The study of sup lattices teaches us the important distinction between the algebraic part of the structure (in this case suprema) and the coincidental part of the structure (in this case infima). While a sup lattice happens to have all infima, only the suprema are part of the algebraic structure.
Extending this idea, we look at posets that happen to have all suprema (and therefore all infima), but we will only declare some of them to be part of the algebraic structure (which we will call joins). We find that a lot of the theory of complete distributivity for sup lattices can be extended to this context. There are a lot of natural examples of completely join-distributive partial lattice complete partial orders, including for example, the lattice of all equivalence relations on a set X, and the lattice of all subgroups of a group G. In both cases we define the join operation as union. This is a partial operation, because for example, the union of subgroups of a group is not necessarily a subgroup. However, sometimes it is, and keeping track of this can help with topics such as the inclusion-exclusion principle.
Another motivation for the study of sup lattices is as a simplified model for the study of presheaf categories. The construction of downsets is a form of the Yoneda embedding, and the study of downset lattices can be a useful guide for the study of presheaf categories. In this context, partial lattices can be viewed as a simplified model for the study of sheaf categories.
Keywords: Partial Sup Lattice, Complete Distributivity
2010 MSC: 18A25
Theory and Applications of Categories, Vol. 30, 2015, No. 9, pp 305-331.