On Jump-Critical Posets with Jump-Number Equal to Width

We investigate the structure of m-jump-critical posets P with w(P) = m. We prove that the size of such posets satisfies |P| 3m ². For the special when the maximum antichain occurs as the maximal (or minimal) elements, we have the sharp upper bound |P| 3m - k; where k = min {|{Max}(P)|, |{Min}(P)|}. We give examples of posets which illustrate the explored structure of these m-jump-critical posets P with width m.