@article { author = {Badr, E. and Moussa, M.}, title = {ON JUMP- CRITICAL ORDERED SETS}, journal = {The International Conference on Mathematics and Engineering Physics}, volume = {5}, number = {International Conference on Mathematics and Engineering Physics (ICMEP-5)}, pages = {1-8}, year = {2010}, publisher = {Military Technical College}, issn = {2636-431X}, eissn = {2636-4328}, doi = {10.21608/icmep.2010.29770}, abstract = {ABSTRACTFor an ordered set P and for a linear extension L of P, Let s (P,L) stand for the number ofordered pairs (x, y) of elements of P such that y is an immediate successor of x in L but y is noteven 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 theoryabout the jump-critical ordered sets with jump number four. Especially, we introduce a completelist 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.}, keywords = {}, url = {https://icmep.journals.ekb.eg/article_29770.html}, eprint = {https://icmep.journals.ekb.eg/article_29770_276af8603e30a55a39d900a2903434f0.pdf} }