A quick question: Given a partially ordered set P, it is easy to associate a simplicial complex to it, called the ordered complex of P, which is essentially the family of chains of P. How about the converse? namely, given a simplicial complex, can we define a partial order on the set of vertices of the complex such that the ordered complex of the defined poset is exactly the simplicial complex?
Please provide some references explicitly, thanks.
You can define a partial order on a certain subset of the power set of the set of vertices of the simplicial complex X. The subsets consist of precisely those vertices that span a simplex in X. The partial order is simply given by containment. You may look at the following for details:
http://www.math.cornell.edu/~levine/18.312/alg-comb-lecture-11.pdf
Thank you for your answer, Mahan. But, what I am asking is slightly different, I am asking a partial order on vertices instead of partial order of inclusion of faces. I was thinking if the partial order of inclusion of faces induces a partial order on vertices, but it seems not. Anyway I think the answer should be yes, I am just surprised I can not find any discussion on this. Maybe it just a trivial problem...
Ricky, I think I understand the problem now, but in that case I don't know how you would give a partial ordering to the following: Take a 2-simplex and attach one cells to all pairs of vertices. So (123)is a 2 simplex. You have new edges (12), (23), (31).
To make it into a simplicial complex, you have to introduce new verts 4, 5, 6 in the 3 new edges. Now, out of 1,2,3 one is maxm by your hypothesis. Say 1
Mahan, it is a good example. I will think about it. A quick thought: when adding these additional 1-simplex, should their vertices 4,5,6 be identified with 1,2,3, I will check out.
I correct my conclusion: it is not always possible to give a partial order on the set of vertices. Just look at the simplicial complex {{1},{2},{3},{1,2},{1,3},{2,3}}. Question closed!
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