On the Lattice of Antichains of Finite Intervals

On the Lattice of Antichains of Finite Intervals 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Order Springer Journals

On the Lattice of Antichains of Finite Intervals

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Publisher
Springer Netherlands
Copyright
Copyright © 2016 by Springer Science+Business Media Dordrecht
Subject
Mathematics; Geometry; Number Theory
ISSN
0167-8094
eISSN
1572-9273
D.O.I.
10.1007/s11083-016-9418-8
Publisher site
See Article on Publisher Site

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.

Journal

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Published: Dec 6, 2016

References

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