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To rephrase your question, what conditions are necessary and sufficient for a simplicial complex to be an order complex?

There are also a few easy necessary conditions. For one, any simplicial complex is the Stanley-Reisner complex of a square-free monomial ideal (label each vertex with a variable, and the minimal non-faces in the simplicial complex are exactly the monomial generators of the ideal.) For all order complexes, their Stanley-Reisner ideal is an edge ideal (i.e. a square free monomial ideal generated in degree 2, called an "edge ideal" because it can be thought of as corresponding to a graph G with an edge for each generator.) This is immediate, because a minimal "non-face" in the order complex is a pair of incomparable elements, so all generators must be of degree 2. This does quickly cut down on the types of simplicial complexes to consider.

Unfortunately, having a 2-generated SR-ideal is also not sufficient. There are numerous forbidden minors subgraphs of a graph which will prevent the Stanley-Reisner complex of its edge ideal from being an order complex. For example, if the graph has an induced cycle of length longer than 7, the complex can't arise as an order complex.

I was working a few months ago on trying to classify the structures in graphs which would prohibit their edge ideals from having SR-complexes which were order complexes, but found the other forbidden structures weren't very easy to characterize. I'd love to see some more answers to this question as well!

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To rephrase your question, what conditions are necessary and sufficient for a simplicial complex to be an order complex?

There are also a few easy necessary conditions. For one, any simplicial complex is the Stanley-Reisner complex of a square-free monomial ideal (label each vertex with a variable, and the minimal non-faces in the simplicial complex are exactly the monomial generators of the ideal.) For all order complexes, their Stanley-Reisner ideal is an edge ideal (i.e. a square free monomial ideal generated in degree 2, called an "edge ideal" because it can be thought of as corresponding to a graph G with an edge for each generator.) This is immediate, because a minimal "non-face" in the order complex is a pair of incomparable elements, so all generators must be of degree 2. This does quickly cut down on the types of simplicial complexes to consider.

Unfortunately, having a 2-generated SR-ideal is also not sufficient. There are numerous forbidden minors of a graph which will prevent the Stanley-Reisner complex of its edge ideal from being an order complex. For example, if the graph has an induced cycle of length longer than 7, the complex can't arise as an order complex.

I was working a few months ago on trying to classify the structures in graphs which would prohibit their edge ideals from having SR-complexes which were order complexes, but found the other forbidden structures weren't very easy to characterize. I'd love to see some more answers to this question as well!