Ordonnés escamotables et points fixes. (Dismantlable orders and fixed points) (Q1074608)

For any ordered set (E,\(\leq)\) the authors consider the set of all selfmappings on E, each preserving the comparability relation \(\sim\) on (E,\(\leq)\), i.e. \(\sim:=\leq \cup \geq\). Inspired by the approach and results of \textit{K. Baclawski} and \textit{A. Björner} [Adv. Math. 31, 263- 287 (1979; Zbl 0417.06002)], they prove theorem 1.1: If E is finite and \(f: E\to E\) preserves \(\sim\), then \(\Lambda (t)=\chi (M_ f)\) (for f increasing [decreasing] this was proved by Baclawski-Björner, loc. cit. Theorems 1.1, 1.2). Theorem 5.2: Every left [right] badly complemented finite (E,\(\leq)\) is contractible and has the f.p.p. (fixed point property); this is a nice generalisation of a nice \textit{K. Baclawski} and \textit{A. Björner} result [J. Comb. Theory, Ser. A 30, 335-338 (1981; Zbl 0459.06004)] that for every finite non complemented lattice L its part \(L\setminus \{0,1\}\) has the f.p.p. Definition: (E,\(\leq)\) is said to be left [right] badly complemented if, for some \(x\in E\) and every \(y\in E\setminus \{x\}\) the set xly [xry] is contractible. For x,y\(\in E\), \(x\neq y\) one puts xly\([xry]:=\{v_{n-1}|\) \(x=v_ 0\geq [\leq]v_ 1\sim v_ 2\sim...\sim v_{n-1}\sim v_ n=y\), n minimal\(\}\).