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.
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References
Aigner, M.: Combinatorial Theory. Springer, Berlin (1979)
Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)
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)
Baker, K.A., Fishburn, P.C., Roberts, F.S.: Partial orders of dimension 2, networks (1972)
Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, New York (1966)
Birkhoff, G.: Lattice Theory, vol. 25. Am. Math. Soc., Providence RI (1940)
Budanitsky, A., Hirst, G.: Evaluating WordNet-based measures of lexical semantic relatedness. Comput. Linguist. 32(1), 13–47 (2006)
Bufardi, A.: An alternative definition for fuzzy interval orders. Fuzzy Set. Syst. 133, 249–259 (2003)
Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)
Diaz, S., De Baets, B., Montes, S.: On the Ferrers property of valued interval orders. TOP 19, 421–447 (2011)
Fellbaum, C (ed.): Wordnet: An Electronic Lexical Database. MIT Press, Cambridge (1998)
Fishburn, P.C.: Interval graphs and interval orders. Discret. Math. 55, 135–149 (1985)
Fishburn, P.C.: Interval Orders and Interval Graphs: Study of Partially Ordered Sets. Wiley-Interscience series in discrete mathematics. Wiley (1985)
Freese, R.: Automated lattice drawing. In: Concept Lattices (ICFCA 04), Lecture Notes in AI, vol. 2961, pp 112–127 (2004)
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)
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)
Joslyn, C., Hogan, E., Pogel, A.: Conjugacy and iteration of standard interval valued rank in finite ordered sets, arXiv:1409.6684 [math.CO] (2014)
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)
Joslyn, C., Mniszewski, S., Fulmer, A., Heaton, G.: The gene ontology categorizer. Bioinformatics 20(s1), 169–177 (2004)
Kaiser, T., Schmidt, S., Joslyn, C.: Adjusting annotated taxonomies. In: International Journal of Foundations of Computer Science, vol. 19:2, pp 345–358 (2008)
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)
Ligozat, G.: Weak representation of interval algebras. In: Proceedings of the 8th National Conference on Artificial Intelligence (AAAI 90), pp 715–720 (1990)
Moore, R.M.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)
Schroder, B.S.W.: Ordered Sets. Birkhauser, Boston (2003)
Tanenbaum, P.J.: Simultaneous represention of interval and Interval-containment orders. Order 13, 339–350 (1996)
Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore (1992)
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)
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)
Wild, M.: On rank functions of lattices. Order 22(4), 357–370 (2005)
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
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Joslyn, C., Pogel, A. & Purvine, E. Interval-Valued Rank in Finite Ordered Sets. Order 34, 491–512 (2017). https://doi.org/10.1007/s11083-016-9411-2
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DOI: https://doi.org/10.1007/s11083-016-9411-2