Some remarks on inp-minimal and finite burden groups

We prove that any left-ordered inp-minimal group is abelian and we provide an example of a non-abelian left-ordered group of dp-rank 2. Furthermore, we establish a necessary condition for group to have finite burden involving normalizers of definable sets, reminiscent of other chain conditions for stable groups. 0 Introduction and preliminaries One of the model-theoretic properties that gained a lot of interest recently is dp-minimality, which, on one hand, significantly strengthens NIP, and on the other hand, is satisfied by all strongly minimal theories, all (weakly) o-minimal theories, algebraically closed valued fields (more generally, by all C-minimal structures), and the valued field of p-adics. Several interesting results were obtained for dp-minimal structures in the algebraic contexts of groups and fields (sometimes with additional structure), see for example [12, 5, 4, 8]. Throughout this note, we work in the context of a complete first-order theory T , and “formula” means a first-order formula in the language of T . We recall some key definitions, which are originally due to Shelah [11], though the precise form of the definitions which we give below seems to come from Usvyatsov [13]. Definition 0.1. 1. An inp-pattern of depth κ (in the partial type π(x)) is a sequence 〈φi(x; yi) : i < κ〉 of formulas and an array {aij : i < κ, j < ω} of parameters (from some model of T ) such that: (a) For each i < κ, there is some ki < ω such that {φi,j(x; ai,j) : j < ω} is ki-inconsistent; and (b) For each η : κ → ω, the partial type π(x) ∪ {φi(x; ai,η(i)) : i < κ} is consistent. 2. The inp-rank (or burden) of a partial type π(x) is the maximal κ such that there is an inppattern of depth κ in π(x), if such a maximum exists. In case there are inp-patterns of depth λ in π(x) for every cardinal λ < κ but no inp-pattern of depth κ, we say that the inp-rank of π(x) is κ−. 3. The inp-rank of T is the inp-rank of x = x, and T is inp-minimal if its inp-rank is 1. ∗The first author was supported by Samsung Science Technology Foundation under Project Number SSTFBA1301-03, and by European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 705410