij ∩ dom(f ) ∩ dom(g) such that for
every i, j ∈ ω, i+1 < d(f (aij ), g(aij )) < i . Then we contradict inp-minimality
with the intervals Ij and the formulas
ϕ(x; i , i+1 ) := i+1 < d(f (x), g(x)) < i .
3.2
Piecewise continuity of the fi ’s and the monotonicity
theorem
In this subsection, we apply the results of the preceding subsection to establish
that the functions fi are continuous almost everywhere on their domains (Theorem 3.27) and then deduce our main monotonicity theorem (Theorem 3.2).
Along the way, we observe that Γ(f ) can have only finitely many points at
which it “branches” (Corollary 3.29).
The first lemma shows that the functions fi are always piecewise continuous
on their domains of definition.
Lemma 3.20. If f : M → M is definable, then for any i between 1 and N (f ),
the set
n
o
Di := a ∈ M : | lim fi (x)| ≥ 2
x→a
is finite.
14
Proof. The set Di is definable, so if it is infinite, then by Lemma 3.3 it is dense
in some interval I. Suppose towards a contradiction that Di is dense on some
interval I. We may further assume that i is minimal (in the sense that for every
j < i, Dj is nowhere dense) and that for any j < i, Dj ∩ I = ∅ (by replacing
I by a slightly smaller subinterval, if necessary). Let fi,1 (a) denote the first
element of limx→a fi (x) and let fi,2 (a) denote the second element, if it exists.
By Lemma 3.19, we may assume that there is some > 0 such that for every
a ∈ I ∩ Di , d(fi,2 (a), fi,1 (a)) > .
Claim 3.21. Without loss of generality, for every a ∈ I, fi (a) ∈ limx→a fi (x).
Proof. By Lemma 3.14 and Lemma 3.17, there are at most finitely many a ∈ I
such that fi (a) ∈
/ limx→a fi (x), so we may shrink I to a subinterval which avoids
these points.
Claim 3.22. The set
{a ∈ I : fi (a) > fi,1 (a)}
is dense in I.
Proof. If not, then there is some interval J ⊆ I such that for every a ∈ J,
fi (a) ≤ fi,1 (a). By Claim 3.21, we may further assume that for every a ∈ J,
fi (a) = fi,1 (a).
Pick any a0 ∈ J. The set S0 := a ∈ J : d(fi (a), fi (a0 )) < 3 is infinite
(since fi (a0 ) = fi,1 (a0 ) ∈ limx→a0 fi (x), and using the fact that fi (a) is not a
valuational cut) and definable, so we can pick some subinterval J1 ⊆ J such that
S0 is dense in J1 . Next, choose any a01 ∈ J1 such that d(fi (a01 ), fi (a0 )) < 3 . Since
d(fi,2 (a01 ), fi,1 (a01 )) > and fi (a01 ) ≤ fi,1 (a01 ), it follows that d(fi,2 (a01 ), fi (a01 )) ≥
. Thus we can pick some point a1 ∈ J1 such that d(fi (a1 ), fi (a01 )) ≥ and
fi (a1 ) > fi (a01 ), and so d(fi (a1 ), fi (a0 )) > 32 and fi (a1 ) > fi (a0 ).
Continuing, we can pick an ω-sequence of elements a1 , a2 , . . . and intervals
J1 ⊇ J2 ⊇ . . . such that for any k > 0,
1. Jk ⊆ Jk−1 ;
2. The set Sk−1 := a ∈ J : d(fi (a), fi (ak−1 )) < 3 is dense in Jk ;
3. ak ∈ Jk ;
4. fi (ak ) > fi (ak−1 ); and
5. d(fi (ak ), fi (ak−1 )) > 23 .
(Given a0 , . . . , ak−1 and J1 , . . . , Jk−1 , we construct Jk so that it satisfies the
first two conditions, and then we pick ak in the same way as we selected a1
above.)
T
Let J∞ := k<ω Jk . Then for any interval J 0 ⊆ J∞ and for any k < ω,
h
i
(∃a ∈ J 0 ) d(fi (a), fi (ak )) <
,
3
15
so it follows immediately from Lemma 3.8 that for any a ∈ J∞ , limx→a fi (x) is
an infinite set. But this contradicts Lemma 3.10.
Claim 3.23. For any a ∈ I and j < i, limx→a fj (x) = fj (a).
Proof. Note that for any j < i, limx→a fj (x) is a single point by Lemma 3.8
plus the minimality assumption on i. So the claim follows immediately from
Claim 3.21.
