Greedy linear extensions with constraints

AbstractLoosely speaking, a greedy linear extension of an ordered set is a linear extension obtained by following the rule: “climb as high as you can”. Given an ordered set P and a partial extension Ṕ of P is there a greedy linear extension of P which satisfies all of the inequalities of Ṕ? We consider special instances of this question. In particular, we impose conditions bearing on the diagram of an ordered set. Our results have applications, to the ‘jump number scheduling problem’ and to the ‘greedy dimension’