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arXiv · 1510.03675

On the lattice of antichains of finite intervals

Paolo Boldi·Sebastiano Vigna
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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 $\land$-representation by $\land$-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in details 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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Paolo Boldi, Sebastiano Vigna. 2016-12-08. On the lattice of antichains of finite intervals. https://arxiv.org/abs/1510.03675

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