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 mini- mality 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 struc- ture 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