On a conjecture about spectral sets

Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Call a partial ordered set spectral if it is order isomorphic to (Spec(R),⊆) for some R. A longstanding open question about spectral sets (since 1976), is that of Lewis and Ohm [Canad. J. Math. 28 (1976) 820, Question 3.4]: "If (X,≤) is an ordered disjoint union of the posets (X_λ,≤_λ), λ∈Λ, and if (X,≤) is spectral, then are the (X_λ,≤_λ) also spectral?".Let (X,≤) be a poset and x∈X. Recall that the D-component of x is defined to be the intersection of all subsets of X containing x that are closed under specialization and generization (i.e., under ≤ and ≥). Let (X,≤) be a spectral set which is an ordered disjoint union of the posets (X_λ, ≤_λ), λ∈Λ. It is clear that (X_λ,≤_λ) is a disjoint union of D-components of X. Thus the conjecture of Lewis and Ohm is equivalent to the following question: "Is a D-component of a spectral set spectral?"This paper deals with topological properties of a D-component of a spectral set, improving the understanding of the conjecture of Lewis and Ohm. The concepts of up-spectral topology and down-spectral topology are introduced and studied.