In the mid 1960's, the incidence algebra was introduced in the seminal paper of Gian-Carlo Rota. He addressed the importance of the Möbius function in combinatorics. In particular, the incidence algebra of a locally finite poset plays an essentially unifying role in the theory of the Möbius function. One of the significant generalizations is the incidence algebra of a Möbius category introduced by Pierre Leroux. With the help from Möbius category, it was exciting to be able to extend the combinatorial results more broadly than just on posets. Before attempting to study this generalization of the Möbius function, we have to begin with the basic concepts needed to define the incidence algebra. In the first chapter, we will see some basic concepts and illustrations of incidence functions in posets. In the second chapter, we will introduce the decomposition-finite category [special characters omitted], the incidence algebra of [special characters omitted], and the Möbius function of the Möbius category [special characters omitted].
Liao, Yi-yu, "Incidence functions" (2010). ETD Collection for University of Texas, El Paso. AAI1477800.