Examples of Nip Structures

Abstract

We classify primitive, rank 1, ω-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear or circular orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures. * Partially supported by NSF (grant no. 1665491) and a Sloan fellowship. PreliminariesAssumption: Throughout this paper, we work in an ω-categorical structure M. That assumption will in general not be recalled, and is implicitely assumed in all statements. Linear orders and their reductsThere is only one countable homogeneous linear order: (Q, ≤). It is also the only ω-categorical primitive linear order. Its reducts follow from Cameron's result on highly homogeneous permutations groups [Cam76]: there are five of them. Apart from the trivial reduct to pure equality, there are three unstable proper reducts:• the generic betweenness relation (Q; B(x, y, z)• the generic circular order (Q; C(x, y, z)1. rk(a/D) = 0 if and only if a ∈ acl(D).2. rk(D 1 ∪ D 2 ) = max(rk(D 1 ), rk(D 2 )).3. If B 1 ⊆ B 2 , then rk(a/B 1 ) ≥ rk(a/B 2 ) 4. If D is definable over B, then there is a ∈ D such that rk(a/B) = rk(D). 5.We have rk(a/b) ≥ n + 1 if and only if there are a ′ , c ∈ M eq with a ′ ∈ acl eq (abc) \ acl eq (ac) and rk(a/a ′ bc) ≥ n.

On Vapnik‐Chervonenkis density over indiscernible sequences

“…This definition is equivalent to the standard definition of dp-rank using ICT-patterns by [18,Proposition 4.20], which is a simple generalization of [17,Lemma 1.4]. This is also equivalent to the definition of dp-rank using mutually indiscernible sequences, as in [8].…”

“…Similarly, when we speak of indiscernible sequences, we mean indiscernible sequences over ∅. However, in light of [18,Lemma 4.13], this distinction will not be too important, as all of the ranks introduced below will be equal regardless of what parameters the indiscernible sequences are taken over.…”

“…In this section, we borrow notation from [18,Chapters 3 & 4]. Let (I ; <) be a linear order and let compl(I ) denote the completion of I .…”

“…We say that a theory is strongly dependent if, for all partial types p(x), the dp-rank of p is finite (cf. [15] or [18,Section 4.3]). There are theories that are dependent but not strongly dependent.…”