Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 1 2026 06 07 Fuzzy Graph Extensions of Sandpile Monoids and Their Connections to Leavitt Path Algebras 130 141 2086 10.22128/ansne.2026.3237.1193 EN Shanookha Ali Department of General Science, Birla Institute of Technology & Science, Pilani, Dubai Campus, Dubai 345055, United Arab Emirates 0000-0001-8720-3764 Farshid Mofidnakhaei Department of Physics, Sari Branch, Islamic Azad University, Sari, Iran 0000-0002-2903-0428 Nitha Niralda PC Department of Mathematics and Statistics, Providence Women's College, Calicut, Kerala, India 0000-0002-5573-3944 Shafeequdheen Palengara Department of Mathematics, SRM University, Andhra Pradesh, India Journal Article 2026 01 31 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. https://ansne.du.ac.ir/article_2086_20b3d4cb3234ee380492f9d4875d7cd7.pdf