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?
Best Answer
If you are making some research on this topic, you can come up with the terminology you think makes better sense to you, preferably without crashing with existing one.
I think you could call these complete semi-lattices.
While some would argue that a complete semi-lattice is just the algebraic reduct of a complete lattice, that can be disputed on the ground that it just have to be complete relative to the relevant operation.
There is a paper by Mai Gehrke and Hilary Priestley where they study a hierarchy of completions of posets:
Canonical extensions and completions of posets and lattices, in Reports on Mathematical Logic, 43 (2008), pages 133-152.
There they introduce concepts such as directed join completion, and duals, and although I only read it some ten years ago, I think they have the concept of completion of a poset which is not a complete lattice, unless that poset was already a lattice. It might worth reading, at least to justify the use of the expression "complete" in this context.