About the properties of a type of inclusion structure

Question

I am interested in learning about the properties of a particular type of set that is relevant to my (non-mathematical) research. I apologize for my totally informal language below (I can read answers in set notation but it's a bad idea for me to try to write with it):

Elements of the set have inclusion relations, they can either overlap (they have an intersection that is less than either element); or one can include the other (they have an intersection that is equal to one element).

There are three clear axioms:

1. Almost every element of the set overlaps with at least one other element.
2. For every pair of overlapping elements, there is an element that exactly includes this pair (there is a "union element" for every such pair).
3. The exception to 1. is that there is a single element that is only a "union element" and that overlaps with no one (and, from 2., therefore is included by no one).

The best that I understand is that this is a specific type of partially-ordered set but that's a pretty general category. It seems very similar to a topological set but it requires unions rather than intersections. Is it familiar to anyone? Or is it nonsense?

If this is a familiar structure, I'd like to know if there are proofs for a few properties that I think must be true; primarily, that every element will have a unique upward and downward inclusion substructure, and that any subset of non-overlapping elements (excluding the Axiom 3 "top" element) will have a common upward including element.

Thanks!

Answer 1

Instead of "almost every", do you mean "all but one" element p?
How can p be a union element if it does not include anything?

You could have a lattice with an order isolated point p or
a downward directed join semilattice and that isolated p.

A downward and upward directed porder and isolated p has
the required properties. That shows if you want a join
semilattice, you will have to assume it.

Answer 2

Is this a correct statement of your problem?

Let K be a collection of subsets of a set S; K subset P(S).

A overlaps B when not empty A $\cap$ B proper subset A,B.

For all multipoint A in K, if A /= S,
. . then some B in K overlaps A.
For all overlaping A,B in K, A $\cup$ B in K.
S in K.