The Problem of Fixed Points in Ordered Sets

Publisher Summary This chapter describes fixed-point problem in ordered sets. An ordered set P has the fixed-point property if every order-preserving map of P to itself has a fixed point; otherwise, P is said to be fixed-point free. The chapter discusses the approaches used for several important classes of ordered sets—that is, lattices and ordered sets of length one. Completeness is the best-known result of fixed points concerned with lattices—that is, ordered sets in which every pair of elements has both supremum and infimum. A lattice has the fixed-point property only if it is complete. Retractions is a fixed point property for ordered sets of length one, which states that an ordered set P of length one has the fixed point property only if (1) P is connected, (2) P contains no crowns, and (3) P contains no infinite fence.

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