A family of subsets with a "gluing" property Somewhat in line with this previous MathOverflow question:
I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call elements of $F$ "connected" subsets. (Think of them like connected sets in a topological space, in a loose way.) This family has to satisfy two properties:


*

*If any two connected sets have non-empty intersection, their union will be connected. (I call this property "glueability".)

*Singletons are always connected. (This property is too vacuous to deserve a name.)
My question is whether any of you have seen anything like this before, or have any idea how to deal with it. This is a sort of awkwardly vague request to make, but I would be very interested to hear if this structure had arisen in any other contexts.
(As an example of what I've come up with: You can easily prove that the maximal connected sets exactly partition the whole set.) 
 A: It's just an exercise in the transitivity of relations? You have the subsets where a given function F takes a given value: in other words any partition, or any equivalence relation. I wouldn't call this a structure.
Edit: As Joshua has remarked, it is not all the subsets of a given partition-set that need be "connected". But do we not get all examples by taking the P that are "maximal connected", and then taking certain subsets of that including P itself and the singletons in it? The only condition seems to be that there are enough subsets to make P by transitive closure.]
A: I can't really follow the discussion in Charles' answer (perhaps, the question got changed?), but there is a clear way to interpret this structure. 
The maximal connected subsets $X_1,\ldots,X_n$ of a finite set $X$ form a partition of $X$. Each connected set $Y$ must be contained in one of the $X_i$s, and so partitioning into maximal connected subsets is repeated for each $X_i,$ each of their maximal connected subsets, and so on, until one arrives at one-element subsets, which are connected by the second axiom. Thus the  "connected structure" on an $n$-element set $X$ may be recorded by a descending chain in the partition lattice $\Pi_n$ whose $k$th element is the partition of $X$ into "depth $k$ maximal connected sets" (a singleton is depth $m,m+1,m+2,\ldots$ ). A subset $Y\subset X$ is connected $\iff$ $Y$ is a block in one of the partitions in the chain. The chains that arise in this way satisfy a certain non-degeneracy condition.
This indeed looks similar to an "exponential structure" mentioned by Martin Rubey (Stanley, Enumerative combinatorics Def 5.5.1), but there is no postulated relation between the decompositions of different maximal connected subsets (no regularity dictated by "size"), so it is different.
