On poset similarity

Abstract

Let P be a poset, and let A be an element of its strict incidence algebra. Saks (SIAM J. Algebraic Discrete Methods 1 (1980) 211–215; Discrete Math. 59 (1986) 135–166) and Gansner (SIAM J. Algebraic Discrete Methods 2 (1981) 429–440) proved that the kth Dilworth number of P is less than or equal to the dimension of the nullspace of A k , and that there is some member of the strict incidence algebra of P for which equality is attained (for all k simultaneously). In this paper we focus attention on the question of when equality is attained with the strict zeta matrix, and proceed under a particular random poset model. We provide an invariant depending only on two measures of nonunimodality of the level structure for the poset that, with probability tending to 1 as the smallest level tends to infinity, takes on the same value as the inequality gap between the width of P and the dimension of the nullspace of its strict zeta matrix. In particular, we characterize the level structures for which the width of P is, with probability tending to 1, equal to the dimension of the nullspace of its strict zeta matrix. As a consequence, by the Kleitman–Rothschild Theorem 5, almost all posets in the Uniform random poset model have width equal to the dimension of the nullspace of their zeta matrices. We hope this is a first step toward a complete characterization of when equality holds in Saks’ and Gansner's inequality for the strict zeta matrix and for all k. New to this paper are also the canonical representatives of the poset similarity classes (where two posets are said to be similar if their strict zeta matrices are similar in the matrix-theoretic sense), and these form the setting for our work on Saks’ and Gansner's inequalities. (Also new are two functions that measure the nonunimodality of a sequence of real numbers.)