Abstract
Dimension theory is a prominent area in ordered sets. In dimension theory, orders are represented using the orders that occur most frequently, namely, total orders. As for Chapter 8 on lattices, it must be said that this chapter can only provide brief exposure to the basics of dimension theory. For a thorough presentation of this subject, consider [311].
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If this is not interesting enough, consider scheduling computations on a processor.
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And, depending on the climate at the institution, the resulting order may be the subject of heated discussion.
References
Arditti, J. C. (1976). Graphes de comparabilité et dimension des ordres. Notes de Recherches, CRM 607. Centre de Recherches Mathématiques Université de Montréal.
Arditti, J. C., & Jung, H. A. (1980). The dimension of finite and infinite comparability graphs. Journal of the London Mathematical Society, 21(2), 31–38.
Brightwell, G. (1988). Linear extensions of infinite posets. Discrete Mathematics, 70, 113–136.
Brightwell, G. R., Felsner, S., & Trotter, W. T. (1995). Balancing pairs and the cross product conjecture. Order, 12, 327–349.
Brualdi, R. A., Jung, H. C., & Trotter, W. T. (1994). On the poset of all posets on n elements. Discrete Applied Mathematics, 50, 111–123.
Dilworth, R. P. (1950). A decomposition theorem for partially ordered sets. Annals of Mathematics, 51, 161–166.
Erné, M. (1981). Open question on p. 843 of [246].
Farley, J. D., & Schröder, B. (2001). Strictly order-preserving maps into \(\mathbb{Z}\), II: A 1979 problem of Erné. Order, 18, 381–385.
Felsner, S., & Trotter, W. T. (2000). Dimension, graph and hypergraph coloring. Order, 17, 167–177.
Gallai, T. (1967). Transitiv orientierbare graphen. Acta Mathematica Hungarica, 18, 25–66.
Golumbic, M. (1980). Algorithmic graph theory and perfect graphs. New York: Academic.
Gysin, R. (1977). Dimension transitiv orientierbarer Graphen. Acta Mathematica Academiae Scientiarum Hungaricae, 29, 313–316.
Hiraguchi, T. (1955). On the dimension of orders. Science Reports of Kanazawa University, 4, 1–20.
Kahn, J., & Saks, M. (1984). Balancing poset extensions. Order, 1, 113–126.
Kelly, D. (1984). Unsolved problems: Removable pairs in dimension theory. Order, 1, 217–218.
Kelly, D., & Trotter, W. T. (1982). Dimension theory for ordered sets. In I. Rival (Ed.), Ordered sets (pp. 171–211). Dordrecht: D. Reidel.
Kierstead, H., & Trotter, W. T. (1991). A note on removable pairs. In Y. Alavi et al. (Eds.), Graph theory, combinatorics and applications (Vol. 2, pp. 739–742). New York: Wiley.
Kimble, R. (1973). Extremal Problems in Dimension Theory for Partially Ordered Sets. Ph. D. dissertation, MIT.
Kislitsin, S. S. (1968). Finite partially ordered sets and their associated sets of permutations. Matematicheskiye Zametki, 4, 511–518.
Rabinovitch, I., & Rival, I. (1979). The rank of a distributive lattice. Discrete Mathematics, 25, 275–279.
Reiner, V., & Welker, V. (1999). A homological lower bound for order dimension of lattices. Order, 16, 165–170.
Reuter, K. (1989). Removing critical Pairs. Order, 6, 107–118.
Saks, M. (1985). Unsolved problems: Balancing linear extensions of ordered sets. Order, 2, 327–330.
Spinrad, J. (1988). Subdivision and lattices. Order, 5, 143–147.
Sysło, M. (1984). A graph theoretic approach to the jump-number problem. In I. Rival (Ed.), Graphs and order (pp. 185–215). Boston: Dordrecht-Reidel.
Szpilrajn, E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16, 386–389.
Trotter, W. T. (1975). Inequalities in dimension theory for posets. Proceedings of the American Mathematical Society, 47, 311–316.
Trotter, W. T. (1976). A forbidden subposet characterization of an order dimension inequality. Mathematical Systems Theory, 10, 91–96.
Trotter, W. T. (1992). Combinatorics and partially ordered sets: Dimension theory. Baltimore: Johns Hopkins University Press.
Trotter, W. T., Moore, J. I., & Sumner, D. P. (1976). The dimension of a comparability graph. Proceedings of the American Mathematical Society, 60, 35–38.
von Rimscha, M. (1983). Reconstructibility and perfect graphs. Discrete Mathematics, 47, 79–90.
Yannakakis, M. (1982). On the complexity of the partial order dimension problem. SIAM Journal of Algebra and Discrete Mathematics, 3, 351–358.
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Schröder, B. (2016). Dimension. In: Ordered Sets. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29788-0_10
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