Fixed point property for 11-element sets
A poset \(P\) is said to have the fixed point property iff each order- preserving map \(f: P\to P\) has a fixed point. An element \(x\in P\) is irreducible in \(P\) iff it has a unique upper cover or a unique lower cover in \(P\). Let \(P\) be a poset and let \(a,b\in P\). Then \(a\) is called retractable (to \(b\)) iff \(a\neq b\) and \((\downarrow a)\backslash \{a\}\subseteq \downarrow b\) and \((\uparrow a)\backslash\{a\}\subseteq \uparrow b\). Main theorems: 1. Let \(P\) be a poset and let \(a\in P\) be retractable to \(b\in P\). Then \(P\) has the fixed point property iff (1) \(P\backslash\{a\}\) has the fixed point property; (2) One of \((\uparrow a)\backslash\{a\}\) and \((\downarrow a)\backslash\{a\}\) has the fixed point property; 2. All posets with 11 elements that have the fixed point property and no retractable point can be described in 8 figures. Finally, the author introduces some lemmas on 11-element posets with no retractable points.
scientific article; zbMATH DE number 549272