Let T be a complete, countable, first-order theory with a finite number of countable models. Assuming that dcl(∅) is infinite we show that T has the strict order property.

We study semi-isolation as a binary relation on the locus of a complete type and prove that—under some additional assumptions—it induces the strict order property.

We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.