\({\mathcal F}\)-multipliers and the localization of distributive lattices
The paper under review introduces the notion of a topology for bounded distributive lattices, modeled after the concept of a Gabriel topology on a commutative ring with 1. If L is such a lattice and \({\mathcal F}^ a \)topology, the localization lattice \(L_{{\mathcal F}}\) of L with respect to \({\mathcal F}\) is constructed in a way formally analogous to localization in (commutative) ring theory. The key device is that of an \({\mathcal F}\)- multiplier which is a clever generalization of a concept studied - in the context of lattices - by \textit{W. H. Cornish} [J. Algebra 32, 339-355 (1974; Zbl 0318.06016)] and then by the reviewer [Houston J. Math. 6, 401-425 (1980; Zbl 0501.06008), and Contributions to lattice theory, Colloq. Math. Soc. János Bolyai 33, 675-696 (1983; Zbl 0526.06009)]. In its simplest form, a multiplier on L is just a map \(f: L\to L\) satisfying \(f(x\wedge y)=f(x)\wedge y\). The author shows that by taking suitable topologies \({\mathcal F}\) on L, well-known extensions of L arise as \(L_{{\mathcal F}}\), e.g. the so-called lattice of fractions of L [cf. \textit{A. Brezuleanu} and \textit{R. Diaconescu}, Rev. Roum. Math. Pures Appl. 14, 311-323 (1969; Zbl 0183.301)], or the maximal lattice of quotients Q(L) studied by the reviewer (loc. cit.). The author also demonstrates that his concept of a topology on L is, in a way, the ''correct'' analogue of a Gabriel topology: Given a commutative ring R with 1, let LR be the (bounded, distributive) lattice of compact open sets of Spec R. \textit{H. Simmons} [J. Algebra 66, 169-192 (1980; Zbl 0462.13002)] constructs a canonical map \(\ell: R\to LR\), the reticulation of R. The author now shows that Gabriel topologies on R are taken to topologies on L by \(\ell\), and that \(\ell\) commutes with localization in important cases. Finally, \({\mathcal F}\)-localization is studied in the context of Łukasiewicz and Post algebras and related to the notion of a lattice of quotients introduced by the reviewer (loc. cit.).
scientific article; zbMATH DE number 3970812
scientific article