Rises for Measuring Local Distributivity in Lattices

TL;DR

This paper introduces the concept of rises as a new measure to quantify distributivity in lattices, especially in Formal Concept Analysis, revealing that real-world concept lattices are mostly join-distributive.

Contribution

The paper defines rises as a novel metric for assessing distributivity in lattices and establishes their relationship with classical distributivity notions.

Findings

Real-world concept lattices are highly join-distributive.

No non-unit rises occur in distributive lattices.

Rises relate to meet- and join-distributivity in lattices.

Abstract

Distributivity is a well-established and extensively studied notion in lattice theory. In the context of data analysis, particularly within Formal Concept Analysis (FCA), lattices are often observed to exhibit a high degree of distributivity. However, no standardized measure exists to quantify this property. In this paper, we introduce the notion of rises in (concept) lattices as a means to assess distributivity. Rises capture how the number of attributes or objects in covering concepts change within the concept lattice. We show that a lattice is distributive if and only if no non-unit rises occur. Furthermore, we relate rises to the classical notion of meet- and join distributivity. We observe that concept lattices from real-world data are to a high degree join-distributive, but much less meet-distributive. We additionally study how join-distributivity manifests on the level of ordered sets.