ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 05, 06 Smoluk, Antoni, Wroclaw University of Economics, Poland Fix-point property of finite partially ordered sets Partially ordered set (poset) X has fixed point property if for every increasing function f : X → X there is a fix-point. Rival has stated the following problem: Characterize the finite ordered sets with fixed point property. Increasing function f is called invariant if X and image f(X) simultaneously have or have not fixed point property. Such image will be also called invariant. All invariant images are ordered by relation of imbedding (isomorphic sets are identified). Let C_{0} = {1} and p, q are different prime numbers; a crown

C_{k+1} = {p, p^{2}q, …, p^{k+1}q^{k}, q, q^{2}p, …, q^{k+1}p^{k}}

is 2(k + 1)-point set of natural numbers ordered by relation of divisibility. Only set C_{0} has the fixed point property. Main result: Poset X has fixed point property if, and only if its minimal invariant image is C_{0}. Minimal invariant image is alw ays unique and a crown. Reference: I. Rival, Unsolved Problems, Order 1(1984), 103-105.