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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?2010-12-12
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    is there a reason you want to work with abstract simplicial complexes instead of simplicial sets?2010-12-12
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    @Sean Tilson: I'm interested in finite set systems with the property of being closed under the superset operation.2010-12-12
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    so there are more of these than abstract simplicial complexes, right?2010-12-12
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    Right. I'd like to consider them from the point of view of coloring (since they could be interpreted as hypergraphs).2010-12-12

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