Classes of lattices (co)generated by a lattice and their global (dual) Krull dimension Original Research Article

The purpose of this paper is to extend some of the results of [4] from modules to classes of upper continuous modular lattices which satisfy a certain generation resp. cogeneration property. The condition satisfied by a module generated by another module can be easily reformulated in a latticial setting [1], which is extended in the present paper to arbitrary posets, and further dualized in a very natural manner in order to define the general concept of a poset (co)generated by another poset. The existence of the supremum of the (dual) Krull dimensions of all right R-modules having (dual) Krull dimension, called in [4] the right global (dual) Krull dimension of R, relies upon the existence of a (co)generator of the category Mod-R of all unital right R-modules. This lead us to consider classes of posets that are (co)generated by a poset and to define and investigate their global (dual) Krull dimension, which are then very easily applied to Grothendieck categories.