Representing an Ordered Set as the Intersection of Super Greedy Linear Extensions

A linear extension [xt <-x2 < 1.1 <xl] of a finite ordered set Y= (P, <) is super greedy if it can be obtained using the following procedure: Choose x, to be a minimal element of 9; suppose .Y, , . . . , xL have been chosen; define p(x) to be the largest j < i such that x, < x if such a j exists and 0 otherwise; choose x,+i to be a minimal element of P{xt , . . . . x,) which maximizes p. Every finite ordered set d can be represented as the intersection of a family of super greedy linear extensions, called a super greedy realizer of 9. The super greedy dimension of 9 is the minimum cardinality of a super greedy realizer of 9. Best possible upper bounds for the super greedy dimension of 9 are derived in terms of IP Al and width (P-A), where A is a maximal antichain. AMS subject classifications (1980). 06A05.