\(pm\)-lattices
Let \(L\) be a bounded distributive lattice; \({\mathcal P}\) and \({\mathcal M}\) are respectively the prime spectrum and maximal spectrum of \(L\). If every prime ideal of \(L\) is contained in a unique maximal ideal of \(L\) then \(L\) is called a \(pm\)-lattice. It is shown that in a \(pm\)-lattice the map which sends every prime ideal into the unique maximal ideal containing it is continuous; and this map is the unique retraction of \({\mathcal P}\) onto \({\mathcal M}\). It is established that the \(pm\)-property is equivalent to the normality of \({\mathcal P}\) and it implies that \({\mathcal M}\) is \(T_2\). The authors have recently succeeded in extending all these considerations to semilattices [``Prime, minimal and maximal spectra of a semilattice'' (to appear)]. In fact we pose and partially solve the following problem: Characterize the poset \({\mathcal P}\) of prime semiideals of a meet semilattice. We also enlist the following results that have been obtained since the publication of the paper under question. 1. Let \(L\) be a bounded distributive lattice. Then the following are all equivalent. a) \(L\) is a \(pm\)-lattice. b) Every minimal prime ideal is contained in a unique maximal ideal. c) Any two distinct maximal ideals of \(L\) are separated by disjoint neighbourhoods in \({\mathcal P}\). d) \(L^*\), the dual lattice of \(L\), is a normal lattice. 2. A pseudocomplemented distributive lattice \(L\) is a Stone lattice if and only if \(L^*\) is a \(pm\)-lattice. 3. If in a distributive lattice \(L\) the maximal ideals intersect trivially then \(L\) is a \(pm\)-lattice.
scientific article; zbMATH DE number 3557852