On the lattice of antichains of finite intervals

TL;DR

This paper explores the mathematical structure of antichains of finite intervals in a totally ordered set, revealing their properties, representations, and applications to information retrieval.

Contribution

It introduces a unique irredundant $ ext{land}$-representation for the lattice of antichains, enabling closed-form expressions for relative pseudo-complements.

Findings

Established a connection with Alexandrov completions.

Derived formulas for element rank and lattice height.

Analyzed properties of operators relevant to information retrieval.

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.