# «ALFRED DOLICH, JOHN GOODRICK, AND DAVID LIPPEL Abstract. We study the notion of dp-minimality, beginning by providing several essential facts about ...»

## DP-MINIMALITY: BASIC FACTS AND EXAMPLES

arXiv:0910.3189v2 [math.LO] 12 Nov 2009

## ALFRED DOLICH, JOHN GOODRICK, AND DAVID LIPPEL

Abstract. We study the notion of dp-minimality, beginning by providing several

essential facts about dp-minimality, establishing several equivalent deﬁnitions for

dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally deﬁnable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the ﬁeld of p-adic numbers is dp-minimal.

1. introduction In this note we study many basic properties of dp-minimality as well as developing several fundamental examples. Dp-minimality—see Deﬁnition 2.2—was introduced by Shelah in [14] as possibly the strongest of a family of notions implying that a theory does not have the independence property—for which see [15]. The study of dp- minimality beyond Shelah’s original work was continued by Onshuus and Usvyatsov in [11] focusing primarily on the stable case and by the second author in [9] primarily in the case of theories expanding the theory of divisible ordered Abelian groups. Our goal in this paper is to provide many basic foundational facts on dp-minimality as well as to explore concrete examples of dp-minimality in the ordered context as well as the valued ﬁeld context. Much of our motivation arises out of the program of attempting to explore the impact of

**Abstract**

model theoretic notions, such as dp-minimality, in concrete situations such as ordered model theory on the reals or the study of valued

these facts are inherent in [14] but we isolate them here and provide straightforward proofs. Section 3 is a brief discussion of the relationship of dp-minimality to some other minimality notions. In Section 4 we focus weak on o-minimality—for which see [10]—and show that a weakly o-minimal theory is dp-minimal as well providing an example of a weakly o-minimal group which is not obtained by expanding on o-minimal structure by convex sets. Work in Section 3 as well as results from [9] indicate that a dp-minimal theory expanding that of divisible ordered Abelian groups has some similarity to a weakly o-minimal theory and we may naturally ask whether any such theory is weakly o-minimal. Section 5 provides a negative answer via an example arising form the valued ﬁeld context. Our ﬁnal section is dedicated to showing that the theory of the p-adic ﬁeld is dp-minimal.

2. basic facts on dp-minimality We develop several basics facts about dp-minimality. The vast majority of the material found below is inherent in Shelah’s paper [14], but typically in the more general context of strong dependence. We provide proofs of these various facts for

**clarity and ease of exposition. Recall:**

Deﬁnition 2.1. Fix a structure M, An ICT pattern in M consists of a pair of formulae φ(x, y) and ψ(x, y); and sequences {ai : i ∈ ω} and {bi : i ∈ ω} from M so that for

**all i, j ∈ ω the following is consistent:**

φ(x, ai ) ∧ ψ(x, bj ) ∧ ¬φ(x, al ) ∧ ¬ψ(x, bk ).

l=i k=j Remark. Deﬁnition 2.1 should more formally be referred to as an ICT pattern of depth two but in this paper we only consider such ICT patterns and thus we omit this extra terminology.

Deﬁnition 2.2. A theory T is said to be dp-minimal if in no model M |= T is there an ICT pattern.

It is often very convenient to use the following deﬁnition and fact.

Deﬁnition 2.3. We say two sequences {ai : i ∈ I} and {bj : j ∈ J} are mutually indiscernible if {ai : i ∈ I} is indiscernible over j∈J bj and {bj : j ∈ J} is indiscernible over i∈I ai. We call an ICT pattern mutually indiscernible if the witnessing sequences are mutually indiscernible.

** Fact 2.4.**

T is dp-minimal if and only if in no model M |= T is there a mutually indiscernible ICT pattern.

Proof. This is a simple application of compactness and Ramsey’s theorem.

Before continuing we should mention another alternative characterization of dpminimality. To this end we have the following deﬁnition.

## DP-MINIMALITY: BASIC FACTS AND EXAMPLES 3

Deﬁnition 2.5. A theory T is said to be inp-minimal if there is no model of M of T formulae φ(x, y) and ψ(x, y), natural numbers k0 and k1, and sequences {ai : i ∈ ω} and {bi : i ∈ ω} so that {φ(x, ai ) : i ∈ ω} is k0 -inconsistent, {ψ(x, bj ) : j ∈ ω} is k1 -inconsistent and for any i, j ∈ ω the formula φ(x, ai ) ∧ ψ(x, bj ) is consistent.**With this deﬁnition we have the following fact:**

** Fact 2.6.**

[11, Lemma 2.11] If T does not have the independence property and is inp-minimal then T is dp-minimal.

The next fact is extremely useful in showing that a theory is not dp-minimal. It is key in establishing the relationship between dp-minimality and indiscernible sequences that follows.

