Some remarks on natural orders for combinatorial cubes (Q1116962)

A total order on a structure C is called natural if the restriction of this total order to an arbitrary substructure is of the same type as the order on C. The author presents a new combinatorial interpretation of the natural total orders on combinatorial cubes over finite sets. (An n-dimensional cube over finite set A is the set \(A^ n\) of n-tuples over A.) The easier description of these orders allows him to simplify the proof of the canonizing ordering theorem for combinatorial cubes (main theorem in the paper of \textit{J. Nešetřil}, the author, \textit{V. Rödl} and \textit{B. Voigt} [J. Comb. Theory, Ser. A 40, 394-408 (1985; Zbl 0582.06001)]). A special case of the canonizing ordering theorem is the description of the natural total orders on finite Boolean lattices.