ISSN:
1572-9273
Keywords:
06A10
;
Ordered set
;
chain
;
core
;
cutset
;
fixed point
;
chain-completeness
;
dismantlability
;
ordered sum
;
fixed point property
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00400289
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