** Fact 2.7.**

Let T be a complete theory, we work in a monster model C. Suppose there are formulae φ0 (x, y) and φ1 (x, y) and mutually indiscernible sequences {ai : i ∈ ω} and {bi : i ∈ ω} so that φ0 (x, a0 ) ∧ ¬φ0 (x, a1 ) ∧ φ1 (x, b0 ) ∧ ¬φ1 (x, b1 ) is consistent then T is not dp-minimal.

Proof. By compactness there are mutually indiscernible sequences ai for i ∈ Z and bi for i ∈ Z and c so that |= φ0 (c, a0 ) ∧ ¬φ0 (c, a1 ) ∧ φ1 (c, b0 ), ∧¬φ1 (c, b1 ).

By applying compactness and Ramsey’s theorem we assume that {ai : i 0} and {ai : i 1} are both indiscernible over i∈ω bi ∪ {c} as well as that {bi : i 0} and {bi : i 1} are both indiscernible over i∈ω ai ∪ {c}. Let di = a2i a2i+1 and ei = b2i b2i+1 for i ∈ Z. Note that these two sequences are mutually indiscernible.

Let ψ0 (x, y 0 y 1 ) be φ0 (x, y 0 ) ↔ ¬φ0 (x, y 1 ) and ψ1 (x, y 0 y 1 ) be φ1 (x, y 0 ) ↔ ¬φ1 (x, y 1 ).

Then C |= ψ0 (c, d0 ) ∧ ψ1 (c, e0 ) and if i = 0 then |= ¬ψ0 (c, di ) ∧ ¬ψ1 (c, ei ) by the indiscernibility assumptions. It follows that ψ0, ψ1, {di : i ∈ ω} and {ei : i ∈ ω} witness that T is not dp-minimal.

**Using the identical proof as above we show:**

**Fact 2.8. The following are equivalent:**

(1) There are formulae φi (x, y) for 1 ≤ i ≤ N and sequences ai for 1 ≤ i ≤ N j

**and j ∈ ω so that for every η : {1,..., N} → ω the type:**

is consistent As in the case of the independence property and strong dependence we have a characterization of dp-minimality in terms of splitting indiscernible sequences.

**Fact 2.9. The following are equivalent for a theory T :**

(1) T is dp-minimal.

(2) If {ai : i ∈ I} is an indiscernible sequence and c is an element then there is a partition of I into ﬁnitely many convex sets I0,..., In, at most two of which are inﬁnite, so that for any 1 ≤ l ≤ n if i, j ∈ Il then tp(ai /c) = tp(aj /c).

(3) If {ai : i ∈ I} is an indiscernible sequence and c is an element then there is a partition of I into ﬁnitely many convex sets I0,..., In, at most two of which are inﬁnite, so that for any 1 ≤ l ≤ n the sequence {ai : i ∈ Il } is indiscernible over c.

Proof. (1) ⇒ (3): Suppose T is dp-minimal and for contradiction suppose that there is an indiscernible sequence (which for notational simplicity we assume consists of singletons) {ai : i ∈ I} and an element c witnessing the failure of (3). Without loss of generality we assume that I is a suﬃciently saturated dense linear order without endpoints. By [1, Corollary 6] there is an initial segment I0 ⊆ I and a ﬁnal segment I1 ⊆ I so that the sequences {ai : i ∈ I0 } and {ai : i ∈ I1 } are indiscernible over c. We may choose I0 to be maximal in the sense that for no convex J with I0 ⊂ J ⊆ I is {aj : j ∈ J} indiscernible over c and similarly for I1. If I \ (I0 ∪ I1 ) is ﬁnite then (3) holds so I \ (I0 ∪ I1 ) is inﬁnite and thus contains an interval. Let J0 and J1 be disjoint convex sets so that I0 ⊂ J0 ⊂ I and I1 ⊂ J1 ⊂ I. We ﬁnd ∗ ∗ j1 · · · jn ∈ J0 and a formula φ(x, y1,..., yn ) so that ¬φ(c, ai1,... ain ) holds for all i1 · · · in ∈ I0 but φ(c, aj1,..., ajm ) holds. We choose a sequence of n-tuples di ∗ ∗

(3) ⇒ (2): Immediate.

(2) ⇒ (1). For contradiction suppose there are formulae φ(x, y) and ψ(x, y) and mutually indiscernible sequences {ai : i ∈ 3 × ω} and {bi : i ∈ 3 × ω} which witness the failure of dp-minimality. Such sequences must exist by compactness. For i ∈ 3×ω

**let ci = ai bi. Note this is an indiscernible sequence. Pick d realizing the type:**

φ(x, aω ) ∧ ψ(x, a2×ω ) ∧ ¬φ(x, ai ) ∧ ¬ψ(x, bj ).

i=ω j=2×ω If i = ω or 2 × ω then |= ¬φ(d, ai ) ∧ ¬ψ(d, bi ), if i = ω then |= φ(d, aω ) ∧ ¬ψ(d, bω ) and if i = 2 × ω then |= ¬φ(d, a2×ω ) ∧ ψ(d, b2×ω ). It follows that the indiscernible sequence {ci : i ∈ 3 × ω} can not be decomposed into convex subsets of which at most two are inﬁnite so that the type over d is constant on each convex set. Hence (2) fails.

