Abstract
We study finite partial orders which have a chain such that every element of the order either belongs to this chain or has all its covers in this chain. We show that such orders are exactly the orders being both interval orders and truncated lattices. We prove that their jump number is polynomially tractable and that their dimension is unbounded. We also show that every order admits a visibility model having such an order as host.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkhoff, G.: Lattice theory. American Mathematical Society Colloquium Publications Volume XXV (1967)
Bogart, K.P., Rabinovich, I., Trotter, W.T.: A bound on the dimension of interval orders. J. Combin. Theory, Ser. A 21(3), 319–328 (1976)
Chein, M., Martin, P.: Sur le nombre de sauts d'une forêt. C. R. Acad. Sci. Paris, Sér. A 275, 159–161 (1972)
Even, S., Kariv, O.: An \(\mathcal{O}(n^{2.5})\) algorithm for maximum matching in general graphs. In: 16th Annual Symposium on Foundations of Computer Science (Berkeley, California), pp. 100–112 (1975)
Freese, R., Ježek, J., Nation, J.B.: Free lattices. Math. Surveys Monogr. 42 (1995)
Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psych. 7, 144–149 (1970)
Fishburn, P.C.: Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. Wiley, Interscience Publication New York (1985)
Gabow, H.N.: A linear time recognition algorithm for interval dags. Inform. Process. Lett. 12, 20–22 (1981)
Guenver, G.-B., Rampon, J.-X.: Split orders. Discrete Math. 276(1–3), 249–267 (2004)
Hiraguchi, T.: On the dimension of orders. Sci. Rep. Kanazawa Univ. 4(1), 1–20 (1955)
Kratsch, D., Rampon, J.-X.: Tree-visibility orders. Discrete Math. 190, 163–175 (1998)
Kuntzmann, J., Verdillon, A.: Recherche d'un ordre minimal compatible avec un ordre partiel donné. Séminaire Institut de Math. Appl. de Grenoble (non publié) (1971)
Mitas, J.: Tackling the jump number of interval orders. Order 8(2), 115–132 (1991)
Mirkin, B.G.: Description of some relations on the set of real-line intervals. J. Math. Psych. 9, 243–252 (1972)
Müller, H., Rampon, J.-X.: Partial orders and their convex subsets. Discrete Math. 165–166, 507–517 (1997)
Müller, H., Rampon, J.-X.: Partial orders on weak orders convex subsets. Order 17(2), 103–123 (2000)
Papadimitriou, C.H., Yannakakis, M.: Scheduling interval ordered tasks. SIAM J. Comput. 8, 405–409 (1979)
Pulleyblank, W.R.: On minimizing setups in precedence constrained scheduling. Report 81105-OR, University of Bonn (1981)
Schröder, B.S.W.: Ordered Sets. An Introduction. Birkhäuser, Boston, Inc., Massachusetts (2003)
Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The John Hopkins University Press, Baltimore, Maryland (1992)
Wiener, N.: A contribution to the theory of relative position. Proc. Camb. Philos. Soc. 17, 441–449 (1914)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guenver, GB., Leblet, J. & Rampon, JX. Chain Dominated Orders. Order 23, 109–127 (2006). https://doi.org/10.1007/s11083-006-9035-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-006-9035-z