, Volume 35, Issue 1, pp 57–81 | Cite as

On the Lattice of Antichains of Finite Intervals

  • Paolo Boldi
  • Sebastiano Vigna


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.


Lattice Antichains Alexandrov completion Information retrieval 


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanoItaly

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