On a conjecture about spectral sets
Let Spec\((R)\) denote the set of prime ideals of a commutative ring \(R\) with identity, ordered by inclusion. A partially ordered set is called spectral if it is order isomorphic to Spec\((R)\) for some \(R.\) The authors present contributions to the better understanding of the following conjecture of \textit{W. J. Lewis} and \textit{J. Ohm} [Can. J. Math. 28, 820--835 (1976; Zbl 0313.13003)]: If \((X,\leq)\) is an ordered disjoint union of the posets \((X_{\lambda},\leq_{\lambda})\), \(\lambda \in \Lambda,\) and if \((X,\leq)\) is spectral, then are the \((X_\lambda,\leq_{\lambda})\) also spectral? They study this problem in the following form: Is a \(D\)-component of a spectral set spectral? For their investigations concepts of up-spectral topology and down-spectral topology (resp. up-spectral set and down-spectral set) are introduced and studied. \{Recall that a topological space is called spectral if it is homeomorphic to the prime spectrum of a ring \(R\) equipped with the Zariski topology. A \(D\)-component of an element \(x\) of a poset \((X,\leq)\) is defined to be the intersection of all subsets of \(X\) containing \(x\) that are closed under \(\leq\) and \(\geq.\)\}
scientific article; zbMATH DE number 2092786
scientific article