On the greedy dimension of a partial order (Q802583)

Let \(\prec\) be a linear extension of a finite partially ordered set (P,\(\leq)\). \(\prec\) can be written in form \(C_ 1\prec C_ 2\prec...\prec C_ m\) where \(C_ i\) are maximal chains in (P,\(\leq)\). This extension is called greedy iff it has the property: min \(C_ j\) covers max \(C_ i\) in (P,\(\leq)\Rightarrow\) there exists \(x\in P- \cup^{i}_{1}C_ k\) with \(x<\min C_ j\) [\textit{O. Cogis} and \textit{M. Habib}; RAIRO, Inf. Théor. 13, 3-18 (1979; Zbl 0413.05013)]. The authors define the greedy dimension \(\dim_ gP\) of P as the minimal number of greedy linear extensions whose intersection is \(<\). Trivially, \(\dim_ gP\geq \dim P\); they show that equality holds in case when P is N-free (its Hasse diagram contains no ''N''), when dim P\(=2\) and when P is a distributive lattice.