Being a proper trapezoid ordered set is a comparability invariant (Q5929973)

A trapezoid ordered set is a finite ordered set for which there exists a standard injective representation in the ordered set of closed trapezoids on common parallel baselines. A property of finite ordered sets is called a comparability invariant if it depends only on the comparability graph. The author proves that if we replace an autonomous subset \(A\) of a finite proper trapezoid ordered set \(P\) with a proper trapezoid ordered set and if \(A\) is not an antichain, then we obtain a proper trapezoid ordered set. An analogous assertion is also true in any \(k\)-dimensional case.