Lattices of fractions and flat morphisms of bounded distributive lattices

Document Type : Original Article


University of Bucharest, Faculty of Mathematics and Computer Science, Bucharest, Romania


The lattices of fractions were introduced by Brezuleanu and Diaconescu in 1969. They used this concept in order to construct a Grothendieck - style duality for the category D_{01} of bounded distributive lattices. Then the lattices of fractions are studied in connection with other themes in lattice theory: lattices schemas, localization of bounded distributive lattices, sheaf representations of normal lattices,etc. This paper continues this research vein. We relate the lattices of fractions to flat lattice morphisms, patch and flat topologies on the spectra of bounded distributive lattices, conormal and Stone lattices, etc. We define the flat morphisms of D_{01} in terms of the residuation operation existing in the frames of lattice ideals. We study how the lattices of fractions preserve the flatness property of morphisms. Two characterization theorems of flat and patch topologies are proved. The lattices of fractions are used for obtaining new characterizations of conormal and Stone lattices.