Badr, E., Moussa, M. (2010). ON JUMP- CRITICAL ORDERED SETS. The International Conference on Mathematics and Engineering Physics, 5(International Conference on Mathematics and Engineering Physics (ICMEP-5)), 1-8. doi: 10.21608/icmep.2010.29770

E. M. Badr; M. I. Moussa. "ON JUMP- CRITICAL ORDERED SETS". The International Conference on Mathematics and Engineering Physics, 5, International Conference on Mathematics and Engineering Physics (ICMEP-5), 2010, 1-8. doi: 10.21608/icmep.2010.29770

Badr, E., Moussa, M. (2010). 'ON JUMP- CRITICAL ORDERED SETS', The International Conference on Mathematics and Engineering Physics, 5(International Conference on Mathematics and Engineering Physics (ICMEP-5)), pp. 1-8. doi: 10.21608/icmep.2010.29770

Badr, E., Moussa, M. ON JUMP- CRITICAL ORDERED SETS. The International Conference on Mathematics and Engineering Physics, 2010; 5(International Conference on Mathematics and Engineering Physics (ICMEP-5)): 1-8. doi: 10.21608/icmep.2010.29770

^{1}Mathematics and computer Science Department, Faculty of Science, Benha University, Benha, Egypt.

^{2}Faculty of computer & information Benha University, Benha, Egypt.

Abstract

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 xP. 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.