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On the Lattice of Antichains of Finite Intervals

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Abstract

Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant ∧-representation by ∧-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in detail properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.

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References

  1. Boldi, P., Vigna, S.: MG4J at TREC 2005. In: Voorhees, E.M., Buckland, L.P. (eds.) The Fourteenth Text REtrieval Conference (TREC 2005) Proceedings, number SP 500–266 in Special Publications. NIST (2005). http://mg4j.di.unimi.it/

  2. Boldi, P., Vigna, S.: Efficient optimally lazy algorithms for minimal-interval semantics. Theor. Comput. Sci. 648, 8–25 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Clarke, C.L.A., Cormack, G.V.: Shortest-substring retrieval and ranking. ACM Trans. Inf. Syst 18(1), 44–78 (2000)

    Article  Google Scholar 

  4. Clarke, C.L.A., Cormack, G.V., Burkowski, F.J.: An algebra for structured text search and a framework for its implementation. Comput. J. 38(1), 43–56 (1995)

    Article  Google Scholar 

  5. Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice Hall (1973)

  6. Engel, K.: Sperner Theory. Cambridge Solid State Science Series. Cambridge University Press (1997)

  7. Erné, M.: Einführung in die Ordnungstheorie. Bibliographisches Institut (1982)

  8. Erné, M.: The ABC of order and topology. In: Category Theory at Work, volume 18 of Res. Expo. Math., pp. 57–83. Heldermann (1991)

  9. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison–Wesley (1994)

  10. OEIS Foundation Inc. The on-line encyclopedia of integer sequences (2011)

  11. Johnstone, P.T.: Stone spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press (1986)

  12. McKinsey, J.C.C., Tarski, A.: On closed elements in closure algebras. Ann. Math. 47(1), 122–162 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ore, O.: Theory of Graphs, volume XXXVIII of American Mathematical Society Colloquium Publications. American Mathematical Society (1962)

  14. Ribenboim, P.: Ordering the set of antichains of an ordered set. Collect. Math. 46(1–2), 159–170 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Smith, D.P.: Meet-irreducible elements in implicative lattices. Proc. Amer. Math. Soc. 34(1), 57–62 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Stanley, R.P.: Catalan Numbers. Cambridge University Press (2015)

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  1. Dipartimento di Informatica, Università degli Studi di Milano, via Comelico 39, 20135, Milano, Italy

    Paolo Boldi & Sebastiano Vigna

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  1. Paolo Boldi
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  2. Sebastiano Vigna
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Correspondence to Sebastiano Vigna.

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Boldi, P., Vigna, S. On the Lattice of Antichains of Finite Intervals. Order 35, 57–81 (2018). https://doi.org/10.1007/s11083-016-9418-8

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  • DOI: https://doi.org/10.1007/s11083-016-9418-8

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