On Jump-Critical Ordered Sets with Jump Number Four

Abstract

For an ordered set P and for a linear extension L of P, let s(P,L) stand for the number of ordered pairs (x, y) of elements of P such that y is an immediate successor of x in L but y is not even above x in P. Put s(P) = min {s(P, L): Llinear extension of P}, the jump number of P. Call an ordered set P jump-critical if s(P - {x}) < s(P) for any x ϵ P. We introduce some theorems about the jump-critical ordered sets with jump number four.