Note that for any i ∈ I, the point (a, fi,1 (a)) is necessarily in Γ(f ) (since it
is an accumulation point of graph(fi ) ⊆ Γ(f ) and Γ(f ) is closed), so a corollary
of Claim 3.22 is that i > 1. Thus i − 1 ≥ 1 and dom(fi−1 ) is dense in I.
Claim 3.24. The set
{a ∈ I : fi−1 (a) = fi,1 (a)}
is dense in I.
Proof. By Claim 3.22, we can pick D ⊆ I which is dense in I such that for any
a ∈ D, we have
fi (a) > fi,1 (a) ≥ lim fi−1 (x).
x→a
By Claim 3.23,
lim fi−1 (x) = fi−1 (a) > fi−2 (a) > . . . > f1 (x).
x→a
Putting all this together, it follows that for any a ∈ D, the (i − 1)th element of
limx→a f (x) must equal fi,1 (a).
Claim 3.25. For any subinterval J ⊆ I and any positive 0 ∈ M , there is an
a ∈ J such that d(fi (a), fi−1 (a)) < 0 .
Proof. Suppose J ⊆ I is any subinterval and 0 > 0. By Claim 3.24, there is an
a0 ∈ J such that
fi−1 (a0 ) = fi,1 (a0 ) ∈ lim f (x);
x→a0
since fi−1 (a0 ) = limx→a0 fi−1 (x) (by Claim 3.23), there is some δ > 0 such that
h
0 i
∀a ∈ (a0 − δ, a0 + δ) \ {a0 } d(fi−1 (a), fi−1 (a0 )) <
.
2
Now pick a ∈ I ∩ (a0 − δ, a0 + δ) such that a 6= a0 and
0
> d(fi (a), fi,1 (a0 )) = d(fi (a), fi−1 (a0 )),
2
and by the triangle inequality,
d(fi (a), fi−1 (a)) ≤ d(fi (a), fi−1 (a0 )) + d(fi−1 (a0 ), fi−1 (a)) < 0 .
16
This last claim contradicts Lemma 3.19, so the lemma is proved.
Corollary 3.26. For each i ≤ N (f ), there are only finitely many a ∈ dom(fi )
such that fi is discontinuous at a.
Proof. By Lemmas 3.14 and 3.17, there are only finitely many a ∈ dom(fi ) such
that fi (a) ∈
/ limx→a fi (x), so the result follows from Lemma 3.20.
Theorem 3.27. Let (M ; <, +, . . .) be a inp-minimal densely ordered group.
Then if f : M → M is any definable unary function, there exists a finite partition M = Y0 ∪ . . . ∪ YN of M into definable subsets such that for each i < N ,
f Yi is continuous.
Proof. Let
Z = {a ∈ M : (a, f (a)) ∈
/ Γ(f )} .
By Lemma 3.14, Z is finite. For each i ≤ N (F ), let Zi be the set of all a ∈
dom(fi ) such that fi is discontinuous at a, which is finite by Corollary 3.26.
Next, let
[
Y0 = Z ∪
Zi .
1≤i≤N (f )
Then Y0 is finite, so f Y0 is trivially continuous. Finally, for each i between 1
and N (f ), let
Yi = {a ∈ (M \ Y0 ) : f (a) = fi (a)} .
By our construction, f Yi is continuous.
As a first application of Theorem 3.27, we will show that for any definable
function f : M → M , there are only finitely many points where Γ(f ) “branches”
(this is the idea behind the merger singularities in the next definition). Although
this result will not be needed for the proof of the monotonicity theorem, it gives
more information about the behavior of definable functions.
Definition 3.28. Suppose f : M → M is a definable function.
1. For any i ≤ N (f ),
Xi := a ∈ M : | b ∈ M : (a, b) ∈ Γ(f ) | = i .
2. An element a ∈ M is a merger singularity if there is an i ≤ N (f ) such
that a is a boundary point of Xi (that is, for every interval I containing
a, I contains both points in Xi and points in M \ Xi ).
Corollary 3.29. For any definable function f : M → M , the set of all merger
singularities of f is finite.
17
Proof. If not, then there are j < k ≤ N (f ) and an interval I such that both
Xj and Xk are dense in I. But this quickly contradicts the fact that Γ(f ) is
closed and the piecewise continuity of the fi ’s (details are left as an exercise to
the reader).
In the remainder of this subsection, we complete the proof of our monotonicity theorem (Theorem 3.2). We use a minor modification of Macpherson,
Marker, and Steinhorn’s proof of the local monotonicity theorem for weakly
o-minimal ordered groups (Theorem 3.4 in [7]).
