Aspects of an internal logic for persistence

TL;DR

This paper explores the algebraic foundations of persistence in topological data analysis, introducing a new logical operation and analyzing its implications for various persistence frameworks.

Contribution

It introduces a novel algebraic operation on persistence modules and analyzes its properties, connecting algebraic logic with persistence theory.

Findings

Defined a new implication operation in persistence algebra

Connected algebraic logic to multidimensional and zig-zag persistence

Provided universal properties and new applications in persistence theory

Abstract

The foundational character of certain algebraic structures as Boolean algebras and Heyting algebras is rooted in their potential to model classical and constructive logic, respectively. In this paper we discuss the contributions of algebraic logic to the study of persistence based on a new operation on the ordered structure of the input diagram of vector spaces and linear maps given by a filtration. Within the context of persistence theory, we give an analysis of the underlying algebra, derive universal properties and discuss new applications. We highlight the definition of the implication operation within this construction, as well as interpret its meaning within persistent homology, multidimensional persistence and zig-zag persistence.