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) : L linear extension of P}, the jump number of P.
Call an ordered set P is jump-critical if s (P-{x}) < s (P) for any xP. We introduce some theory
about the jump-critical ordered sets with jump number four. Especially, we introduce a complete
list of the jump-critical ordered sets with jump number four ( it has four maximal elements).
Finally, we prove that a k-critical ordered set is a k-tower ( its width is 2, k >1).
KEYWORDS: Jump number, jump-critical ordered sets.