The interval inclusion number of a partially ordered set

Abstract

AbstractA containment representation for a poset P is a map ƒ such that x<y in P if and only if ƒ(x) ⊂ ƒ(y). We introduce the interval inclusion number (or interval number) i(P) as the smallest t such that P has a containment representation f in which each f(x) is the union of at most t intervals. Trivially, i(P)=1 if and only if dim(P)⩽2. Posets with i(P)=2 include the standard n-dimensional poset and all interval orders; i.e. posets of arbitrarily high dimension. In general we have the upper bound i(P) ⩽ ⌈;dim(P)2⌉, with equality holding for the Boolean algebras. For lexicographic composition, i(P) = k and dim(Q) = 2k + 1 imply i(P[Q]) = k + 1. This result and i(B2k) = k imply that testing i(P) ⩽ k for any fixed k ⩾ 2 is NP-complete. Concerning removal theorems, we prove that i(P − x) ⩾ i(P) − 1 when x is a maximal or minimal element of P, and in general i(P − x) ⩾ i(P)2