I have a list of sets that I would like to sort into a partial order based on the subset relation.

In fact, I do not require the complete ordering, only the minimal elements.

If I am not mistaken, each minimal elements should define one separate component of the respective graph - and this component should be a meet-semilattice.

What would be the most convenient space and time efficient way to solve this problem? Perhaps there is a way that does not require to build the entire graph? Perhaps there is a known algorithm under a better terminology than what I have naively described above?

I am aware that the time and space requirements are underspecified above, but I would be happy about any suggestions, whether they are proven to be optimal or not...

**Background**: I am currently building an entire graph database that holds all the edges between the sets and then look for the nodes that have no generalizations, but this is quite complicated, slow and requires a lot of (disk) space. The list mentioned above contains *roughly 100 million sets*.