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 Conjecture (stating that a countable first-order structure must have infinitely many countable, pairwise non-isomorphic elementary extensions). In this article a connection between ...

A weakly o-minimal structure M = (M,≤,+, . . .) expanding an ordered group (M,≤, +) is called non-valuational iff for every cut 〈C,D〉 of (M,≤) definable in M, we have that inf{y − x : x ∈ C, y ∈ D} = 0. The study of non-valuational weakly o-minimal expansions of real closed fields carried out in [MMS] suggests that this class is very close to the class of o-minimal expansions of real closed fie...

We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.