Abstract
It is proved that if we replace an autonomous subset of a finite proper trapezoid ordered set with a proper trapezoid ordered set, then we obtain a proper trapezoid ordered set provided the autonomous subset is not an antichain, and analogously in the k-dimensional case. As corollaries we obtain that being a proper trapezoid ordered set is a comparability invariant, more generally, proper interval dimension is a comparability invariant.
Similar content being viewed by others
References
Bogart, K. P., Rabinovich, I. and Trotter, W. T. (1976) A bound on the dimension of interval orders, J. Combin. Theory Ser. A 21, 319–328.
Bogart, K. P., Möhring, R. H. and Ryan, S. P. (1998) Proper and unit trapezoid orders and graphs, Order 15, 325–340.
Felsner, S. and Möhring, R. H. (1998) Note: Semi-order dimension two is a comparability invariant, Order 15, 385–390.
Gallai, T. (1967) Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18, 25–66.
Habib, M., Kelly, D. and Möhring, R. H. (1991) Interval dimension is a comparability invariant, Discrete Math. 88, 211–229.
Rabinovich, I. (1978) The dimension of semiorders, J. Combin. Theory Ser. A 25, 274–279.
Roberts, F. S. (1969) Indifference graphs, in F. Harary (ed.), Proof Techniques in Graph Theory, Academic Press, pp. 139–146.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Niederle, J. Being a Proper Trapezoid Ordered Set Is a Comparability Invariant. Order 17, 301–308 (2000). https://doi.org/10.1023/A:1026717230082
Issue Date:
DOI: https://doi.org/10.1023/A:1026717230082