Comparability invariance of the fixed point property.

*(English)*Zbl 0577.06005The comparability graph of an ordered set P is the undirected graph whose vertex set is P and in which two vertices x,y are adjacent if and only if \(x<y\) or \(x>y\). A property of an ordered set is called comparability invariant, if two arbitrary ordered sets with isomorphic comparability graphs either both have this property, or both have not. If P, Q are ordered sets and \(a\in P\), then P(a,Q) is the ordered set obtained from P by replacing a by Q. An ordered set P is said to have the fixed point property, if every order-preserving mapping from P to P has at least one fixed point.

Three theorems are proved. The first of them has an auxiliary character; the further ones are the following. Theorem 2. Let P and Q be finite ordered sets with \(a\in P\). Then P(a,Q) has the fixed point property if and only if either (i) P and Q both have the fixed point property; or (ii) P has the fixed point property, Q does not, and there is no order- preserving map \(f: P\to P\) which fixes only a and sends no element above or below a to a. Theorem 3. The fixed point property is comparability invariant for finite ordered sets.

Three theorems are proved. The first of them has an auxiliary character; the further ones are the following. Theorem 2. Let P and Q be finite ordered sets with \(a\in P\). Then P(a,Q) has the fixed point property if and only if either (i) P and Q both have the fixed point property; or (ii) P has the fixed point property, Q does not, and there is no order- preserving map \(f: P\to P\) which fixes only a and sends no element above or below a to a. Theorem 3. The fixed point property is comparability invariant for finite ordered sets.

Reviewer: B.Zelinka