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A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation.

What's the name for the following:

$D=(S,D)$ defined as above except that $D$ is closed under the superset operation?

Crossposted from MO.

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An abstract simplicial mplex? – Qiaochu Yuan Dec 12 '10 at 0:02
is there a reason you want to work with abstract simplicial complexes instead of simplicial sets? – Sean Tilson Dec 12 '10 at 1:17
@Sean Tilson: I'm interested in finite set systems with the property of being closed under the superset operation. – Oleksandr Bondarenko Dec 12 '10 at 1:22
so there are more of these than abstract simplicial complexes, right? – Sean Tilson Dec 12 '10 at 1:39
Right. I'd like to consider them from the point of view of coloring (since they could be interpreted as hypergraphs). – Oleksandr Bondarenko Dec 12 '10 at 1:48
up vote 0 down vote accepted

I've found here for simplicial compleces - down closed set system and, respectively, for the objects I asked about - up closed set system.

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