\(p\)-systems in local Noether lattices
Let \(\mathcal L\) be a local Noether lattice, \(M\) its maximal element. The radical of an element \(A \in {\mathcal L}\) is \(\text{Rad}(A) = V\{X \in {\mathcal L}\); \(X^ s \leq A\) for some integer \(s\)\}. If \(Q \in {\mathcal L}\) and \(\text{Rad}(Q) = M\), then the element \(Q\) is called \(M\)-primary. An element \(A \in {\mathcal L}\) which is not of the greatest element of \(\mathcal L\) is a \(p\)-system in \(\mathcal L\) if for every \(M\)-primary element \(Q' \in {\mathcal L}\) such that \(A \leq Q'\) there exists a meet-irreducible \(M\)- primary element \(Q \in {\mathcal L}\) with \(A \leq Q \leq Q'\). The authors give some characterizations of \(p\)-systems in \(\mathcal L\). One of them is topological using a certain metric given on \(\mathcal L\), another one needs the existence of a special decreasing sequence of meet-irreducible \(M\)- primary elements in \(\mathcal L\). Also, \(p\)-systems in quotient lattices and in the metric completion of \(\mathcal L\) are characterized.
scientific article; zbMATH DE number 697103