The fixed point property for small sets (Q1262328)

An ordered set P has the fixed point property if every order-preserving self-mapping of P has a fixed point. An element \(x\equiv P\) is irreducible in P if it has unique upper cover or unique lower cover. Main theorem: There exist exactly eleven (up to isomorphism and duality) ordered sets of size \(\leq 10\) with a fixed point property and containing no irreducible elements. These are the singleton and ten sets which the author exactly describes by figures.