Tame Parts of Free Summands in Coproducts of Priestley Spaces
نویسندگان
چکیده
It is well known that a sum (coproduct) of a family {Xi : 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 Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully understood. In this article we study a certain dense subset X u ⊆ Xu satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the Xi’s. 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 Xi. In this case, all the Xu’s with u free turn out to be isomorphic under the Continuum Hypothesis.
منابع مشابه
Configurations in Coproducts of Priestley Spaces
Let P be a configuration, i.e., a finite poset with top element. Let Forb(P ) be the class of bounded distributive lattices L whose Priestley space P(L) contains no copy of P . We show that the following are equivalent: Forb(P ) is first-order definable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes membership in Forb(P ...
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