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.
