L-fuzzy simplicial homology

TL;DR

This paper introduces L-fuzzy simplicial homology, a generalization of classical simplicial homology for L-fuzzy subcomplexes, enabling new applications in topological data analysis.

Contribution

It defines L-fuzzy simplicial homology, explores its properties, and connects it to filtrations and datasets, expanding topological invariants to fuzzy settings.

Findings

Defines L-fuzzy simplicial homology and its main properties.

Provides methods for computing the structure of L-fuzzy homology.

Interprets filtrations and datasets within the L-fuzzy framework.

Abstract

Simplicial homology is a classical tool that assigns a sequence of modules to a simplicial complex, providing invariants for the study of its topological properties. In this article, we introduce the notion of L-fuzzy simplicial homology, a generalization of simplicial homology for L-fuzzy subcomplexes, in which each simplex is assigned a value from a completely distributive lattice L. We present its definition and main properties and describe methods to compute its structure. In addition, we interpret filtrations over a poset and chromatic datasets in this setting, opening a door to further applications in topological data analysis.