First, we adapt some useful notation from that proof. We fix some definable
function f : M → M , and let Y0 , . . . , YN as in the conclusion of Corollary 3.27
above. For each k < N , we define the following formulas:
ϕk0 (x) := (x ∈ Yk ) ∧ ∃x1 [x1 > x ∧ ∀y ∈ (x, x1 ) (y ∈ Yk → f (y) < f (x))] ,
ϕk1 (x) := (x ∈ Yk ) ∧ ∃x1 [x1 > x ∧ ∀y ∈ (x, x1 ) (y ∈ Yk → f (y) = f (x))] ,
ϕk2 (x) := (x ∈ Yk ) ∧ ∃x1 [x1 > x ∧ ∀y ∈ (x, x1 ) (y ∈ Yk → f (y) > f (x))] ,
ψ0k (x) := (x ∈ Yk ) ∧ ∃x0 [x0 < x ∧ ∀y ∈ (x0 , x) (y ∈ Yk → f (y) < f (x))] ,
ψ1k (x) := (x ∈ Yk ) ∧ ∃x0 [x0 < x ∧ ∀y ∈ (x0 , x) (y ∈ Yk → f (y) = f (x))] ,
ψ2k (x) := (x ∈ Yk ) ∧ ∃x0 [x0 < x ∧ ∀y ∈ (x0 , x) (y ∈ Yk → f (y) > f (x))] ,
and for i, j ∈ {0, 1, 2},
k
θij
:= ψik (x) ∧ ϕkj (x).
k
Lemma 3.30. If a ∈ Yk , then there exist i, j ∈ {0, 1, 2} such that M |= θij
(a).
Proof. Fix such an a ∈ Yk . If there is some a0 > a such that (a, a0 ) ∩ Yk = ∅,
then it is vacuously true that M |= ϕk0 (a). Otherwise, we define the following
sets:
S0 := {a0 ∈ Yk : a0 > a, a0 ∈ Yk , and f (a0 ) < f (a)} ,
S1 := {a0 ∈ Yk : a0 > a, a0 ∈ Yk , and f (a0 ) = f (a)} ,
S2 := {a0 ∈ Yk : a0 > a, a0 ∈ Yk , and f (a0 ) > f (a)} .
Then these three sets form a partition of Yk ∩ (a, +∞) into disjoint definable
sets, so by Lemma 3.3, there must be an i ∈ {0, 1, 2} and an a1 > a such that
Si ∩Yk is dense on (a, a1 ). In the case where i = 1, it follows from the continuity
of f Yk that f (a0 ) = f (a) for every a0 ∈ (a, a1 ) ∩ Yk , so M |= ϕk1 (a). If i = 0,
then the continuity of f Yk implies that S2 ∩ (a, a1 ) = ∅ and S1 is nowhere
dense in (a, a1 ). So by Lemma 3.3, S1 ∩ (a, a1 ) is finite, and M |= ϕk0 (a).
Similarly, if i = 2 then M |= ϕk2 (a).
A similar argument shows that M |= ψ0k (a) ∨ ψ1k (a) ∨ ψ2k (a).
18
Corollary 3.31. If i = 0, 1, 2 and ϕki (x) (or ψik (x)) holds for densely many x
in an interval I, then it holds for every x in Yk ∩ I.
Proof. If ϕk0 (x), for instance, holds densely often in I, then it follows directly
from the definition that for any x ∈ Yk ∩ I, neither ϕk1 (x) nor ϕk2 (x) can hold,
and the result follows by Lemma 3.30.
We now show how the lemmas used to prove Theorem 3.4 in [7] can be
generalized to our context. We emphasize that the remaining proofs in this
section are quite similar to the proofs of the corresponding lemmas in that
paper, but we will spell out the details for ease of reading.
Lemma 3.32. If I ⊆ M is an interval and k < N , then it cannot happen that
k
k
k
k
any one of the formulas θ01
, θ10
, θ12
, or θ21
holds on a dense subset of I.
Proof. The idea is the same as in the proof of Lemma 3.6 of [7]. Namely,
k
suppose that for some k, θ01
holds densely often in I (and the other three cases
are similar). Pick a ∈ I such that ϕk1 (a) holds. Then a ∈ Yk , and there is an
element a1 > a such that
∀y ∈ (a, a1 ) ∩ Yk [f (y) = f (a)] .
