- Call an ordered set a poset when the ordering is transitive and antisymmetric.
- Call a poset bounded when it has a top and a bottom element (i.e. a greatest and least element).
- Call a poset cocomplete when it has least upper bounds of arbitrary (possibly infinite) sets of elements.
Question: What applications and examples can people point me towards, of bounded cocomplete posets? Are there known canonical (or exotic) examples of these structures?
(To forestall comments: I know that a cocomplete lattice is also complete -- just take the join of all the lower bounds -- but in the context I'm interested in, the meets thus formed are not particularly natural.)