Abstract
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.