Next, pick b ∈ (a, a1 ) such that ψ0k (b) holds; this means that b ∈ Yk , and there
is a b0 < b such that
∀y ∈ (b0 , b) ∩ Yk [f (y) < f (b)] .
k
Now since Yk is dense in I (since θ01
is true densely often in I), we can pick an
element c ∈ Yk such that max(a, b0 ) < c < b, and it follows that f (c) = f (a) =
f (b) but also f (c) < f (b), a contradiction.
k
Lemma 3.33. Let S ⊆ Yk be a definable set such that θ02
(x) holds for every
x ∈ S. Then there is a definable partition S = X ∪ S1 ∪ . . . ∪ St such that X
is finite and each function f Sj is locally increasing. Similarly, if S ⊆ Yk and
k
θ20
(x) holds for every x ∈ S, the same conclusion holds with “locally increasing”
replaced by “locally decreasing.”
k
Proof. (Similar to Lemma 3.9 of [7].) If θ02
(x) holds for every x ∈ S, we define
formulas
χk0 (x) := (∀x1 > x) ∃y ∃z [y ∈ Yk ∧ z ∈ Yk ∧ x < y < z < x1 ∧ f (z) ≤ f (y)] ,
χk2 (x) := (∀x1 < x) ∃y ∃z [y ∈ Yk ∧ z ∈ Yk ∧ x1 < y < z < x ∧ f (z) ≤ f (y)] .
Claim 3.34. For i = 0, 2, the set χki (M ) ∩ S cannot hold on a set of points in
M that is dense in an interval.
Proof. Suppose i = 0 (the case i = 2 is similar). Assume towards a contradiction
that there is an interval J ⊆ M such that χk0 (M ) ∩ S is dense in J.
19
For any point x ∈ J, let
Akx := {z − x : z ∈ (J ∩ Yk ) ∧ x < z ∧ ∀y ∈ (x, z) ∩ Yk [f (y) > f (x)]} ,
and define the function g : J → M by g(x) = sup Ax . For every x ∈ J ∩ S, since
ϕ2k (x) holds, g(x) > 0. We claim that for any a, b ∈ J ∩ Yk such that a < b, 0 is
a limit point of the set {g(x) : x ∈ (a, b) ∩ S}: for if 0 < < b − a, then χk0 (x)
holds on some point in (a, a + ), so there are y, z ∈ (a, a + ) ∩ Yk such that
y < z and f (z) ≤ f (y), which means that if we pick w ∈ S such that w ∈ (y, z),
then g(w) < . But the previous two sentences contradict Lemma 3.19, proving
the claim.
So by the Claim and Lemma 3.3, the formula χk0 ∨ χk2 holds of only finitely
many points a ∈ S. Let X be the set of points in S on which χk0 ∨ χk2 holds, and
let I1 , . . . , It be all the intervals bounded by adjecent points in X ∪ {−∞, +∞}.
If we let Sj = S ∩ Ij , then each function f Sj is locally increasing.
k
The case when θ20
(x) holds for every x ∈ S is similar.
k
Lemma 3.35. For any k < N , there is no interval J such that either θ00
(a) or
k
θ22 (a) holds for every a ∈ J ∩ Yk
Proof. (This is like the argument for “point (i)” in the proof of Theorem 3.4 of
k
[7].) Suppose towards a contradiction that θ22
(a) holds for every a ∈ J ∩ Yk
k
(and a similar proof takes care of θ00 ). Define the sets Ax (for x ∈ J) and
the function g : J → M as in the proof of Claim 3.34 above. Then for every
a ∈ J ∩ Yk , g(a) > 0 since ϕk2 (a) holds.
Claim 3.36. For every x, y ∈ J ∩ Yk such that x < y, 0 is an accumulation point
of {g(x) : x ∈ (x, y) ∩ Yk }.
k
Proof. Let be any positive element of M . Then since θ22
(y) holds, y is a
local minimum of f Yk , and so there is a z ∈ (x, y) such that y − z < and
f (z) > f (y). It follows that g(z) < .
From the Claim, we contradict Lemma 3.19, so the lemma is proved.
Proof of Theorem 3.2: We show that for each k < N , the set Yk can be
further subdivided into finitely many subsets Z0 , . . . , Zm such that for each
` ≤ m, f Zi is either locally increasing, locally decreasing, or locally constant.
k
By Lemma 3.30, one of the nine formulas θij
(i, j ∈ {0, 1, 2}) holds of every
point in Yk . By Lemmas 3.32 and 3.35, the only ones of these formulas which
k
k
k
, and θ20
. We let Z0 be the set
can hold for infinitely many x ∈ Yk are θ11
, θ02
k
k
k
of all realizations in Yk of ¬(θ11 ∨ θ02 ∨ θ20 ) (so that f Z0 is trivially locally
k
constant). If we let Z1 be the set of all realizations in Yk of θ11
, then f Z1
is clearly locally constant as well. Finally, Lemma 3.33 implies that we can
partition Yk \ (Z0 ∪ Z1 ) into finitely many subsets Z2 , . . . , Zm such that for each
` between 2 and m, f Z` is either locally increasing or locally decreasing.
This completes the proof of Theorem 3.2. 20
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