Facets of order polytopes

The order polytopes we consider here are the linear order polytope, the interval order polytope, the semiorder polytope and the partial order polytope. Among their known facet defining inequalities (FDIs), many have their coefficients in {-1, 0, 1}. We consider the problem of finding all of these particular FDIs. The problem is easy for the partial order polytope. For the interval order polytope, we prove that the solution consists of the so-called io-clique inequalities of Müller and Schulz [5]. We present a characterisation of the {-1, 0, 1}-FDIs for the semiorder polytope. The similar problem for the linear order polytope remains open and seems harder to solve because of the large variety of known examples of FDIs which fall in this class.