Tame parts of free summands in coproducts of Priestley spaces

It is well known that a sum (coproduct) of a family { X i : i ∈ I } of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces X u , indexed by the ultrafilters u on the index set I. The nature of those subspaces X u indexed by the free ultrafilters u is not yet fully understood. s article we study a certain dense subset X u ∂ ⊆ X u satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the X i ʹs. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a Čech–Stone compactification of the disjoint union ⋃ I X i . In this case, all the X u ʹs with u free turn out to be isomorphic under the Continuum Hypothesis.