Facets of linear signed order polytopes Original Research Article

Self-reflecting signed orders have been proposed to aid assessment of preferences between subsets of an n-item set {1,2,…,n} by considering desirabilities of excluding as well as including items in a set. A linear signed order for n is a linear order ≻ on the 2n-element set {1,…,n}∪{1∗,…,n∗}, where (x∗)∗=x, which satisfies the self-reflection property x≻y⇔y∗≻x∗. The linear signed order polytope Qn for n is defined in a standard way as a polytope in [0,1]2n(2n−1). It has dimension n2. We note a complete equation system for Qn and specify all facet defining inequalities for n⩽4. Additional classes of facets for larger n that are not induced by a lifting lemma are identified. Comparisons to linear ordering polytopes are included.