Is there a name or anything else known for a semi-lattice whose Hasse Diagram becomes a tree after applying transitive reduction?
Trying to find more about it since it comes up in an optimization problem (need to find the optimal way to distribute * over + in a sum containing terms with this kind of lattice structure)
For instance consider the following subsets of real line ordered by subset inclusion: $$[0, 5], [1,3], [2,3], [1,2], [3,5]$$
Its Hasse Diagram:
Two gray lines disappear after transitive reduction, so the result is a tree.
I found this paper on semi-lattices whose diagrams are trees, but that's a much more restrictive set, doesn't include my example.
** edit ** the paper doesn't define "diagram", but we assume the same as Hasse Diagram from wikipedia, it sounds like it's saying these semi-lattices would be called "convex semi-lattices"
In general, the only diagrammatic representation for a partially ordered set (poset) one cares for is the Hasse diagram. Drawing more (valid) edges is redundant, as the poset will still be the same and the diagram will be more cluttered. As such, all diagrammatic characterizations of a poset apply to its Hasse diagram.