On dp-minimal ordered structures

Simon, Pierre

doi:10.2178/jsl/1305810758

Cited by 63 publications

(104 citation statements)

References 9 publications

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“…Proof. All we need to repeat the proof from [12] is to prove that all cosets of H contained in C are cofinal in C. So take any c ∈ C and fix h ∈ H. Choose h 1 ∈ H such that h 1 < c. Then, by the left-invariance of the order, h = (hh −1 1 )h 1 < hh −1 1 c ∈ Hc, so Hc is cofinal in C.…”

Section: Inp-minimal Left-ordered Groupsmentioning

confidence: 99%

“…In this section, we prove that every inp-minimal left-ordered group is abelian. For a left-ordered group G and a subset A ⊆ G, by h(A) we will denote the convex hull of A in G. Note that (in contrast to the bi-ordered groups), in a left-ordered group G, the convex hull of a subgroup H need not be a subgroup of G: Nevertheless, in left-ordered groups, h(H) is always a union of right H-cosets, and, as an analogue of Lemma 3.2 from [12], we obtain: Fact 2.3. Let G be an inp-minimal left-ordered group.…”

Section: Inp-minimal Left-ordered Groupsmentioning

confidence: 99%

“…The example from Section 1, as every group definable in the Presburger artithmetic, is abelianby-finite (see [9]). It seems natural to ask the following general question: To apply some ideas from the proof of Theorem 2.6, it seems necessary to prove some variant of the following property, which was essentially observed in the proof of Proposition 3.1 from [12]: Below, we make an observation of this kind in the context of finite inp-rank, but we need to work with normal subgroups.…”

Section: Some Observations On Groups Of Finite Inp-rankmentioning

confidence: 99%

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Some remarks on inp-minimal and finite burden groups

Dobrowolski

Goodrick

2018

Arch. Math. Logic

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Section: Inp-minimal Left-ordered Groupsmentioning

confidence: 99%

Section: Inp-minimal Left-ordered Groupsmentioning

confidence: 99%

Section: Some Observations On Groups Of Finite Inp-rankmentioning

confidence: 99%

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Some remarks on inp-minimal and finite burden groups

Dobrowolski

Goodrick

2018

Arch. Math. Logic

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“…, P -minimal [HM97] or C-minimal [HM94], and more generally if it is a dp-minimal ordered or valued field [Sim11], [JSW17]. The same holds true if K is any Henselian (non-trivially) valued field of characteristic 0, or any algebraically bounded expansion of such [vdD89].…”

Section: (Bfin)mentioning

confidence: 99%

Defining integer-valued functions in rings of continuous definable functions over a topological field

Darnière

Tressl

2020

J. Math. Log.

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Let K be an expansion of either an ordered field (K, ), or a valued field (K, v). Given a definable set X ⊆ K m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P -minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P -minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f ∈ C(X) which take values in Z is definable in C(X).

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“…The final notion of rank discussed in the introduction is dp-rank, which we calculate for type-definable sets in the same way. In particular, if X is type-definable then dp(X) = sup{dp(p) : p |= X} where we set dp(p) to be the supremum over cardinals κ such that the relation "dp(p) ≥ κ" holds, as defined in [23,Chapter 4]. We are justified in avoiding the full definition of dp-rank because of the following fact about stable theories.…”

Section: Preliminariesmentioning

confidence: 99%

Stable groups and expansions of $(\mathbb Z,+,0)$

Conant

Pillay

2018

Fund. Math.

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We show that if G is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U -rank. Combined with recent work of Palacín and Sklinos, we conclude that (Z, +, 0) has no proper stable expansions of finite weight. A corollary of this result is that if P ⊆ Z n is definable in a finite dp-rank expansion of (Z, +, 0), and (Z, +, 0, P ) is stable, then P is definable in (Z, +, 0). In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.Theorem 1.2. If P ⊆ Z n is definable in a finite dp-rank expansion of (Z, +, 0), and (Z, +, 0, P ) is stable, then P is definable in (Z, +, 0).The notion of dp-rank in NIP theories has been an important tool in extending the work of stability theory to the unstable setting (see, e.g., [23]), and so Theorem 1.2 establishes a fundamental fact about the behavior of NIP expansions of (Z, +, 0). The proof of this theorem will be obtained from a more general result on stable

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On dp-minimal ordered structures

Abstract: We show basic facts about dp-minimal ordered structures. The main results are : dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dpminimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.

Cited by 63 publications

References 9 publications

Some remarks on inp-minimal and finite burden groups

Some remarks on inp-minimal and finite burden groups

Defining integer-valued functions in rings of continuous definable functions over a topological field

Stable groups and expansions of $(\mathbb Z,+,0)$

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