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On hybrid order dimension - MaRDI portal

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On hybrid order dimension

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Publication:6503920

arXiv2002.02005MaRDI QIDQ6503920

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Abstract: The notion of interval order was introduced by Norbert Wiener cite{wie} in order to clarify the relation between the notion of an instant of time and that of a period of time. This was a problem on which Bertrand Russell cite{rus} worked at the time. Interval orders play an important role in many areas of pure and applied mathematics, graph theory, computer science and engineering. Special cases of interval order are the semiorder and linear order. All of these notions are especially important in the study of linear-interval and linear-semiorder dimension of a binary relation. This kind of dimension, which we call {it hybric order dimension}, gives a common generalization of linear order and interval order (semiorder) dimension and is arguably the most important measure of ordered set complexity. In this paper, we give three main results of the theory of hybric order dimension. More precicely, we obtain necessary and sufficient conditions for a binary relation to have an interval order (resp. linear-interval order, linear-simiorder) extension as well as a realizer of interval orders (resp. linear-interval orders, linear-simiorders). We also obtain a characterization of the interval order (resp. linear-interval order, linear-simiorder) dimension. Since the hybric order dimension of a binary relation is less than its (linear) order dimension, these results will be able to improve known results in graph theory and computer science by finding more efficient algorithms.





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