Cores, cutsets and the fixed point property

Abstract

The purpose of this paper is the analysis and application of the concepts of a core (a pair of chains) and cutset in the fixed point theory for posets. The main results are:

• (1) (Theorem 3) If P is chain-complete and (*), it contains a cutset S such that every nonempty subset of S has a join or a meet in P, then P has the fixed point property (FPP),

• (2) (Theorem 5) If P or Q is chain-complete, Q satisfies (*) and both P and Q have the FPP, then P x Q has the FPP.

• (3) (Theorem 6) Let P or Q be chain-complete and there exist p∈P and a finite sequence f ₁, f ₂, ..., f _n of order-preserving mappings of P into P such that

  $$\left( {\forall x\varepsilon P} \right)x \leqslant f_1 \left( x \right) \geqslant f_2 \left( x \right) \leqslant \cdots \geqslant f_n \left( x \right) \leqslant p$$

  If P and Q have the FPP then P x Q has the FPP.

• (4) (Theorem 7) If T is an ordered set with the FPP and {P _t :t∈T} is a disjoint family of ordered sets with the FPP then its ordered sum ∪{P _t :t∈T} has the FPP.