Representing an ordered set as the intersection of super greedy linear extensions

A linear extension [x₁<x₂<...<x_t] of a finite ordered set P=(P, <) is super greedy if it can be obtained using the following procedure: Choose x₁ to be a minimal element of P; suppose x₁,..., x_i have been chosen; define p(x) to be the largest j≤i such that x_j<x if such a j exists and 0 otherwise; choose x_i+1 to be a minimal element of P-{x₁,..., x_i} which maximizes p. Every finite ordered set P can be represented as the intersection of a family of super greedy linear extensions, called a super greedy realizer of P. The super greedy dimension of P is the minimum cardinality of a super greedy realizer of P. Best possible upper bounds for the super greedy dimension of P are derived in terms of |P-A| and width (P-A), where A is a maximal antichain.