Some questions concerning minimal structures

An infinite first-order structure is minimal if its each definable subset is either finite or co-finite. We formulate three questions concerning order properties of minimal structures which are motivated by Pillay's Con- jecture (stating that a countable first-order structure must have infinitely many countable, pairwise non-isomorphic elementary extensions). In this article a connection between articles (7) and (8) is explained in order to motivate some questions concerning minimal, first-order structures which I could not answer. On the way, a minor gap which appeared in (8) will be fixed; thanks to Enrique Casanovas for pointing it out to me. The original motivation for this work comes from Pillay's work on countable elementary extensions of first-order structures. If M0 =( M0 ,... ) is a countable first-order structure and M0 ≺ M1, M0 ≺ M2 then we say that M1 and M2 are isomorphic over M0 if there is an isomorphism between them fixing M0 pointwise. Pillay's Conjecture. Any countable first-order structure M0 has infinitely many countable elementary extensions which are pairwise non-isomorphic over M0. There are a few results partially confirming Pillay's Conjecture. The initial result of Pillay's is in (3) where he proved that there are at least four nonisomorphic countable elementary extensions of M0. There he also reduces the general case to the case when M0 is minimal and has small theory (|S(M0)| = ℵ0); recall that an infinite first-order structure is minimal if its each definable (possibly with parameters) subset is either finite or co-finite. By a well known result of Baldwin and Lachlan, see (1), any countable strongly minimal structure has infinitely many countable pairwise non-isomorphic elementary extensions, so the conjecture is true for strongly minimal structures (a minimal structure is strongly minimal if the minimality is preserved in all elementarily equivalent structures). Belegradek in (2) found a pattern for constructing minimal, but not strongly minimal structures; other examples of such structures are (ω, <) and (ω + ω ∗ ,< ) (where ω ∗ is reversely