# What's the name for a dual of an abstract simplicial complex

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