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  • 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.)

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  • $\begingroup$ Thanks for the response @Kaind but it doesn't address the question. I'm seeking to collect examples of bounded cocomplete semilattices. $\endgroup$
    – Jim
    Mar 28, 2022 at 18:20
  • $\begingroup$ If it has least upper bounds of all sets, then it's a complete lattice, where the greatest lower bound of a set $A$ is the least upper bound of the set $L$ of lower bounds of $A$. Also, it follows that it's bounded, where the top element is the least upper bound of the whole poset, and the bottom is the least upper bound of the empty set. If that doesn't suit you, then you have to give another definition of these structures. (I'm assuming that what you call a poset is a set with a binary relation which is reflexive, symmetric and transitive.) $\endgroup$
    – amrsa
    Mar 28, 2022 at 19:31
  • $\begingroup$ @amrsa Thanks for your response. By "poset" I mean a set with a binary relation that is reflexive antisymmetric, and transitive, yes. $\endgroup$
    – Jim
    Mar 28, 2022 at 20:55
  • $\begingroup$ Your observation that cocomplete => complete is accurate. See the comment in brackets "To forestall comments" above. However, existence of meets does not imply existence of useful meets. For instance: if the bounded cocomplete semilattice is a sublattice of a powerset, then meets need not correspond to sets intersections. So the fact that limits exist does not a priori mean they are helpful (it depends). $\endgroup$
    – Jim
    Mar 28, 2022 at 21:01
  • $\begingroup$ Yes, I mean antisymmetric, of course; otherwise it would be an equivalence relation. Regarding that notion of useful meets, I suppose you should clearly state what you mean with that. Is it just meets which coincide with intersections, when applied to power sets? I think that even that is not very clear, since a poset may be embedded into a power set in different ways. And notice the comment that co-complete implies complete doesn't apply just to lattices, but to all posets. Unless having least upper bounds of arbitrary sets doesn't actually apply to the empty set (which is also a subset). $\endgroup$
    – amrsa
    Mar 29, 2022 at 7:11

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