# How to name a poset in which all nonempty subsets have a supremum?

Let $$(X,\leq)$$ be a partially ordered set such that each nonempty subset of $$X$$ has a supremum in $$X$$.

I have two related questions regarding such structures as $$(X,\leq)$$:

Question 1: Is $$(X,\leq)$$ equivalent to being a directed-complete partial order that is also a join-semilattice?

Since any nonempty subset of $$X$$ has a supremum, this in particular holds for any nonempty finite subset of $$X$$, and so it is a join-semilattice.

Let $$U$$ be an upward directed set in $$X$$. Then it is a nonempty subset of $$X$$, and so it has a supremum. It follows that $$(X,\leq)$$ is a directed-complete partial order.

Is this reasoning correct?

As the converse also holds, a partially ordered set in which all nonempty sets have a supremum must be equivalent to a directed-complete partial order that is also a join-semilattice.

Question 2: How to name such a poset in which all nonempty subsets have a supremum?

The structure of $$(X,\leq)$$ is dual to that of posets in which all nonempty subsets have an infimum. The latter is equivalent to bounded completeness, and according to Wikipedia "there is no common name for the dual property" of bounded completeness, i.e., for the structure of $$(X,\leq)$$ considered here. So, what would be a good name for the structure of $$(X,\leq)$$? Is there a better name than a directed-complete partial order (a.k.a. dcpo) that is also a join-semilattice?

• Complete lattice? Aug 8, 2021 at 9:11
• @Berci I think that is strictly stronger. In a complete lattice also the empty set has a supremum, which is the least element of the lattice. But I do not require the existence of a least element. For example, take for $X$ the collection of all nonempty subsets of the set $\{1,2\}$ and set inclusion as ordering, then each nonempty subset of $X$ has a supremum, but $X$ has no least element.
– Bart
Aug 8, 2021 at 9:38
• @Ali Do I understand correctly that a suplattice from a categorical prespective is like a complete lattice, except that the morphisms are different? In my (perhaps ignorant) way of thinking I only look at objects, not morphisms, so it's basically the same? Sorry, my understanding of category theory is very limited.
– Bart
Aug 8, 2021 at 10:04