On Minimal Ordered Structures

We partially describe minimal, first-order structures which have a strong form of the strict order property. An infinite first-order structure is minimal if its each definable (possibly with parameters) subset is either finite or co-finite. It is strongly minimal if the minimality is preserved in elementarily equivalent structures. While strongly minimal structures were investigated more closely in a number of papers beginning with [4] and [1], there are a very few results on minimal but not strongly minimal structures. For some examples see [2] and [3]. In this paper we shall consider minimal, ordered structures. A first-order structure M0 = (M0, . . . ) is ordered if there is a binary relation < on M0, which is definable possibly with parameters from M0, irreflexive, antisymmetric, transitive and has arbitrarily large finite chains. We usually distinguish (one) such relation by absorbing the involved parameters into the language and assuming that < is an interpretation of a relation symbol from the language, in which case we write M0 = (M0, <, . . . ). Two basic examples of minimal, ordered structures are (ω, <) and (ω + ω∗, <) (where ω∗ is reversely ordered ω and + denotes the (ordinal) sum of partial orders). We can modify a basic example by replacing the original order by a new one, so that the structure remains minimal, ordered. For example, we can change < by taking a finite set of elements from the domain and rearranging them arbitrarily, leaving the order of the other elements unchanged; e.g. we can take 0 ∈ ω and make it bigger than all other elements, or incompatible to the others. Also, we can simply reverse the original order. Note that the ’new’ order obtained in either way is inter-definable with the original one, so that the structure remains unchanged. Further, we can enlarge basic structure as follows: starting with (ω, <) we can replace each n ∈ ω by some (large enough) finite set L(n) and define for a ∈ L(n) and b ∈ L(m): a < b iff n < m. For example (details are left to the reader): 1991 Mathematics Subject Classification: Primary 03C50; Secondary 03C15. The author is supported by Ministry of Science and Technology of Serbia.