On ordered minimal structures

Recall that a structure M is minimal if every definable (with parameters) subset of M is either finite or cofinite. The study of minimal partially ordered structures was initiated by Tanović in [4, 5]. The results from [4, 5] were essential in the proof by Tanović of the Pillay’s conjecture [1], which states that every countable structure in a countable language has infinitely many nonisomorphic, countable extensions. The analysis of minimal ordered structures continued in [6] and produced further results. An example is a partial answer to Kueker’s Conjecture, which states that if a theory T is not א0-categorical and its every uncountable model is א0-saturated, then T is א1-categorical. In [3] the conjecture is proved under the additional assumption that T has the NIP property and dcl(∅) is infinite. Another example is a reduction of Podewski’s Conjecture which states that every minimal field is algebraically closed. The paper [7] proves that this reduces Received 21 November 2013 The second author is supported by the Polish Government grant no. N201 545938 36 GRZEGORZ JAGIELLA AND LUDOMIR NEWELSKI to showing that there are no minimal partially ordered fields of characteristic 0 such that the order has an infinite chain. The results of Tanović justify our interest in minimal ordered structures as an independent object of study. In this paper we deal with pure minimal partially ordered structures, that is minimal structures of the form (M,<) where < is a partial order. In this situation it is natural to expect that in the nontrivial case, where < has arbitrarily long chains (subsets of M linearly ordered by <), (M,<) should resemble a linear ordering. Despite the simple setup it is not easy to obtain a full description of such structures. We obtained some partial results in this direction. In Theorem 6 we prove that if (M,<) is minimal then both its lower and upper parts L = (L(M), < ),U = (U(M), <) (defined below) are minimal and stably embedded in M . In Theorem 9 we show that M is almost linear if and only if L or U is almost linear. We begin by setting up the notation and recalling the fundamental results from [4, 5]. All structures in the paper are models of an arbitrary first-order theory in the language L = {<} where < is interpreted as an ordering. Given a set of parameters A we write L(A) for the set of formulas in L with parameters from A. By definable sets we mean sets definable with parameters. Likewise, an interpretable structure also means interpretable with parameters. Definition 1. Let (M,<) be a minimal ordered structure. Let p be the unique non-algebraic type over M . We define L(M) = {m ∈M : (m < x) ∈ p}, U(M) = {m ∈M : (m > x) ∈ p}, I(M) = {m ∈M : (m ⊥ x) ∈ p}. The following result appears in [4, Proposition 1.1, Proposition 1.2] and [5, Theorem 2]. Theorem 2. Let (M,<) be a minimal ordered structure with an infinite chain. Then