Interval-valued rank in finite ordered sets. (English) Zbl 1414.06002
The authors consider the rank as a measure of the vertical levels and positions of elements of an ordered set. Representing semantic hierarchies as finite, bounded ordered sets, they recognize the duality of ordered structures to motivate rank functions with respect to verticality both from bottom and from the top. Their rank functions are interval-valuated even for non-graded ordered sets. The concept of rank width arises naturaly, allowing to identify the ordered set region with point-valued width as longest graded portion. The properties of standard interval rank function are examined, including the relationship to traditional grading and rank functions.
Reviewer: Ivan Chajda (Přerov)
References:
[1] | Aigner, M.: Combinatorial Theory. Springer, Berlin (1979) · Zbl 0415.05001 · doi:10.1007/978-1-4615-6666-3 |
[2] | Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832-843 (1983) · Zbl 0519.68079 · doi:10.1145/182.358434 |
[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) · doi:10.1038/75556 |
[4] | Baker, K.A., Fishburn, P.C., Roberts, F.S.: Partial orders of dimension 2, networks (1972) · Zbl 0247.06002 |
[5] | Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, New York (1966) · Zbl 0187.44103 |
[6] | Birkhoff, G.: Lattice Theory, vol. 25. Am. Math. Soc., Providence RI (1940) · Zbl 0063.00402 |
[7] | Budanitsky, A., Hirst, G.: Evaluating WordNet-based measures of lexical semantic relatedness. Comput. Linguist. 32(1), 13-47 (2006) · Zbl 1234.68399 · doi:10.1162/coli.2006.32.1.13 |
[8] | Bufardi, A.: An alternative definition for fuzzy interval orders. Fuzzy Set. Syst. 133, 249-259 (2003) · Zbl 1009.06001 · doi:10.1016/S0165-0114(02)00135-5 |
[9] | Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990) · Zbl 0701.06001 |
[10] | Diaz, S., De Baets, B., Montes, S.: On the Ferrers property of valued interval orders. TOP 19, 421-447 (2011) · Zbl 1284.91126 · doi:10.1007/s11750-010-0134-z |
[11] | Fellbaum, C (ed.): Wordnet: An Electronic Lexical Database. MIT Press, Cambridge (1998) · Zbl 0913.68054 |
[12] | Fishburn, P.C.: Interval graphs and interval orders. Discret. Math. 55, 135-149 (1985) · Zbl 0568.05047 · doi:10.1016/0012-365X(85)90042-1 |
[13] | Fishburn, P.C.: Interval Orders and Interval Graphs: Study of Partially Ordered Sets. Wiley-Interscience series in discrete mathematics. Wiley (1985) · Zbl 0874.06003 |
[14] | Freese, R.: Automated lattice drawing. In: Concept Lattices (ICFCA 04), Lecture Notes in AI, vol. 2961, pp 112-127 (2004) · Zbl 1198.06001 |
[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) · Zbl 1104.68738 |
[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) |
[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] | 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] | Joslyn, C., Mniszewski, S., Fulmer, A., Heaton, G.: The gene ontology categorizer. Bioinformatics 20(s1), 169-177 (2004) · doi:10.1093/bioinformatics/bth921 |
[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) · Zbl 1156.68584 |
[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] | Ligozat, G.: Weak representation of interval algebras. In: Proceedings of the 8th National Conference on Artificial Intelligence (AAAI 90), pp 715-720 (1990) |
[23] | Moore, R.M.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979) · Zbl 0417.65022 · doi:10.1137/1.9781611970906 |
[24] | Schroder, B.S.W.: Ordered Sets. Birkhauser, Boston (2003) · Zbl 1010.06001 · doi:10.1007/978-1-4612-0053-6 |
[25] | Tanenbaum, P.J.: Simultaneous represention of interval and Interval-containment orders. Order 13, 339-350 (1996) · Zbl 0874.06003 · doi:10.1007/BF00405593 |
[26] | Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore (1992) · Zbl 0764.05001 |
[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) · Zbl 0877.06001 |
[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) · doi:10.1110/ps.062184006 |
[29] | Wild, M.: On rank functions of lattices. Order 22(4), 357-370 (2005) · Zbl 1105.06007 · doi:10.1007/s11083-005-9025-6 |
[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 · Zbl 1326.06005 · doi:10.1007/s00500-013-1010-1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.