The fixed point property in ordered sets of width two (Q1095161)

The authors study the fixed point property in partially ordered sets of width two (every antichain contains at most two elements). Four-crown towers and belfries are defined, where a four-crown tower is a lexicographic sum of two-element antichains over a well-ordered index- set, and a belfry is - loosely speaking - a four-crown tower with a top to it. Theorem 1. Let P be an ordered set of width two. (1.1) If P has the fixed point property, then it contains a belfry (in P). (1.2) If P is chain- complete, then the following statements are equivalent: (a) P has the fixed point property, (b) P contains a belfry (in P), (c) no four-crown tower in P is a retract of P, (d) P is dismantlable. - Theorem 2. Let P be an ordered set of width two. If P or Q is chain-complete (Q is a poset of arbitrary width), and both P and Q have the fixed point property, then \(P\times Q\) has the fixed point property.