Fuzzy Graph Extensions of Sandpile Monoids and Their Connections to Leavitt Path Algebras

Document Type : Research Article

Authors

1 Department of General Science, Birla Institute of Technology & Science, Pilani, Dubai Campus, Dubai 345055, United Arab Emirates

2 Department of Physics, Sari Branch, Islamic Azad University, Sari, Iran

3 Department of Mathematics and Statistics, Providence Women's College, Calicut, Kerala, India

4 Department of Mathematics, SRM University, Andhra Pradesh, India

Abstract

We broaden the framework of sandpile monoids and weighted Leavitt path algebras to encompass fuzzy graph structures, establishing key structural relationships within this extended setting. Specifically, we prove that idempotent components in fuzzy sandpile monoids $\text{FSP}(E, \mu, \gamma)$ correspond to fuzzy hereditary saturated subsets $(E, \mu, \gamma)$. Additionally, we demonstrate that these idempotent structures exhibit a lattice organization governed by order ideals within $(\bar{E}, \mu, \gamma)$. Furthermore, this lattice structure aligns with the lattice formed by vertex-generated ideals in the fuzzy weighted Leavitt path algebra $L_1(\bar{E}, \omega, \mu, \gamma)$. We characterize the fuzzy sandpile group through Archimedean equivalence classes and establish that optimal subgroups align exactly with Grothendieck groupoids of these equivalence classes. Our analysis reveals how the lattice of idempotents in $\text{FSP}(E, \mu, \gamma)$ forms a system of graded ideals that preserve invariance under graded automorphisms.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 1
June 2026
Pages 130-141

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History

  • Receive Date: 31 January 2026
  • Accept Date: 18 February 2026
  • Publish Date: 07 June 2026

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