# The Distributive, Graded Lattice of EL Concept Descriptions and its Neighborhood Relation (Extended Version)

## TL;DR

This paper explores the structural properties of the lattice of EL concept descriptions, demonstrating it is distributive, modular, graded, and metric, which facilitates the definition of rank and distance functions.

## Contribution

It establishes the distributive, graded, and metric nature of the EL concept description lattice, providing new insights into its mathematical structure.

## Key findings

- The lattice of EL concept descriptions is distributive.
- The lattice is modular and graded.
- A rank and a distance function exist for the lattice.

## Abstract

For the description logic EL, we consider the neighborhood relation which is induced by the subsumption order, and we show that the corresponding lattice of EL concept descriptions is distributive, modular, graded, and metric. In particular, this implies the existence of a rank function as well as the existence of a distance function.

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Source: https://tomesphere.com/paper/1901.05919