We use the preceding fact to provide a characterization of dp-minimality which allows us to consider formulae with more than one free variable in a variant of Definition 2.2 as well as to consider sets of parameters of arbitrary size in a variant of Fact 2.9. The proof is immediate using Facts 2.8 and 2.9.

Given a dp-minimal theory T it is reasonable to ask if the bound in Fact 2.10(4) may be improved from 2n to n + 1. This holds in the stable case and we sketch the proof.

** Fact 2.11.**

If T is dp-minimal and stable then there is no sequence of formulae

is consistent.

Proof. (Sketch) By [11, Theorem 3.5] T is stable and dp-minimal if and only if every 1-type has weight 1. Thus if T is stable and dp-minimal every n-type has weight at most n. Apply [11, Lemma 2.3 and Lemma 2.11] and the result follows.

This fact also holds for weakly o-minimal T (we defer discussion of this to section 3).

In fact it is tempting to restate the result with stable replaced by rosy and use þweight as in [12] but at the moment it is not clear if the result follows. Overall we do not know whether these improved bounds hold for a general dp-minimal theory.

We ﬁnish with two facts which are useful in studying speciﬁc examples. The ﬁrst of these is particularly useful when studying theories which admit some type of cell decomposition—viz. the p-adics.

** Fact 2.12.**

Suppose that φ(x, y) and ψ(x, y) are formulae witnessing that a theory T is not dp-minimal. Further suppose that φ(x, y) is φ1 (x, y) ∨ · · · ∨ φn (x, y). Then for some 1 ≤ l ≤ n φl (x, y) and ψ(x, y) witness that T is not dp-minimal.

Proof. There are mutually indiscernible sequences {ai : i ∈ Z} and bi : i ∈ Z} so that for any i∗, j ∗ ∈ Z the type φ(x, ai∗ ) ∧ ψ(x, bj ∗ ) ∧ ¬φ(x, ai ) ∧ ¬ψ(x, bj ) i=i∗ j=j ∗

Thus α realizes θ(x, ci ) and θ(x, dj ). If k = i then α realizes ¬φ(x, ai ) and ¬ψ(x, bi ) and hence realizes ¬θ(x, ci ). Finally if l = j then α realizes ¬θ(x, aω+l ) and ¬θ(x, bω+l ) and hence ¬θ(x, dj ) as desired.

3. Relationship with similar notions In this brief section we examine the relationship between dp-minimality and various other strong forms of dependence.

We begin with the notion of VC-density as studied in [3]. For the ensuing deﬁnition we ﬁx ∆(x, y) a ﬁnite set of formulae where we consider y as the parameter variables. If A is a set of |y|-tuples we write S ∆ (A) for the set of complete ∆-types with parameters from A.

Deﬁnition 3.1. A theory T has VC-density one if for any ﬁnite set of formulae ∆(x, y) there is a constant C so that for any ﬁnite set, A, of |y|-tuples S ∆ (A) ≤ C |A||x|.

For example in [3] it is shown that any weakly o-minimal theory and any quasi o-minimal theory with deﬁnable bounds (for which see [4]) has VC-density one. We

**have a strong relationship between VC-density one and dp-minimality:**

Proposition 3.2. If T has VC-density one then T is dp-minimal.

Proof. Suppose that T is not dp-minimal. Apply Fact 2.13 to ﬁnd a formula φ(x, y) and sequences ai and bj as described there. For N ∈ N let AN = {ai : i ≤ N} ∪ {bj : j ≤ N}.

|AN |2 for all 1 By the failure of dp-minimality we immediately see that S {φ} (AN ) ≥ 4 N. Thus T does not have VC-density one.

Note that the above proof only requires that we have that S ∆ (A) ≤ C |A| for ∆ consisting of formulae of the form φ(x, y), i.e. with only one free variable. We may use this fact to provide a novel proof for the dp-minimality of the theory of algebraically closed valued ﬁelds. This fact was already observed in [2] and as the proof is somewhat technical we do not include it here.

Corollary 3.3. Any weakly o-minimal theory as well as any quasi o-minimal theory with deﬁnable bounds is dp-minimal.

## 8 ALFRED DOLICH, JOHN GOODRICK, AND DAVID LIPPEL

is consistent.

We consider the notion of VC-minimality as introduced in [2].

Deﬁnition 3.5. Fix a theory T and a monster model C |= T. T is VC-minimal if

**there is a family of formulae Φ of the the form φ(x, y) so that:**

• If φ(x, y), ψ(x, y) ∈ Φ and a, b ∈ C then one of:

– φ(C, a) ⊆ ψ(C, b), – ψ(C, b) ⊆ φ(C, a), – ¬φ(C, a) ⊆ ψ(C, b), – ψ(C, b) ⊆ ¬φ(C, a).

• If X ⊆ C then there are a ﬁnite collection of φi (x, y) from Φ and tuples ai ∈ C so that X is a Boolean combination of the sets φi (C, ai ).