Analyzing the Structure of Groupoids of Order 3, 4, and 5 Using PCA

Abstract

Groupoids are algebraic structures, which generalize groups by allowing partial symmetries, and are useful in various fields, including topology, category theory, and algebraic geometry. Understanding the variance explained by Principal Component Analysis (PCA) components and the correlations among variables within groupoids can provide valuable insights into their structures and relationships. This study aims to explore the use of PCA as a dimensionality reduction technique to understand the variance explained by different components in the context of groupoids. Additionally, we examine the interrelationships among variables through a color-coded correlation matrix, facilitating insights into the structure and dependencies within groupoid datasets. The findings contribute to the broader understanding of data representation and analysis in mathematical and computational frameworks. © 2025 Elsevier B.V., All rights reserved.