Classes of lattices (co)generated by a lattice and their global (dual) Krull dimension
✍ Scribed by Toma Albu
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 825 KB
- Volume
- 185
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
✦ Synopsis
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 fight R-modules having (dual) Krull dimension, called in [4] the riyht 9lobal (dual) Krull dimension of R, relies upon the existence of a (co)generator of the category Mod-R of all unital fight 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.