• Dr. Ramkrishna Thakur
• Dr. Ramanand Raman

This research explores the algebraic structure of incidence algebras arising from partially ordered sets (posets), emphasizing their foundational properties and applications in representation theory. Beginning with definitions and axioms of posets and incidence algebras, the study delves into the convolution operation and its role in defining associative algebras over commutative rings. Central to this discussion is the Möbius function, introduced by Gian-Carlo Rota, whose inversion formula plays a critical role in enumerative combinatorics and algebraic analysis. Through detailed examples - including chains, Boolean lattices, and divisor posets - the article demonstrates the computational aspects of Möbius functions. Further, the paper examines how incidence algebras naturally interface with representation theory via quiver representations, module categories, and homological tools such as projective resolutions. Advanced generalizations are also considered, including extensions to infinite posets, topological incidence algebras, Hopf algebra structures, and categorical interpretations. These developments reveal the incidence algebra as a unifying framework across combinatorics, algebra, topology, and category theory. The study concludes by outlining future directions in quantum and non-commutative generalizations.

Incidence algebra, poset, Möbius function, Möbius inversion, quiver representation, associative algebra, Hopf algebra, representation theory, category theory, topological combinatorics

"INCIDENCE ALGEBRAS OF POSETS: ALGEBRAIC STRUCTURES AND APPLICATIONS IN REPRESENTATION THEORY", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.12, Issue 2, page no.i330-i336, February-2025, Available :http://www.jetir.org/papers/JETIR2502834.pdf

"INCIDENCE ALGEBRAS OF POSETS: ALGEBRAIC STRUCTURES AND APPLICATIONS IN REPRESENTATION THEORY", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.12, Issue 2, page no. ppi330-i336, February-2025, Available at : http://www.jetir.org/papers/JETIR2502834.pdf