Order

pp 1–22

Interval-Valued Rank in Finite Ordered Sets

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Abstract

We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions with respect to verticality both from the bottom and from the top. Our rank functions are thus interval-valued, and always exist, even for non-graded posets, providing order homomorphisms to an interval order on the interval-valued ranks. The concept of rank width arises naturally, allowing us to identify the poset region with point-valued width as its longest graded portion (which we call the “spindle”). A standard interval rank function is naturally motivated both in terms of its extremality and on pragmatic grounds. Its properties are examined, including the relationship to traditional grading and rank functions, and methods to assess comparisons of standard interval-valued ranks.

Keywords

Ordered sets Ordered set rank Graded ordered sets Interval-valued rank 

References

  1. 1.
    Aigner, M.: Combinatorial Theory. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
  2. 2.
    Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)CrossRefMATHGoogle Scholar
  3. 3.
    Ashburner, M., Ball, C.A., Blake, J.A., et al.: Gene ontology: Tool for the unification of biology. Nature Genetics 25(1), 25–29 (2000)CrossRefGoogle Scholar
  4. 4.
    Baker, K.A., Fishburn, P.C., Roberts, F.S.: Partial orders of dimension 2, networks (1972)Google Scholar
  5. 5.
    Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, New York (1966)MATHGoogle Scholar
  6. 6.
    Birkhoff, G.: Lattice Theory, vol. 25. Am. Math. Soc., Providence RI (1940)Google Scholar
  7. 7.
    Budanitsky, A., Hirst, G.: Evaluating WordNet-based measures of lexical semantic relatedness. Comput. Linguist. 32(1), 13–47 (2006)CrossRefMATHGoogle Scholar
  8. 8.
    Bufardi, A.: An alternative definition for fuzzy interval orders. Fuzzy Set. Syst. 133, 249–259 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)Google Scholar
  10. 10.
    Diaz, S., De Baets, B., Montes, S.: On the Ferrers property of valued interval orders. TOP 19, 421–447 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fellbaum, C (ed.): Wordnet: An Electronic Lexical Database. MIT Press, Cambridge (1998)Google Scholar
  12. 12.
    Fishburn, P.C.: Interval graphs and interval orders. Discret. Math. 55, 135–149 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fishburn, P.C.: Interval Orders and Interval Graphs: Study of Partially Ordered Sets. Wiley-Interscience series in discrete mathematics. Wiley (1985)Google Scholar
  14. 14.
    Freese, R.: Automated lattice drawing. In: Concept Lattices (ICFCA 04), Lecture Notes in AI, vol. 2961, pp 112–127 (2004)Google Scholar
  15. 15.
    Joslyn, C.: Poset ontologies and concept lattices as semantic hierarchies. In: Wolff, P, Delugach (eds.) Conceptual Structures at Work, Lecture Notes in Artificial Intelligence, vol. 3127, pp 287–302. Springer, Berlin (2004)Google Scholar
  16. 16.
    Joslyn, C., Hogan, E.: Order metrics for semantic knowledge systems. In: Corchado Rogriguez, E S et al. (eds.) 5th International Conference on Hybrid Artificial Intelligence System (HAIS 2010), Lecture Notes in Artificial Intelligence, vol. 6077, pp 399–409. Springer, Berlin (2010)Google Scholar
  17. 17.
    Joslyn, C., Hogan, E., Pogel, A.: Conjugacy and iteration of standard interval valued rank in finite ordered sets, arXiv:1409.6684 [math.CO] (2014)
  18. 18.
    Joslyn, C., Mniszewski, S.M., Smith, S.A., Weber, P.M.: Spindleviz: A three dimensional, order theoretical visualization environment for the gene ontology. In: Joint BioLINK and 9th Bio-Ontologies Meeting (JBB 06). http://bio-ontologies.org.uk/2006/download/Joslyn2EtAlSpindleviz.pdf (2006)
  19. 19.
    Joslyn, C., Mniszewski, S., Fulmer, A., Heaton, G.: The gene ontology categorizer. Bioinformatics 20(s1), 169–177 (2004)CrossRefGoogle Scholar
  20. 20.
    Kaiser, T., Schmidt, S., Joslyn, C.: Adjusting annotated taxonomies. In: International Journal of Foundations of Computer Science, vol. 19:2, pp 345–358 (2008)Google Scholar
  21. 21.
    Ladkin, P.: Maddux R. Algebra of Convex Time Intervals. Tech. rep., http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.681 (1987)
  22. 22.
    Ligozat, G.: Weak representation of interval algebras. In: Proceedings of the 8th National Conference on Artificial Intelligence (AAAI 90), pp 715–720 (1990)Google Scholar
  23. 23.
    Moore, R.M.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)CrossRefMATHGoogle Scholar
  24. 24.
    Schroder, B.S.W.: Ordered Sets. Birkhauser, Boston (2003)CrossRefMATHGoogle Scholar
  25. 25.
    Tanenbaum, P.J.: Simultaneous represention of interval and Interval-containment orders. Order 13, 339–350 (1996)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore (1992)MATHGoogle Scholar
  27. 27.
    Trotter, W.T.: New perspectives on interval orders and interval graphs. In: Bailey, R.Ã. (ed.) Surveys in Combinatorics, London Mathematical Society Lecture Note Series, vol. 241, pp 237–286. London Math. Society, London (1997)Google Scholar
  28. 28.
    Verspoor, K.M., Cohn, J.D., Mniszewski, S.M., Joslyn, C.A.: A categorization approach to automated ontological function annotation. Protein Sci. 15, 1544–1549 (2006)CrossRefGoogle Scholar
  29. 29.
    Wild, M.: On rank functions of lattices. Order 22(4), 357–370 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zapata, F., Kreinovich, V., Joslyn, C.A., Hogan, E.: Orders on intervals over partially ordered sets: extending Allen’s algebra and interval graph results. Soft. Comput. (2013). doi:http://dx.doi.org/10.1007/s00500-013-1010-1 MATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Data Sciences and AnalyticsPacific Northwest National LaboratorySeattleUSA
  2. 2.Physical Science LaboratoryNew Mexico State UniversityLas CrucesUSA

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