A Lattice for Persistence

TL;DR

This paper introduces a novel lattice-theoretic interpretation of persistent homology, representing the structure as a complete Heyting algebra, which offers new insights and potential applications in computational topology.

Contribution

It presents an alternative lattice-based framework for persistent homology, generalizing Boolean algebra to Heyting algebra, with algorithmic methods and exploration of its properties.

Findings

Lattice structure of persistence forms a complete Heyting algebra.

Algorithmic construction of the lattice enables new operations on homology groups.

Potential applications in computational topology and data analysis.

Abstract

The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the most prominent areas of research in computational topology in the past 20 years. In this paper we will introduce an alternative interpretation of persistence based on the study of the order structure of its correspondent lattice. Its algorithmic construction leads to two operations on homology groups which describe a diagram of spaces as a complete Heyting algebra, which is a generalization of a Boolean algebra. We investigate some of the properties of this lattice, the algorithmic implications of it, and some possible applications.